up follow livre

venv/lib/python3.13/site-packages/scipy/signal/__init__.py

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+"""
+=======================================
+Signal processing (:mod:`scipy.signal`)
+=======================================
+
+Convolution
+===========
+
+.. autosummary::
+   :toctree: generated/
+
+   convolve           -- N-D convolution.
+   correlate          -- N-D correlation.
+   fftconvolve        -- N-D convolution using the FFT.
+   oaconvolve         -- N-D convolution using the overlap-add method.
+   convolve2d         -- 2-D convolution (more options).
+   correlate2d        -- 2-D correlation (more options).
+   sepfir2d           -- Convolve with a 2-D separable FIR filter.
+   choose_conv_method -- Chooses faster of FFT and direct convolution methods.
+   correlation_lags   -- Determines lag indices for 1D cross-correlation.
+
+B-splines
+=========
+
+.. autosummary::
+   :toctree: generated/
+
+   gauss_spline   -- Gaussian approximation to the B-spline basis function.
+   cspline1d      -- Coefficients for 1-D cubic (3rd order) B-spline.
+   qspline1d      -- Coefficients for 1-D quadratic (2nd order) B-spline.
+   cspline2d      -- Coefficients for 2-D cubic (3rd order) B-spline.
+   qspline2d      -- Coefficients for 2-D quadratic (2nd order) B-spline.
+   cspline1d_eval -- Evaluate a cubic spline at the given points.
+   qspline1d_eval -- Evaluate a quadratic spline at the given points.
+   spline_filter  -- Smoothing spline (cubic) filtering of a rank-2 array.
+
+Filtering
+=========
+
+.. autosummary::
+   :toctree: generated/
+
+   order_filter  -- N-D order filter.
+   medfilt       -- N-D median filter.
+   medfilt2d     -- 2-D median filter (faster).
+   wiener        -- N-D Wiener filter.
+
+   symiirorder1  -- 2nd-order IIR filter (cascade of first-order systems).
+   symiirorder2  -- 4th-order IIR filter (cascade of second-order systems).
+   lfilter       -- 1-D FIR and IIR digital linear filtering.
+   lfiltic       -- Construct initial conditions for `lfilter`.
+   lfilter_zi    -- Compute an initial state zi for the lfilter function that
+                 -- corresponds to the steady state of the step response.
+   filtfilt      -- A forward-backward filter.
+   savgol_filter -- Filter a signal using the Savitzky-Golay filter.
+
+   deconvolve    -- 1-D deconvolution using lfilter.
+
+   sosfilt       -- 1-D IIR digital linear filtering using
+                 -- a second-order sections filter representation.
+   sosfilt_zi    -- Compute an initial state zi for the sosfilt function that
+                 -- corresponds to the steady state of the step response.
+   sosfiltfilt   -- A forward-backward filter for second-order sections.
+   hilbert       -- Compute 1-D analytic signal, using the Hilbert transform.
+   hilbert2      -- Compute 2-D analytic signal, using the Hilbert transform.
+   envelope      -- Compute the envelope of a real- or complex-valued signal.
+
+   decimate      -- Downsample a signal.
+   detrend       -- Remove linear and/or constant trends from data.
+   resample      -- Resample using Fourier method.
+   resample_poly -- Resample using polyphase filtering method.
+   upfirdn       -- Upsample, apply FIR filter, downsample.
+
+Filter design
+=============
+
+.. autosummary::
+   :toctree: generated/
+
+   bilinear      -- Digital filter from an analog filter using
+                    -- the bilinear transform.
+   bilinear_zpk  -- Digital filter from an analog filter using
+                    -- the bilinear transform.
+   findfreqs     -- Find array of frequencies for computing filter response.
+   firls         -- FIR filter design using least-squares error minimization.
+   firwin        -- Windowed FIR filter design, with frequency response
+                    -- defined as pass and stop bands.
+   firwin2       -- Windowed FIR filter design, with arbitrary frequency
+                    -- response.
+   firwin_2d        -- Windowed FIR filter design, with frequency response for 
+                    -- 2D using 1D design.
+   freqs         -- Analog filter frequency response from TF coefficients.
+   freqs_zpk     -- Analog filter frequency response from ZPK coefficients.
+   freqz         -- Digital filter frequency response from TF coefficients.
+   sosfreqz      -- Digital filter frequency response for SOS format filter (legacy).
+   freqz_sos     -- Digital filter frequency response for SOS format filter.
+   freqz_zpk     -- Digital filter frequency response from ZPK coefficients.
+   gammatone     -- FIR and IIR gammatone filter design.
+   group_delay   -- Digital filter group delay.
+   iirdesign     -- IIR filter design given bands and gains.
+   iirfilter     -- IIR filter design given order and critical frequencies.
+   kaiser_atten  -- Compute the attenuation of a Kaiser FIR filter, given
+                    -- the number of taps and the transition width at
+                    -- discontinuities in the frequency response.
+   kaiser_beta   -- Compute the Kaiser parameter beta, given the desired
+                    -- FIR filter attenuation.
+   kaiserord     -- Design a Kaiser window to limit ripple and width of
+                    -- transition region.
+   minimum_phase -- Convert a linear phase FIR filter to minimum phase.
+   savgol_coeffs -- Compute the FIR filter coefficients for a Savitzky-Golay
+                    -- filter.
+   remez         -- Optimal FIR filter design.
+
+   unique_roots  -- Unique roots and their multiplicities.
+   residue       -- Partial fraction expansion of b(s) / a(s).
+   residuez      -- Partial fraction expansion of b(z) / a(z).
+   invres        -- Inverse partial fraction expansion for analog filter.
+   invresz       -- Inverse partial fraction expansion for digital filter.
+   BadCoefficients  -- Warning on badly conditioned filter coefficients.
+
+Lower-level filter design functions:
+
+.. autosummary::
+   :toctree: generated/
+
+   abcd_normalize -- Check state-space matrices and ensure they are rank-2.
+   band_stop_obj  -- Band Stop Objective Function for order minimization.
+   besselap       -- Return (z,p,k) for analog prototype of Bessel filter.
+   buttap         -- Return (z,p,k) for analog prototype of Butterworth filter.
+   cheb1ap        -- Return (z,p,k) for type I Chebyshev filter.
+   cheb2ap        -- Return (z,p,k) for type II Chebyshev filter.
+   ellipap        -- Return (z,p,k) for analog prototype of elliptic filter.
+   lp2bp          -- Transform a lowpass filter prototype to a bandpass filter.
+   lp2bp_zpk      -- Transform a lowpass filter prototype to a bandpass filter.
+   lp2bs          -- Transform a lowpass filter prototype to a bandstop filter.
+   lp2bs_zpk      -- Transform a lowpass filter prototype to a bandstop filter.
+   lp2hp          -- Transform a lowpass filter prototype to a highpass filter.
+   lp2hp_zpk      -- Transform a lowpass filter prototype to a highpass filter.
+   lp2lp          -- Transform a lowpass filter prototype to a lowpass filter.
+   lp2lp_zpk      -- Transform a lowpass filter prototype to a lowpass filter.
+   normalize      -- Normalize polynomial representation of a transfer function.
+
+
+
+Matlab-style IIR filter design
+==============================
+
+.. autosummary::
+   :toctree: generated/
+
+   butter -- Butterworth
+   buttord
+   cheby1 -- Chebyshev Type I
+   cheb1ord
+   cheby2 -- Chebyshev Type II
+   cheb2ord
+   ellip -- Elliptic (Cauer)
+   ellipord
+   bessel -- Bessel (no order selection available -- try butterod)
+   iirnotch      -- Design second-order IIR notch digital filter.
+   iirpeak       -- Design second-order IIR peak (resonant) digital filter.
+   iircomb       -- Design IIR comb filter.
+
+Continuous-time linear systems
+==============================
+
+.. autosummary::
+   :toctree: generated/
+
+   lti              -- Continuous-time linear time invariant system base class.
+   StateSpace       -- Linear time invariant system in state space form.
+   TransferFunction -- Linear time invariant system in transfer function form.
+   ZerosPolesGain   -- Linear time invariant system in zeros, poles, gain form.
+   lsim             -- Continuous-time simulation of output to linear system.
+   impulse          -- Impulse response of linear, time-invariant (LTI) system.
+   step             -- Step response of continuous-time LTI system.
+   freqresp         -- Frequency response of a continuous-time LTI system.
+   bode             -- Bode magnitude and phase data (continuous-time LTI).
+
+Discrete-time linear systems
+============================
+
+.. autosummary::
+   :toctree: generated/
+
+   dlti             -- Discrete-time linear time invariant system base class.
+   StateSpace       -- Linear time invariant system in state space form.
+   TransferFunction -- Linear time invariant system in transfer function form.
+   ZerosPolesGain   -- Linear time invariant system in zeros, poles, gain form.
+   dlsim            -- Simulation of output to a discrete-time linear system.
+   dimpulse         -- Impulse response of a discrete-time LTI system.
+   dstep            -- Step response of a discrete-time LTI system.
+   dfreqresp        -- Frequency response of a discrete-time LTI system.
+   dbode            -- Bode magnitude and phase data (discrete-time LTI).
+
+LTI representations
+===================
+
+.. autosummary::
+   :toctree: generated/
+
+   tf2zpk        -- Transfer function to zero-pole-gain.
+   tf2sos        -- Transfer function to second-order sections.
+   tf2ss         -- Transfer function to state-space.
+   zpk2tf        -- Zero-pole-gain to transfer function.
+   zpk2sos       -- Zero-pole-gain to second-order sections.
+   zpk2ss        -- Zero-pole-gain to state-space.
+   ss2tf         -- State-pace to transfer function.
+   ss2zpk        -- State-space to pole-zero-gain.
+   sos2zpk       -- Second-order sections to zero-pole-gain.
+   sos2tf        -- Second-order sections to transfer function.
+   cont2discrete -- Continuous-time to discrete-time LTI conversion.
+   place_poles   -- Pole placement.
+
+Waveforms
+=========
+
+.. autosummary::
+   :toctree: generated/
+
+   chirp        -- Frequency swept cosine signal, with several freq functions.
+   gausspulse   -- Gaussian modulated sinusoid.
+   max_len_seq  -- Maximum length sequence.
+   sawtooth     -- Periodic sawtooth.
+   square       -- Square wave.
+   sweep_poly   -- Frequency swept cosine signal; freq is arbitrary polynomial.
+   unit_impulse -- Discrete unit impulse.
+
+Window functions
+================
+
+For window functions, see the `scipy.signal.windows` namespace.
+
+In the `scipy.signal` namespace, there is a convenience function to
+obtain these windows by name:
+
+.. autosummary::
+   :toctree: generated/
+
+   get_window -- Return a window of a given length and type.
+
+Peak finding
+============
+
+.. autosummary::
+   :toctree: generated/
+
+   argrelmin        -- Calculate the relative minima of data.
+   argrelmax        -- Calculate the relative maxima of data.
+   argrelextrema    -- Calculate the relative extrema of data.
+   find_peaks       -- Find a subset of peaks inside a signal.
+   find_peaks_cwt   -- Find peaks in a 1-D array with wavelet transformation.
+   peak_prominences -- Calculate the prominence of each peak in a signal.
+   peak_widths      -- Calculate the width of each peak in a signal.
+
+Spectral analysis
+=================
+
+.. autosummary::
+   :toctree: generated/
+
+   periodogram    -- Compute a (modified) periodogram.
+   welch          -- Compute a periodogram using Welch's method.
+   csd            -- Compute the cross spectral density, using Welch's method.
+   coherence      -- Compute the magnitude squared coherence, using Welch's method.
+   spectrogram    -- Compute the spectrogram (legacy).
+   lombscargle    -- Computes the Lomb-Scargle periodogram.
+   vectorstrength -- Computes the vector strength.
+   ShortTimeFFT   -- Interface for calculating the \
+                     :ref:`Short Time Fourier Transform <tutorial_stft>` and \
+                     its inverse.
+   closest_STFT_dual_window -- Calculate the STFT dual window of a given window \
+                               closest to a desired dual window.
+   stft           -- Compute the Short Time Fourier Transform (legacy).
+   istft          -- Compute the Inverse Short Time Fourier Transform (legacy).
+   check_COLA     -- Check the COLA constraint for iSTFT reconstruction (legacy).
+   check_NOLA     -- Check the NOLA constraint for iSTFT reconstruction.
+
+Chirp Z-transform and Zoom FFT
+============================================
+
+.. autosummary::
+   :toctree: generated/
+
+   czt - Chirp z-transform convenience function
+   zoom_fft - Zoom FFT convenience function
+   CZT - Chirp z-transform function generator
+   ZoomFFT - Zoom FFT function generator
+   czt_points - Output the z-plane points sampled by a chirp z-transform
+
+The functions are simpler to use than the classes, but are less efficient when
+using the same transform on many arrays of the same length, since they
+repeatedly generate the same chirp signal with every call.  In these cases,
+use the classes to create a reusable function instead.
+
+"""
+# bring in the public functionality from private namespaces
+
+# mypy: ignore-errors
+
+from ._support_alternative_backends import *
+from . import _support_alternative_backends
+__all__ = _support_alternative_backends.__all__
+del _support_alternative_backends, _signal_api, _delegators  # noqa: F821
+
+
+# Deprecated namespaces, to be removed in v2.0.0
+from . import (
+    bsplines, filter_design, fir_filter_design, lti_conversion, ltisys,
+    spectral, signaltools, waveforms, wavelets, spline
+)
+
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

venv/lib/python3.13/site-packages/scipy/signal/_arraytools.py

@@ -0,0 +1,264 @@
+"""
+Functions for acting on a axis of an array.
+"""
+import numpy as np
+
+
+def axis_slice(a, start=None, stop=None, step=None, axis=-1):
+    """Take a slice along axis 'axis' from 'a'.
+
+    Parameters
+    ----------
+    a : numpy.ndarray
+        The array to be sliced.
+    start, stop, step : int or None
+        The slice parameters.
+    axis : int, optional
+        The axis of `a` to be sliced.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._arraytools import axis_slice
+    >>> a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    >>> axis_slice(a, start=0, stop=1, axis=1)
+    array([[1],
+           [4],
+           [7]])
+    >>> axis_slice(a, start=1, axis=0)
+    array([[4, 5, 6],
+           [7, 8, 9]])
+
+    Notes
+    -----
+    The keyword arguments start, stop and step are used by calling
+    slice(start, stop, step). This implies axis_slice() does not
+    handle its arguments the exactly the same as indexing. To select
+    a single index k, for example, use
+        axis_slice(a, start=k, stop=k+1)
+    In this case, the length of the axis 'axis' in the result will
+    be 1; the trivial dimension is not removed. (Use numpy.squeeze()
+    to remove trivial axes.)
+    """
+    a_slice = [slice(None)] * a.ndim
+    a_slice[axis] = slice(start, stop, step)
+    b = a[tuple(a_slice)]
+    return b
+
+
+def axis_reverse(a, axis=-1):
+    """Reverse the 1-D slices of `a` along axis `axis`.
+
+    Returns axis_slice(a, step=-1, axis=axis).
+    """
+    return axis_slice(a, step=-1, axis=axis)
+
+
+def odd_ext(x, n, axis=-1):
+    """
+    Odd extension at the boundaries of an array
+
+    Generate a new ndarray by making an odd extension of `x` along an axis.
+
+    Parameters
+    ----------
+    x : ndarray
+        The array to be extended.
+    n : int
+        The number of elements by which to extend `x` at each end of the axis.
+    axis : int, optional
+        The axis along which to extend `x`. Default is -1.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._arraytools import odd_ext
+    >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]])
+    >>> odd_ext(a, 2)
+    array([[-1,  0,  1,  2,  3,  4,  5,  6,  7],
+           [-4, -1,  0,  1,  4,  9, 16, 23, 28]])
+
+    Odd extension is a "180 degree rotation" at the endpoints of the original
+    array:
+
+    >>> t = np.linspace(0, 1.5, 100)
+    >>> a = 0.9 * np.sin(2 * np.pi * t**2)
+    >>> b = odd_ext(a, 40)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(np.arange(-40, 140), b, 'b', lw=1, label='odd extension')
+    >>> plt.plot(np.arange(100), a, 'r', lw=2, label='original')
+    >>> plt.legend(loc='best')
+    >>> plt.show()
+    """
+    if n < 1:
+        return x
+    if n > x.shape[axis] - 1:
+        raise ValueError(("The extension length n (%d) is too big. " +
+                         "It must not exceed x.shape[axis]-1, which is %d.")
+                         % (n, x.shape[axis] - 1))
+    left_end = axis_slice(x, start=0, stop=1, axis=axis)
+    left_ext = axis_slice(x, start=n, stop=0, step=-1, axis=axis)
+    right_end = axis_slice(x, start=-1, axis=axis)
+    right_ext = axis_slice(x, start=-2, stop=-(n + 2), step=-1, axis=axis)
+    ext = np.concatenate((2 * left_end - left_ext,
+                          x,
+                          2 * right_end - right_ext),
+                         axis=axis)
+    return ext
+
+
+def even_ext(x, n, axis=-1):
+    """
+    Even extension at the boundaries of an array
+
+    Generate a new ndarray by making an even extension of `x` along an axis.
+
+    Parameters
+    ----------
+    x : ndarray
+        The array to be extended.
+    n : int
+        The number of elements by which to extend `x` at each end of the axis.
+    axis : int, optional
+        The axis along which to extend `x`. Default is -1.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._arraytools import even_ext
+    >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]])
+    >>> even_ext(a, 2)
+    array([[ 3,  2,  1,  2,  3,  4,  5,  4,  3],
+           [ 4,  1,  0,  1,  4,  9, 16,  9,  4]])
+
+    Even extension is a "mirror image" at the boundaries of the original array:
+
+    >>> t = np.linspace(0, 1.5, 100)
+    >>> a = 0.9 * np.sin(2 * np.pi * t**2)
+    >>> b = even_ext(a, 40)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(np.arange(-40, 140), b, 'b', lw=1, label='even extension')
+    >>> plt.plot(np.arange(100), a, 'r', lw=2, label='original')
+    >>> plt.legend(loc='best')
+    >>> plt.show()
+    """
+    if n < 1:
+        return x
+    if n > x.shape[axis] - 1:
+        raise ValueError(("The extension length n (%d) is too big. " +
+                         "It must not exceed x.shape[axis]-1, which is %d.")
+                         % (n, x.shape[axis] - 1))
+    left_ext = axis_slice(x, start=n, stop=0, step=-1, axis=axis)
+    right_ext = axis_slice(x, start=-2, stop=-(n + 2), step=-1, axis=axis)
+    ext = np.concatenate((left_ext,
+                          x,
+                          right_ext),
+                         axis=axis)
+    return ext
+
+
+def const_ext(x, n, axis=-1):
+    """
+    Constant extension at the boundaries of an array
+
+    Generate a new ndarray that is a constant extension of `x` along an axis.
+
+    The extension repeats the values at the first and last element of
+    the axis.
+
+    Parameters
+    ----------
+    x : ndarray
+        The array to be extended.
+    n : int
+        The number of elements by which to extend `x` at each end of the axis.
+    axis : int, optional
+        The axis along which to extend `x`. Default is -1.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._arraytools import const_ext
+    >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]])
+    >>> const_ext(a, 2)
+    array([[ 1,  1,  1,  2,  3,  4,  5,  5,  5],
+           [ 0,  0,  0,  1,  4,  9, 16, 16, 16]])
+
+    Constant extension continues with the same values as the endpoints of the
+    array:
+
+    >>> t = np.linspace(0, 1.5, 100)
+    >>> a = 0.9 * np.sin(2 * np.pi * t**2)
+    >>> b = const_ext(a, 40)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(np.arange(-40, 140), b, 'b', lw=1, label='constant extension')
+    >>> plt.plot(np.arange(100), a, 'r', lw=2, label='original')
+    >>> plt.legend(loc='best')
+    >>> plt.show()
+    """
+    if n < 1:
+        return x
+    left_end = axis_slice(x, start=0, stop=1, axis=axis)
+    ones_shape = [1] * x.ndim
+    ones_shape[axis] = n
+    ones = np.ones(ones_shape, dtype=x.dtype)
+    left_ext = ones * left_end
+    right_end = axis_slice(x, start=-1, axis=axis)
+    right_ext = ones * right_end
+    ext = np.concatenate((left_ext,
+                          x,
+                          right_ext),
+                         axis=axis)
+    return ext
+
+
+def zero_ext(x, n, axis=-1):
+    """
+    Zero padding at the boundaries of an array
+
+    Generate a new ndarray that is a zero-padded extension of `x` along
+    an axis.
+
+    Parameters
+    ----------
+    x : ndarray
+        The array to be extended.
+    n : int
+        The number of elements by which to extend `x` at each end of the
+        axis.
+    axis : int, optional
+        The axis along which to extend `x`. Default is -1.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._arraytools import zero_ext
+    >>> a = np.array([[1, 2, 3, 4, 5], [0, 1, 4, 9, 16]])
+    >>> zero_ext(a, 2)
+    array([[ 0,  0,  1,  2,  3,  4,  5,  0,  0],
+           [ 0,  0,  0,  1,  4,  9, 16,  0,  0]])
+    """
+    if n < 1:
+        return x
+    zeros_shape = list(x.shape)
+    zeros_shape[axis] = n
+    zeros = np.zeros(zeros_shape, dtype=x.dtype)
+    ext = np.concatenate((zeros, x, zeros), axis=axis)
+    return ext
+
+
+def _validate_fs(fs, allow_none=True):
+    """
+    Check if the given sampling frequency is a scalar and raises an exception
+    otherwise. If allow_none is False, also raises an exception for none
+    sampling rates. Returns the sampling frequency as float or none if the
+    input is none.
+    """
+    if fs is None:
+        if not allow_none:
+            raise ValueError("Sampling frequency can not be none.")
+    else:  # should be float
+        if not np.isscalar(fs):
+            raise ValueError("Sampling frequency fs must be a single scalar.")
+        fs = float(fs)
+    return fs

venv/lib/python3.13/site-packages/scipy/signal/_czt.py

@@ -0,0 +1,575 @@
+# This program is public domain
+# Authors: Paul Kienzle, Nadav Horesh
+"""
+Chirp z-transform.
+
+We provide two interfaces to the chirp z-transform: an object interface
+which precalculates part of the transform and can be applied efficiently
+to many different data sets, and a functional interface which is applied
+only to the given data set.
+
+Transforms
+----------
+
+CZT : callable (x, axis=-1) -> array
+   Define a chirp z-transform that can be applied to different signals.
+ZoomFFT : callable (x, axis=-1) -> array
+   Define a Fourier transform on a range of frequencies.
+
+Functions
+---------
+
+czt : array
+   Compute the chirp z-transform for a signal.
+zoom_fft : array
+   Compute the Fourier transform on a range of frequencies.
+"""
+
+import cmath
+import numbers
+import numpy as np
+from numpy import pi, arange
+from scipy.fft import fft, ifft, next_fast_len
+
+__all__ = ['czt', 'zoom_fft', 'CZT', 'ZoomFFT', 'czt_points']
+
+
+def _validate_sizes(n, m):
+    if n < 1 or not isinstance(n, numbers.Integral):
+        raise ValueError('Invalid number of CZT data '
+                         f'points ({n}) specified. '
+                         'n must be positive and integer type.')
+
+    if m is None:
+        m = n
+    elif m < 1 or not isinstance(m, numbers.Integral):
+        raise ValueError('Invalid number of CZT output '
+                         f'points ({m}) specified. '
+                         'm must be positive and integer type.')
+
+    return m
+
+
+def czt_points(m, w=None, a=1+0j):
+    """
+    Return the points at which the chirp z-transform is computed.
+
+    Parameters
+    ----------
+    m : int
+        The number of points desired.
+    w : complex, optional
+        The ratio between points in each step.
+        Defaults to equally spaced points around the entire unit circle.
+    a : complex, optional
+        The starting point in the complex plane.  Default is 1+0j.
+
+    Returns
+    -------
+    out : ndarray
+        The points in the Z plane at which `CZT` samples the z-transform,
+        when called with arguments `m`, `w`, and `a`, as complex numbers.
+
+    See Also
+    --------
+    CZT : Class that creates a callable chirp z-transform function.
+    czt : Convenience function for quickly calculating CZT.
+
+    Examples
+    --------
+    Plot the points of a 16-point FFT:
+
+    >>> import numpy as np
+    >>> from scipy.signal import czt_points
+    >>> points = czt_points(16)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(points.real, points.imag, 'o')
+    >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3))
+    >>> plt.axis('equal')
+    >>> plt.show()
+
+    and a 91-point logarithmic spiral that crosses the unit circle:
+
+    >>> m, w, a = 91, 0.995*np.exp(-1j*np.pi*.05), 0.8*np.exp(1j*np.pi/6)
+    >>> points = czt_points(m, w, a)
+    >>> plt.plot(points.real, points.imag, 'o')
+    >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3))
+    >>> plt.axis('equal')
+    >>> plt.show()
+    """
+    m = _validate_sizes(1, m)
+
+    k = arange(m)
+
+    a = 1.0 * a  # at least float
+
+    if w is None:
+        # Nothing specified, default to FFT
+        return a * np.exp(2j * pi * k / m)
+    else:
+        # w specified
+        w = 1.0 * w  # at least float
+        return a * w**-k
+
+
+class CZT:
+    """
+    Create a callable chirp z-transform function.
+
+    Transform to compute the frequency response around a spiral.
+    Objects of this class are callables which can compute the
+    chirp z-transform on their inputs.  This object precalculates the constant
+    chirps used in the given transform.
+
+    Parameters
+    ----------
+    n : int
+        The size of the signal.
+    m : int, optional
+        The number of output points desired.  Default is `n`.
+    w : complex, optional
+        The ratio between points in each step.  This must be precise or the
+        accumulated error will degrade the tail of the output sequence.
+        Defaults to equally spaced points around the entire unit circle.
+    a : complex, optional
+        The starting point in the complex plane.  Default is 1+0j.
+
+    Returns
+    -------
+    f : CZT
+        Callable object ``f(x, axis=-1)`` for computing the chirp z-transform
+        on `x`.
+
+    See Also
+    --------
+    czt : Convenience function for quickly calculating CZT.
+    ZoomFFT : Class that creates a callable partial FFT function.
+
+    Notes
+    -----
+    The defaults are chosen such that ``f(x)`` is equivalent to
+    ``fft.fft(x)`` and, if ``m > len(x)``, that ``f(x, m)`` is equivalent to
+    ``fft.fft(x, m)``.
+
+    If `w` does not lie on the unit circle, then the transform will be
+    around a spiral with exponentially-increasing radius.  Regardless,
+    angle will increase linearly.
+
+    For transforms that do lie on the unit circle, accuracy is better when
+    using `ZoomFFT`, since any numerical error in `w` is
+    accumulated for long data lengths, drifting away from the unit circle.
+
+    The chirp z-transform can be faster than an equivalent FFT with
+    zero padding.  Try it with your own array sizes to see.
+
+    However, the chirp z-transform is considerably less precise than the
+    equivalent zero-padded FFT.
+
+    As this CZT is implemented using the Bluestein algorithm, it can compute
+    large prime-length Fourier transforms in O(N log N) time, rather than the
+    O(N**2) time required by the direct DFT calculation.  (`scipy.fft` also
+    uses Bluestein's algorithm'.)
+
+    (The name "chirp z-transform" comes from the use of a chirp in the
+    Bluestein algorithm.  It does not decompose signals into chirps, like
+    other transforms with "chirp" in the name.)
+
+    References
+    ----------
+    .. [1] Leo I. Bluestein, "A linear filtering approach to the computation
+           of the discrete Fourier transform," Northeast Electronics Research
+           and Engineering Meeting Record 10, 218-219 (1968).
+    .. [2] Rabiner, Schafer, and Rader, "The chirp z-transform algorithm and
+           its application," Bell Syst. Tech. J. 48, 1249-1292 (1969).
+
+    Examples
+    --------
+    Compute multiple prime-length FFTs:
+
+    >>> from scipy.signal import CZT
+    >>> import numpy as np
+    >>> a = np.random.rand(7)
+    >>> b = np.random.rand(7)
+    >>> c = np.random.rand(7)
+    >>> czt_7 = CZT(n=7)
+    >>> A = czt_7(a)
+    >>> B = czt_7(b)
+    >>> C = czt_7(c)
+
+    Display the points at which the FFT is calculated:
+
+    >>> czt_7.points()
+    array([ 1.00000000+0.j        ,  0.62348980+0.78183148j,
+           -0.22252093+0.97492791j, -0.90096887+0.43388374j,
+           -0.90096887-0.43388374j, -0.22252093-0.97492791j,
+            0.62348980-0.78183148j])
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(czt_7.points().real, czt_7.points().imag, 'o')
+    >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3))
+    >>> plt.axis('equal')
+    >>> plt.show()
+    """
+
+    def __init__(self, n, m=None, w=None, a=1+0j):
+        m = _validate_sizes(n, m)
+
+        k = arange(max(m, n), dtype=np.min_scalar_type(-max(m, n)**2))
+
+        if w is None:
+            # Nothing specified, default to FFT-like
+            w = cmath.exp(-2j*pi/m)
+            wk2 = np.exp(-(1j * pi * ((k**2) % (2*m))) / m)
+        else:
+            # w specified
+            wk2 = w**(k**2/2.)
+
+        a = 1.0 * a  # at least float
+
+        self.w, self.a = w, a
+        self.m, self.n = m, n
+
+        nfft = next_fast_len(n + m - 1)
+        self._Awk2 = a**-k[:n] * wk2[:n]
+        self._nfft = nfft
+        self._Fwk2 = fft(1/np.hstack((wk2[n-1:0:-1], wk2[:m])), nfft)
+        self._wk2 = wk2[:m]
+        self._yidx = slice(n-1, n+m-1)
+
+    def __call__(self, x, *, axis=-1):
+        """
+        Calculate the chirp z-transform of a signal.
+
+        Parameters
+        ----------
+        x : array
+            The signal to transform.
+        axis : int, optional
+            Axis over which to compute the FFT. If not given, the last axis is
+            used.
+
+        Returns
+        -------
+        out : ndarray
+            An array of the same dimensions as `x`, but with the length of the
+            transformed axis set to `m`.
+        """
+        x = np.asarray(x)
+        if x.shape[axis] != self.n:
+            raise ValueError(f"CZT defined for length {self.n}, not "
+                             f"{x.shape[axis]}")
+        # Calculate transpose coordinates, to allow operation on any given axis
+        trnsp = np.arange(x.ndim)
+        trnsp[[axis, -1]] = [-1, axis]
+        x = x.transpose(*trnsp)
+        y = ifft(self._Fwk2 * fft(x*self._Awk2, self._nfft))
+        y = y[..., self._yidx] * self._wk2
+        return y.transpose(*trnsp)
+
+    def points(self):
+        """
+        Return the points at which the chirp z-transform is computed.
+        """
+        return czt_points(self.m, self.w, self.a)
+
+
+class ZoomFFT(CZT):
+    """
+    Create a callable zoom FFT transform function.
+
+    This is a specialization of the chirp z-transform (`CZT`) for a set of
+    equally-spaced frequencies around the unit circle, used to calculate a
+    section of the FFT more efficiently than calculating the entire FFT and
+    truncating.
+
+    Parameters
+    ----------
+    n : int
+        The size of the signal.
+    fn : array_like
+        A length-2 sequence [`f1`, `f2`] giving the frequency range, or a
+        scalar, for which the range [0, `fn`] is assumed.
+    m : int, optional
+        The number of points to evaluate.  Default is `n`.
+    fs : float, optional
+        The sampling frequency.  If ``fs=10`` represented 10 kHz, for example,
+        then `f1` and `f2` would also be given in kHz.
+        The default sampling frequency is 2, so `f1` and `f2` should be
+        in the range [0, 1] to keep the transform below the Nyquist
+        frequency.
+    endpoint : bool, optional
+        If True, `f2` is the last sample. Otherwise, it is not included.
+        Default is False.
+
+    Returns
+    -------
+    f : ZoomFFT
+        Callable object ``f(x, axis=-1)`` for computing the zoom FFT on `x`.
+
+    See Also
+    --------
+    zoom_fft : Convenience function for calculating a zoom FFT.
+
+    Notes
+    -----
+    The defaults are chosen such that ``f(x, 2)`` is equivalent to
+    ``fft.fft(x)`` and, if ``m > len(x)``, that ``f(x, 2, m)`` is equivalent to
+    ``fft.fft(x, m)``.
+
+    Sampling frequency is 1/dt, the time step between samples in the
+    signal `x`.  The unit circle corresponds to frequencies from 0 up
+    to the sampling frequency.  The default sampling frequency of 2
+    means that `f1`, `f2` values up to the Nyquist frequency are in the
+    range [0, 1). For `f1`, `f2` values expressed in radians, a sampling
+    frequency of 2*pi should be used.
+
+    Remember that a zoom FFT can only interpolate the points of the existing
+    FFT.  It cannot help to resolve two separate nearby frequencies.
+    Frequency resolution can only be increased by increasing acquisition
+    time.
+
+    These functions are implemented using Bluestein's algorithm (as is
+    `scipy.fft`). [2]_
+
+    References
+    ----------
+    .. [1] Steve Alan Shilling, "A study of the chirp z-transform and its
+           applications", pg 29 (1970)
+           https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf
+    .. [2] Leo I. Bluestein, "A linear filtering approach to the computation
+           of the discrete Fourier transform," Northeast Electronics Research
+           and Engineering Meeting Record 10, 218-219 (1968).
+
+    Examples
+    --------
+    To plot the transform results use something like the following:
+
+    >>> import numpy as np
+    >>> from scipy.signal import ZoomFFT
+    >>> t = np.linspace(0, 1, 1021)
+    >>> x = np.cos(2*np.pi*15*t) + np.sin(2*np.pi*17*t)
+    >>> f1, f2 = 5, 27
+    >>> transform = ZoomFFT(len(x), [f1, f2], len(x), fs=1021)
+    >>> X = transform(x)
+    >>> f = np.linspace(f1, f2, len(x))
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(f, 20*np.log10(np.abs(X)))
+    >>> plt.show()
+    """
+
+    def __init__(self, n, fn, m=None, *, fs=2, endpoint=False):
+        m = _validate_sizes(n, m)
+
+        k = arange(max(m, n), dtype=np.min_scalar_type(-max(m, n)**2))
+
+        if np.size(fn) == 2:
+            f1, f2 = fn
+        elif np.size(fn) == 1:
+            f1, f2 = 0.0, fn
+        else:
+            raise ValueError('fn must be a scalar or 2-length sequence')
+
+        self.f1, self.f2, self.fs = f1, f2, fs
+
+        if endpoint:
+            scale = ((f2 - f1) * m) / (fs * (m - 1))
+        else:
+            scale = (f2 - f1) / fs
+        a = cmath.exp(2j * pi * f1/fs)
+        wk2 = np.exp(-(1j * pi * scale * k**2) / m)
+
+        self.w = cmath.exp(-2j*pi/m * scale)
+        self.a = a
+        self.m, self.n = m, n
+
+        ak = np.exp(-2j * pi * f1/fs * k[:n])
+        self._Awk2 = ak * wk2[:n]
+
+        nfft = next_fast_len(n + m - 1)
+        self._nfft = nfft
+        self._Fwk2 = fft(1/np.hstack((wk2[n-1:0:-1], wk2[:m])), nfft)
+        self._wk2 = wk2[:m]
+        self._yidx = slice(n-1, n+m-1)
+
+
+def czt(x, m=None, w=None, a=1+0j, *, axis=-1):
+    """
+    Compute the frequency response around a spiral in the Z plane.
+
+    Parameters
+    ----------
+    x : array
+        The signal to transform.
+    m : int, optional
+        The number of output points desired.  Default is the length of the
+        input data.
+    w : complex, optional
+        The ratio between points in each step.  This must be precise or the
+        accumulated error will degrade the tail of the output sequence.
+        Defaults to equally spaced points around the entire unit circle.
+    a : complex, optional
+        The starting point in the complex plane.  Default is 1+0j.
+    axis : int, optional
+        Axis over which to compute the FFT. If not given, the last axis is
+        used.
+
+    Returns
+    -------
+    out : ndarray
+        An array of the same dimensions as `x`, but with the length of the
+        transformed axis set to `m`.
+
+    See Also
+    --------
+    CZT : Class that creates a callable chirp z-transform function.
+    zoom_fft : Convenience function for partial FFT calculations.
+
+    Notes
+    -----
+    The defaults are chosen such that ``signal.czt(x)`` is equivalent to
+    ``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.czt(x, m)`` is
+    equivalent to ``fft.fft(x, m)``.
+
+    If the transform needs to be repeated, use `CZT` to construct a
+    specialized transform function which can be reused without
+    recomputing constants.
+
+    An example application is in system identification, repeatedly evaluating
+    small slices of the z-transform of a system, around where a pole is
+    expected to exist, to refine the estimate of the pole's true location. [1]_
+
+    References
+    ----------
+    .. [1] Steve Alan Shilling, "A study of the chirp z-transform and its
+           applications", pg 20 (1970)
+           https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf
+
+    Examples
+    --------
+    Generate a sinusoid:
+
+    >>> import numpy as np
+    >>> f1, f2, fs = 8, 10, 200  # Hz
+    >>> t = np.linspace(0, 1, fs, endpoint=False)
+    >>> x = np.sin(2*np.pi*t*f2)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(t, x)
+    >>> plt.axis([0, 1, -1.1, 1.1])
+    >>> plt.show()
+
+    Its discrete Fourier transform has all of its energy in a single frequency
+    bin:
+
+    >>> from scipy.fft import rfft, rfftfreq
+    >>> from scipy.signal import czt, czt_points
+    >>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x)))
+    >>> plt.margins(0, 0.1)
+    >>> plt.show()
+
+    However, if the sinusoid is logarithmically-decaying:
+
+    >>> x = np.exp(-t*f1) * np.sin(2*np.pi*t*f2)
+    >>> plt.plot(t, x)
+    >>> plt.axis([0, 1, -1.1, 1.1])
+    >>> plt.show()
+
+    the DFT will have spectral leakage:
+
+    >>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x)))
+    >>> plt.margins(0, 0.1)
+    >>> plt.show()
+
+    While the DFT always samples the z-transform around the unit circle, the
+    chirp z-transform allows us to sample the Z-transform along any
+    logarithmic spiral, such as a circle with radius smaller than unity:
+
+    >>> M = fs // 2  # Just positive frequencies, like rfft
+    >>> a = np.exp(-f1/fs)  # Starting point of the circle, radius < 1
+    >>> w = np.exp(-1j*np.pi/M)  # "Step size" of circle
+    >>> points = czt_points(M + 1, w, a)  # M + 1 to include Nyquist
+    >>> plt.plot(points.real, points.imag, '.')
+    >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3))
+    >>> plt.axis('equal'); plt.axis([-1.05, 1.05, -0.05, 1.05])
+    >>> plt.show()
+
+    With the correct radius, this transforms the decaying sinusoid (and others
+    with the same decay rate) without spectral leakage:
+
+    >>> z_vals = czt(x, M + 1, w, a)  # Include Nyquist for comparison to rfft
+    >>> freqs = np.angle(points)*fs/(2*np.pi)  # angle = omega, radius = sigma
+    >>> plt.plot(freqs, abs(z_vals))
+    >>> plt.margins(0, 0.1)
+    >>> plt.show()
+    """
+    x = np.asarray(x)
+    transform = CZT(x.shape[axis], m=m, w=w, a=a)
+    return transform(x, axis=axis)
+
+
+def zoom_fft(x, fn, m=None, *, fs=2, endpoint=False, axis=-1):
+    """
+    Compute the DFT of `x` only for frequencies in range `fn`.
+
+    Parameters
+    ----------
+    x : array
+        The signal to transform.
+    fn : array_like
+        A length-2 sequence [`f1`, `f2`] giving the frequency range, or a
+        scalar, for which the range [0, `fn`] is assumed.
+    m : int, optional
+        The number of points to evaluate.  The default is the length of `x`.
+    fs : float, optional
+        The sampling frequency.  If ``fs=10`` represented 10 kHz, for example,
+        then `f1` and `f2` would also be given in kHz.
+        The default sampling frequency is 2, so `f1` and `f2` should be
+        in the range [0, 1] to keep the transform below the Nyquist
+        frequency.
+    endpoint : bool, optional
+        If True, `f2` is the last sample. Otherwise, it is not included.
+        Default is False.
+    axis : int, optional
+        Axis over which to compute the FFT. If not given, the last axis is
+        used.
+
+    Returns
+    -------
+    out : ndarray
+        The transformed signal.  The Fourier transform will be calculated
+        at the points f1, f1+df, f1+2df, ..., f2, where df=(f2-f1)/m.
+
+    See Also
+    --------
+    ZoomFFT : Class that creates a callable partial FFT function.
+
+    Notes
+    -----
+    The defaults are chosen such that ``signal.zoom_fft(x, 2)`` is equivalent
+    to ``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.zoom_fft(x, 2, m)``
+    is equivalent to ``fft.fft(x, m)``.
+
+    To graph the magnitude of the resulting transform, use::
+
+        plot(linspace(f1, f2, m, endpoint=False), abs(zoom_fft(x, [f1, f2], m)))
+
+    If the transform needs to be repeated, use `ZoomFFT` to construct
+    a specialized transform function which can be reused without
+    recomputing constants.
+
+    Examples
+    --------
+    To plot the transform results use something like the following:
+
+    >>> import numpy as np
+    >>> from scipy.signal import zoom_fft
+    >>> t = np.linspace(0, 1, 1021)
+    >>> x = np.cos(2*np.pi*15*t) + np.sin(2*np.pi*17*t)
+    >>> f1, f2 = 5, 27
+    >>> X = zoom_fft(x, [f1, f2], len(x), fs=1021)
+    >>> f = np.linspace(f1, f2, len(x))
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(f, 20*np.log10(np.abs(X)))
+    >>> plt.show()
+    """
+    x = np.asarray(x)
+    transform = ZoomFFT(x.shape[axis], fn, m=m, fs=fs, endpoint=endpoint)
+    return transform(x, axis=axis)

venv/lib/python3.13/site-packages/scipy/signal/_delegators.py

@@ -0,0 +1,568 @@
+"""Delegators for alternative backends in scipy.signal.
+
+The signature of `func_signature` must match the signature of signal.func.
+The job of a `func_signature` is to know which arguments of `signal.func`
+are arrays.
+
+* signatures are generated by
+
+--------------
+ import inspect
+ from scipy import signal
+
+ names = [x for x in dir(signal) if not x.startswith('_')]
+ objs = [getattr(signal, name) for name in names]
+ funcs = [obj for obj in objs if inspect.isroutine(obj)]
+
+ for func in funcs:
+     try:
+        sig = inspect.signature(func)
+     except ValueError:
+         sig = "( FIXME )"
+     print(f"def {func.__name__}_signature{sig}:\n\treturn array_namespace(...
+ )\n\n")
+---------------
+
+* which arguments to delegate on: manually trawled the documentation for
+  array-like and array arguments
+
+"""
+import numpy as np
+from scipy._lib._array_api import array_namespace, np_compat
+
+
+def _skip_if_lti(arg):
+    """Handle `system` arg overloads.
+
+    ATM, only pass tuples through. Consider updating when cupyx.lti class
+    is supported.
+    """
+    if isinstance(arg, tuple):
+        return arg
+    else:
+        return (None,)
+
+
+def _skip_if_str_or_tuple(window):
+    """Handle `window` being a str or a tuple or an array-like.
+    """
+    if isinstance(window, str) or isinstance(window, tuple) or callable(window):
+        return None
+    else:
+        return window
+
+
+def _skip_if_poly1d(arg):
+    return None if isinstance(arg, np.poly1d) else arg
+
+
+###################
+
+def abcd_normalize_signature(A=None, B=None, C=None, D=None):
+    return array_namespace(A, B, C, D)
+
+
+def argrelextrema_signature(data, *args, **kwds):
+    return array_namespace(data)
+
+argrelmax_signature = argrelextrema_signature
+argrelmin_signature = argrelextrema_signature
+
+
+def band_stop_obj_signature(wp, ind, passb, stopb, gpass, gstop, type):
+    return array_namespace(passb, stopb)
+
+
+def bessel_signature(N, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+butter_signature = bessel_signature
+
+
+def cheby2_signature(N, rs, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+def cheby1_signature(N, rp, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+def ellip_signature(N, rp, rs, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+########################## XXX: no arrays in, arrays out
+def besselap_signature(N, norm='phase'):
+    return np
+
+def buttap_signature(N):
+    return np
+
+def cheb1ap_signature(N, rp):
+    return np
+
+
+def cheb2ap_signature(N, rs):
+    return np
+
+def ellipap_signature(N, rp, rs):
+    return np
+
+def correlation_lags_signature(in1_len, in2_len, mode='full'):
+    return np
+
+
+def czt_points_signature(m, w=None, a=(1+0j)):
+    return np
+
+
+def gammatone_signature(freq, ftype, order=None, numtaps=None, fs=None):
+    return np
+
+
+def iircomb_signature(w0, Q, ftype='notch', fs=2.0, *, pass_zero=False):
+    return np
+
+
+def iirnotch_signature(w0, Q, fs=2.0):
+    return np
+
+
+def iirpeak_signature(w0, Q, fs=2.0):
+    return np
+
+
+def savgol_coeffs_signature(
+    window_length, polyorder, deriv=0, delta=1.0, pos=None, use='conv'
+):
+    return np
+
+
+def unit_impulse_signature(shape, idx=None, dtype=float):
+    return np
+############################
+
+
+####################### XXX: no arrays, maybe arrays out
+def buttord_signature(wp, ws, gpass, gstop, analog=False, fs=None):
+    return np
+
+def cheb1ord_signature(wp, ws, gpass, gstop, analog=False, fs=None):
+    return np
+
+def cheb2ord_signature(wp, ws, gpass, gstop, analog=False, fs=None):
+    return np
+
+def ellipord_signature(wp, ws, gpass, gstop, analog=False, fs=None):
+    return np
+###########################################
+
+
+########### NB: scalars in, scalars out
+def kaiser_atten_signature(numtaps, width):
+    return np
+
+def kaiser_beta_signature(a):
+    return np
+
+def kaiserord_signature(ripple, width):
+    return np
+
+def get_window_signature(window, Nx, fftbins=True, *, xp=None, device=None):
+    return np if xp is None else xp
+#################################
+
+
+def bode_signature(system, w=None, n=100):
+    return array_namespace(*_skip_if_lti(system), w)
+
+dbode_signature = bode_signature
+
+
+def freqresp_signature(system, w=None, n=10000):
+    return array_namespace(*_skip_if_lti(system), w)
+
+dfreqresp_signature = freqresp_signature
+
+
+def impulse_signature(system, X0=None, T=None, N=None):
+    return array_namespace(*_skip_if_lti(system), X0, T)
+
+
+def dimpulse_signature(system, x0=None, t=None, n=None):
+    return array_namespace(*_skip_if_lti(system), x0, t)
+
+
+def lsim_signature(system, U, T, X0=None, interp=True):
+    return array_namespace(*_skip_if_lti(system), U, T, X0)
+
+
+def dlsim_signature(system, u, t=None, x0=None):
+    return array_namespace(*_skip_if_lti(system), u, t, x0)
+
+
+def step_signature(system, X0=None, T=None, N=None):
+    return array_namespace(*_skip_if_lti(system), X0, T)
+
+def dstep_signature(system, x0=None, t=None, n=None):
+    return array_namespace(*_skip_if_lti(system), x0, t)
+
+
+def cont2discrete_signature(system, dt, method='zoh', alpha=None):
+    return array_namespace(*_skip_if_lti(system))
+
+
+def bilinear_signature(b, a, fs=1.0):
+    return array_namespace(b, a)
+
+
+def bilinear_zpk_signature(z, p, k, fs):
+    return array_namespace(z, p)
+
+
+def chirp_signature(t,*args, **kwds):
+    return array_namespace(t)
+
+
+############## XXX: array-likes in, str out
+def choose_conv_method_signature(in1, in2, *args, **kwds):
+    return array_namespace(in1, in2)
+############################################
+
+
+def convolve_signature(in1, in2, *args, **kwds):
+    return array_namespace(in1, in2)
+
+fftconvolve_signature = convolve_signature
+oaconvolve_signature = convolve_signature
+correlate_signature = convolve_signature
+correlate_signature = convolve_signature
+convolve2d_signature = convolve_signature
+correlate2d_signature = convolve_signature
+
+
+def coherence_signature(x, y, fs=1.0, window='hann', *args, **kwds):
+    return array_namespace(x, y, _skip_if_str_or_tuple(window))
+
+
+def csd_signature(x, y, fs=1.0, window='hann', *args, **kwds):
+    return array_namespace(x, y, _skip_if_str_or_tuple(window))
+
+
+def periodogram_signature(x, fs=1.0, window='boxcar'):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def welch_signature(x, fs=1.0, window='hann', *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def spectrogram_signature(x, fs=1.0, window=('tukey', 0.25), *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def stft_signature(x, fs=1.0, window='hann', *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def istft_signature(Zxx, fs=1.0, window='hann', *args, **kwds):
+    return array_namespace(Zxx, _skip_if_str_or_tuple(window))
+
+
+def resample_signature(x, num, t=None, axis=0, window=None, domain='time'):
+    return array_namespace(x, t, _skip_if_str_or_tuple(window))
+
+
+def resample_poly_signature(x, up, down, axis=0, window=('kaiser', 5.0), *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def check_COLA_signature(window, nperseg, noverlap, tol=1e-10):
+    return array_namespace(_skip_if_str_or_tuple(window))
+
+
+def check_NOLA_signature(window, nperseg, noverlap, tol=1e-10):
+    return array_namespace(_skip_if_str_or_tuple(window))
+
+
+def czt_signature(x, *args, **kwds):
+    return array_namespace(x)
+
+decimate_signature = czt_signature
+gauss_spline_signature = czt_signature
+
+
+def deconvolve_signature(signal, divisor):
+    return array_namespace(signal, divisor)
+
+
+def detrend_signature(data, axis=1, type='linear', bp=0, *args, **kwds):
+    return array_namespace(data, bp)
+
+
+def filtfilt_signature(b, a, x, *args, **kwds):
+    return array_namespace(b, a, x)
+
+
+def lfilter_signature(b, a, x, axis=-1, zi=None):
+    return array_namespace(b, a, x, zi)
+
+
+def envelope_signature(z, *args, **kwds):
+    return array_namespace(z)
+
+
+def find_peaks_signature(
+    x, height=None, threshold=None, distance=None, prominence=None, width=None,
+    wlen=None, rel_height=0.5, plateau_size=None
+):
+    return array_namespace(x, height, threshold, prominence, width, plateau_size)
+
+
+def find_peaks_cwt_signature(
+    vector, widths, wavelet=None, max_distances=None, *args, **kwds
+):
+    return array_namespace(vector, widths, max_distances)
+
+
+def findfreqs_signature(num, den, N, kind='ba'):
+    return array_namespace(num, den)
+
+
+def firls_signature(numtaps, bands, desired, *, weight=None, fs=None):
+    return array_namespace(bands, desired, weight)
+
+
+def firwin_signature(numtaps, cutoff, *args, **kwds):
+    if isinstance(cutoff, int | float):
+        xp = np_compat
+    else:
+        xp = array_namespace(cutoff)
+    return xp
+
+
+def firwin2_signature(numtaps, freq, gain, *args, **kwds):
+    return array_namespace(freq, gain)
+
+
+def freqs_zpk_signature(z, p, k, worN, *args, **kwds):
+    return array_namespace(z, p, worN)
+
+freqz_zpk_signature = freqs_zpk_signature
+
+
+def freqs_signature(b, a, worN=200, *args, **kwds):
+    return array_namespace(b, a, worN)
+
+freqz_signature = freqs_signature
+
+
+def freqz_sos_signature(sos, worN=512, *args, **kwds):
+    return array_namespace(sos, worN)
+
+sosfreqz_signature = freqz_sos_signature
+
+
+def gausspulse_signature(t, *args, **kwds):
+    arr_t = None if isinstance(t, str) else t
+    return array_namespace(arr_t)
+
+
+def group_delay_signature(system, w=512, whole=False, fs=6.283185307179586):
+    return array_namespace(_skip_if_str_or_tuple(system), w)
+
+
+def hilbert_signature(x, N=None, axis=-1):
+    return array_namespace(x)
+
+hilbert2_signature = hilbert_signature
+
+
+def iirdesign_signature(wp, ws, *args, **kwds):
+    return array_namespace(wp, ws)
+
+
+def iirfilter_signature(N, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+def invres_signature(r, p, k, tol=0.001, rtype='avg'):
+    return array_namespace(r, p, k)
+
+invresz_signature = invres_signature
+
+
+############################### XXX: excluded, blacklisted on CuPy (mismatched API)
+def lfilter_zi_signature(b, a):
+    return array_namespace(b, a)
+
+def sosfilt_zi_signature(sos):
+    return array_namespace(sos)
+
+# needs to be blacklisted on CuPy (is not implemented)
+def remez_signature(numtaps, bands, desired, *, weight=None, **kwds):
+    return array_namespace(bands, desired, weight)
+#############################################
+
+def lfiltic_signature(b, a, y, x=None):
+    return array_namespace(b, a, y, x)
+
+
+def lombscargle_signature(
+    x, y, freqs, precenter=False, normalize=False, *,
+    weights=None, floating_mean=False
+):
+    return array_namespace(x, y, freqs, weights)
+
+
+def lp2bp_signature(b, a, *args, **kwds):
+    return array_namespace(b, a)
+
+lp2bs_signature = lp2bp_signature
+lp2hp_signature = lp2bp_signature
+lp2lp_signature = lp2bp_signature
+
+tf2zpk_signature = lp2bp_signature
+tf2sos_signature = lp2bp_signature
+
+normalize_signature = lp2bp_signature
+residue_signature = lp2bp_signature 
+residuez_signature = residue_signature
+
+
+def lp2bp_zpk_signature(z, p, k, *args, **kwds):
+    return array_namespace(z, p)
+
+lp2bs_zpk_signature = lp2bp_zpk_signature
+lp2hp_zpk_signature = lp2bs_zpk_signature
+lp2lp_zpk_signature = lp2bs_zpk_signature
+
+
+def zpk2sos_signature(z, p, k, *args, **kwds):
+    return array_namespace(z, p)
+
+zpk2ss_signature = zpk2sos_signature
+zpk2tf_signature = zpk2sos_signature
+
+
+def max_len_seq_signature(nbits, state=None, length=None, taps=None):
+    return array_namespace(state, taps)
+
+
+def medfilt_signature(volume, kernel_size=None):
+    return array_namespace(volume)
+
+
+def medfilt2d_signature(input, kernel_size=3):
+    return array_namespace(input)
+
+
+def minimum_phase_signature(h, *args, **kwds):
+    return array_namespace(h)
+
+
+def order_filter_signature(a, domain, rank):
+    return array_namespace(a, domain)
+
+
+def peak_prominences_signature(x, peaks, *args, **kwds):
+    return array_namespace(x, peaks)
+
+
+peak_widths_signature = peak_prominences_signature
+
+
+def place_poles_signature(A, B, poles, method='YT', rtol=0.001, maxiter=30):
+    return array_namespace(A, B, poles)
+
+
+def savgol_filter_signature(x, *args, **kwds):
+    return array_namespace(x)
+
+
+def sawtooth_signature(t, width=1):
+    return array_namespace(t)
+
+
+def sepfir2d_signature(input, hrow, hcol):
+    return array_namespace(input, hrow, hcol)
+
+
+def sos2tf_signature(sos):
+    return array_namespace(sos)
+
+
+sos2zpk_signature = sos2tf_signature
+
+
+def sosfilt_signature(sos, x, axis=-1, zi=None):
+    return array_namespace(sos, x, zi)
+
+
+def sosfiltfilt_signature(sos, x, *args, **kwds):
+    return array_namespace(sos, x)
+
+
+def spline_filter_signature(Iin, lmbda=5.0):
+    return array_namespace(Iin)
+
+
+def square_signature(t, duty=0.5):
+    return array_namespace(t)
+
+
+def ss2tf_signature(A, B, C, D, input=0):
+    return array_namespace(A, B, C, D)
+
+ss2zpk_signature = ss2tf_signature
+
+
+def sweep_poly_signature(t, poly, phi=0):
+    return array_namespace(t, _skip_if_poly1d(poly))
+
+
+def symiirorder1_signature(signal, c0, z1, precision=-1.0):
+    return array_namespace(signal)
+
+
+def symiirorder2_signature(input, r, omega, precision=-1.0):
+    return array_namespace(input, r, omega)
+
+
+def cspline1d_signature(signal, *args, **kwds):
+    return array_namespace(signal)
+
+qspline1d_signature = cspline1d_signature
+cspline2d_signature = cspline1d_signature
+qspline2d_signature = qspline1d_signature
+
+
+def cspline1d_eval_signature(cj, newx, *args, **kwds):
+    return array_namespace(cj, newx)
+
+qspline1d_eval_signature = cspline1d_eval_signature
+
+
+def tf2ss_signature(num, den):
+    return array_namespace(num, den)
+
+
+def unique_roots_signature(p, tol=0.001, rtype='min'):
+    return array_namespace(p)
+
+
+def upfirdn_signature(h, x, up=1, down=1, axis=-1, mode='constant', cval=0):
+    return array_namespace(h, x)
+
+
+def vectorstrength_signature(events, period):
+    return array_namespace(events, period)
+
+
+def wiener_signature(im, mysize=None, noise=None):
+    return array_namespace(im)
+
+
+def zoom_fft_signature(x, fn, m=None, *, fs=2, endpoint=False, axis=-1):
+    return array_namespace(x, fn)
+

venv/lib/python3.13/site-packages/scipy/signal/_lti_conversion.py

@@ -0,0 +1,534 @@
+"""
+ltisys -- a collection of functions to convert linear time invariant systems
+from one representation to another.
+"""
+
+import numpy as np
+from numpy import (r_, eye, atleast_2d, poly, dot,
+                   asarray, zeros, array, outer)
+from scipy import linalg
+
+from ._filter_design import tf2zpk, zpk2tf, normalize
+
+
+__all__ = ['tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk',
+           'cont2discrete']
+
+
+def tf2ss(num, den):
+    r"""Transfer function to state-space representation.
+
+    Parameters
+    ----------
+    num, den : array_like
+        Sequences representing the coefficients of the numerator and
+        denominator polynomials, in order of descending degree. The
+        denominator needs to be at least as long as the numerator.
+
+    Returns
+    -------
+    A, B, C, D : ndarray
+        State space representation of the system, in controller canonical
+        form.
+
+    Examples
+    --------
+    Convert the transfer function:
+
+    .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
+
+    >>> num = [1, 3, 3]
+    >>> den = [1, 2, 1]
+
+    to the state-space representation:
+
+    .. math::
+
+        \dot{\textbf{x}}(t) =
+        \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
+
+        \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
+
+    >>> from scipy.signal import tf2ss
+    >>> A, B, C, D = tf2ss(num, den)
+    >>> A
+    array([[-2., -1.],
+           [ 1.,  0.]])
+    >>> B
+    array([[ 1.],
+           [ 0.]])
+    >>> C
+    array([[ 1.,  2.]])
+    >>> D
+    array([[ 1.]])
+    """
+    # Controller canonical state-space representation.
+    #  if M+1 = len(num) and K+1 = len(den) then we must have M <= K
+    #  states are found by asserting that X(s) = U(s) / D(s)
+    #  then Y(s) = N(s) * X(s)
+    #
+    #   A, B, C, and D follow quite naturally.
+    #
+    num, den = normalize(num, den)   # Strips zeros, checks arrays
+    nn = len(num.shape)
+    if nn == 1:
+        num = asarray([num], num.dtype)
+    M = num.shape[1]
+    K = len(den)
+    if M > K:
+        msg = "Improper transfer function. `num` is longer than `den`."
+        raise ValueError(msg)
+    if M == 0 or K == 0:  # Null system
+        return (array([], float), array([], float), array([], float),
+                array([], float))
+
+    # pad numerator to have same number of columns has denominator
+    num = np.hstack((np.zeros((num.shape[0], K - M), dtype=num.dtype), num))
+
+    if num.shape[-1] > 0:
+        D = atleast_2d(num[:, 0])
+
+    else:
+        # We don't assign it an empty array because this system
+        # is not 'null'. It just doesn't have a non-zero D
+        # matrix. Thus, it should have a non-zero shape so that
+        # it can be operated on by functions like 'ss2tf'
+        D = array([[0]], float)
+
+    if K == 1:
+        D = D.reshape(num.shape)
+
+        return (zeros((1, 1)), zeros((1, D.shape[1])),
+                zeros((D.shape[0], 1)), D)
+
+    frow = -array([den[1:]])
+    A = r_[frow, eye(K - 2, K - 1)]
+    B = eye(K - 1, 1)
+    C = num[:, 1:] - outer(num[:, 0], den[1:])
+    D = D.reshape((C.shape[0], B.shape[1]))
+
+    return A, B, C, D
+
+
+def _none_to_empty_2d(arg):
+    if arg is None:
+        return zeros((0, 0))
+    else:
+        return arg
+
+
+def _atleast_2d_or_none(arg):
+    if arg is not None:
+        return atleast_2d(arg)
+
+
+def _shape_or_none(M):
+    if M is not None:
+        return M.shape
+    else:
+        return (None,) * 2
+
+
+def _choice_not_none(*args):
+    for arg in args:
+        if arg is not None:
+            return arg
+
+
+def _restore(M, shape):
+    if M.shape == (0, 0):
+        return zeros(shape)
+    else:
+        if M.shape != shape:
+            raise ValueError("The input arrays have incompatible shapes.")
+        return M
+
+
+def abcd_normalize(A=None, B=None, C=None, D=None):
+    """Check state-space matrices and ensure they are 2-D.
+
+    If enough information on the system is provided, that is, enough
+    properly-shaped arrays are passed to the function, the missing ones
+    are built from this information, ensuring the correct number of
+    rows and columns. Otherwise a ValueError is raised.
+
+    Parameters
+    ----------
+    A, B, C, D : array_like, optional
+        State-space matrices. All of them are None (missing) by default.
+        See `ss2tf` for format.
+
+    Returns
+    -------
+    A, B, C, D : array
+        Properly shaped state-space matrices.
+
+    Raises
+    ------
+    ValueError
+        If not enough information on the system was provided.
+
+    """
+    A, B, C, D = map(_atleast_2d_or_none, (A, B, C, D))
+
+    MA, NA = _shape_or_none(A)
+    MB, NB = _shape_or_none(B)
+    MC, NC = _shape_or_none(C)
+    MD, ND = _shape_or_none(D)
+
+    p = _choice_not_none(MA, MB, NC)
+    q = _choice_not_none(NB, ND)
+    r = _choice_not_none(MC, MD)
+    if p is None or q is None or r is None:
+        raise ValueError("Not enough information on the system.")
+
+    A, B, C, D = map(_none_to_empty_2d, (A, B, C, D))
+    A = _restore(A, (p, p))
+    B = _restore(B, (p, q))
+    C = _restore(C, (r, p))
+    D = _restore(D, (r, q))
+
+    return A, B, C, D
+
+
+def ss2tf(A, B, C, D, input=0):
+    r"""State-space to transfer function.
+
+    A, B, C, D defines a linear state-space system with `p` inputs,
+    `q` outputs, and `n` state variables.
+
+    Parameters
+    ----------
+    A : array_like
+        State (or system) matrix of shape ``(n, n)``
+    B : array_like
+        Input matrix of shape ``(n, p)``
+    C : array_like
+        Output matrix of shape ``(q, n)``
+    D : array_like
+        Feedthrough (or feedforward) matrix of shape ``(q, p)``
+    input : int, optional
+        For multiple-input systems, the index of the input to use.
+
+    Returns
+    -------
+    num : 2-D ndarray
+        Numerator(s) of the resulting transfer function(s). `num` has one row
+        for each of the system's outputs. Each row is a sequence representation
+        of the numerator polynomial.
+    den : 1-D ndarray
+        Denominator of the resulting transfer function(s). `den` is a sequence
+        representation of the denominator polynomial.
+
+    Examples
+    --------
+    Convert the state-space representation:
+
+    .. math::
+
+        \dot{\textbf{x}}(t) =
+        \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
+
+        \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
+
+    >>> A = [[-2, -1], [1, 0]]
+    >>> B = [[1], [0]]  # 2-D column vector
+    >>> C = [[1, 2]]    # 2-D row vector
+    >>> D = 1
+
+    to the transfer function:
+
+    .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
+
+    >>> from scipy.signal import ss2tf
+    >>> ss2tf(A, B, C, D)
+    (array([[1., 3., 3.]]), array([ 1.,  2.,  1.]))
+    """
+    # transfer function is C (sI - A)**(-1) B + D
+
+    # Check consistency and make them all rank-2 arrays
+    A, B, C, D = abcd_normalize(A, B, C, D)
+
+    nout, nin = D.shape
+    if input >= nin:
+        raise ValueError("System does not have the input specified.")
+
+    # make SIMO from possibly MIMO system.
+    B = B[:, input:input + 1]
+    D = D[:, input:input + 1]
+
+    try:
+        den = poly(A)
+    except ValueError:
+        den = 1
+
+    if (B.size == 0) and (C.size == 0):
+        num = np.ravel(D)
+        if (D.size == 0) and (A.size == 0):
+            den = []
+        return num, den
+
+    num_states = A.shape[0]
+    type_test = A[:, 0] + B[:, 0] + C[0, :] + D + 0.0
+    num = np.empty((nout, num_states + 1), type_test.dtype)
+    for k in range(nout):
+        Ck = atleast_2d(C[k, :])
+        num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den
+
+    return num, den
+
+
+def zpk2ss(z, p, k):
+    """Zero-pole-gain representation to state-space representation
+
+    Parameters
+    ----------
+    z, p : sequence
+        Zeros and poles.
+    k : float
+        System gain.
+
+    Returns
+    -------
+    A, B, C, D : ndarray
+        State space representation of the system, in controller canonical
+        form.
+
+    """
+    return tf2ss(*zpk2tf(z, p, k))
+
+
+def ss2zpk(A, B, C, D, input=0):
+    """State-space representation to zero-pole-gain representation.
+
+    A, B, C, D defines a linear state-space system with `p` inputs,
+    `q` outputs, and `n` state variables.
+
+    Parameters
+    ----------
+    A : array_like
+        State (or system) matrix of shape ``(n, n)``
+    B : array_like
+        Input matrix of shape ``(n, p)``
+    C : array_like
+        Output matrix of shape ``(q, n)``
+    D : array_like
+        Feedthrough (or feedforward) matrix of shape ``(q, p)``
+    input : int, optional
+        For multiple-input systems, the index of the input to use.
+
+    Returns
+    -------
+    z, p : sequence
+        Zeros and poles.
+    k : float
+        System gain.
+
+    """
+    return tf2zpk(*ss2tf(A, B, C, D, input=input))
+
+
+def cont2discrete(system, dt, method="zoh", alpha=None):
+    """
+    Transform a continuous to a discrete state-space system.
+
+    Parameters
+    ----------
+    system : a tuple describing the system or an instance of `lti`
+        The following gives the number of elements in the tuple and
+        the interpretation:
+
+            * 1: (instance of `lti`)
+            * 2: (num, den)
+            * 3: (zeros, poles, gain)
+            * 4: (A, B, C, D)
+
+    dt : float
+        The discretization time step.
+    method : str, optional
+        Which method to use:
+
+            * gbt: generalized bilinear transformation
+            * bilinear: Tustin's approximation ("gbt" with alpha=0.5)
+            * euler: Euler (or forward differencing) method ("gbt" with alpha=0)
+            * backward_diff: Backwards differencing ("gbt" with alpha=1.0)
+            * zoh: zero-order hold (default)
+            * foh: first-order hold (*versionadded: 1.3.0*)
+            * impulse: equivalent impulse response (*versionadded: 1.3.0*)
+
+    alpha : float within [0, 1], optional
+        The generalized bilinear transformation weighting parameter, which
+        should only be specified with method="gbt", and is ignored otherwise
+
+    Returns
+    -------
+    sysd : tuple containing the discrete system
+        Based on the input type, the output will be of the form
+
+        * (num, den, dt)   for transfer function input
+        * (zeros, poles, gain, dt)   for zeros-poles-gain input
+        * (A, B, C, D, dt) for state-space system input
+
+    Notes
+    -----
+    By default, the routine uses a Zero-Order Hold (zoh) method to perform
+    the transformation. Alternatively, a generalized bilinear transformation
+    may be used, which includes the common Tustin's bilinear approximation,
+    an Euler's method technique, or a backwards differencing technique.
+
+    The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear
+    approximation is based on [2]_ and [3]_, the First-Order Hold (foh) method
+    is based on [4]_.
+
+    References
+    ----------
+    .. [1] https://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models
+
+    .. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf
+
+    .. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized
+        bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754,
+        2009.
+        (https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf)
+
+    .. [4] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control
+        of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley,
+        pp. 204-206, 1998.
+
+    Examples
+    --------
+    We can transform a continuous state-space system to a discrete one:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.signal import cont2discrete, lti, dlti, dstep
+
+    Define a continuous state-space system.
+
+    >>> A = np.array([[0, 1],[-10., -3]])
+    >>> B = np.array([[0],[10.]])
+    >>> C = np.array([[1., 0]])
+    >>> D = np.array([[0.]])
+    >>> l_system = lti(A, B, C, D)
+    >>> t, x = l_system.step(T=np.linspace(0, 5, 100))
+    >>> fig, ax = plt.subplots()
+    >>> ax.plot(t, x, label='Continuous', linewidth=3)
+
+    Transform it to a discrete state-space system using several methods.
+
+    >>> dt = 0.1
+    >>> for method in ['zoh', 'bilinear', 'euler', 'backward_diff', 'foh', 'impulse']:
+    ...    d_system = cont2discrete((A, B, C, D), dt, method=method)
+    ...    s, x_d = dstep(d_system)
+    ...    ax.step(s, np.squeeze(x_d), label=method, where='post')
+    >>> ax.axis([t[0], t[-1], x[0], 1.4])
+    >>> ax.legend(loc='best')
+    >>> fig.tight_layout()
+    >>> plt.show()
+
+    """
+    if hasattr(system, 'to_discrete') and callable(system.to_discrete):
+        return system.to_discrete(dt=dt, method=method, alpha=alpha)
+
+    if len(system) == 2:
+        sysd = cont2discrete(tf2ss(system[0], system[1]), dt, method=method,
+                             alpha=alpha)
+        return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
+    elif len(system) == 3:
+        sysd = cont2discrete(zpk2ss(system[0], system[1], system[2]), dt,
+                             method=method, alpha=alpha)
+        return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
+    elif len(system) == 4:
+        a, b, c, d = system
+    else:
+        raise ValueError("First argument must either be a tuple of 2 (tf), "
+                         "3 (zpk), or 4 (ss) arrays.")
+
+    if method == 'gbt':
+        if alpha is None:
+            raise ValueError("Alpha parameter must be specified for the "
+                             "generalized bilinear transform (gbt) method")
+        elif alpha < 0 or alpha > 1:
+            raise ValueError("Alpha parameter must be within the interval "
+                             "[0,1] for the gbt method")
+
+    if method == 'gbt':
+        # This parameter is used repeatedly - compute once here
+        ima = np.eye(a.shape[0]) - alpha*dt*a
+        ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a)
+        bd = linalg.solve(ima, dt*b)
+
+        # Similarly solve for the output equation matrices
+        cd = linalg.solve(ima.transpose(), c.transpose())
+        cd = cd.transpose()
+        dd = d + alpha*np.dot(c, bd)
+
+    elif method == 'bilinear' or method == 'tustin':
+        return cont2discrete(system, dt, method="gbt", alpha=0.5)
+
+    elif method == 'euler' or method == 'forward_diff':
+        return cont2discrete(system, dt, method="gbt", alpha=0.0)
+
+    elif method == 'backward_diff':
+        return cont2discrete(system, dt, method="gbt", alpha=1.0)
+
+    elif method == 'zoh':
+        # Build an exponential matrix
+        em_upper = np.hstack((a, b))
+
+        # Need to stack zeros under the a and b matrices
+        em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])),
+                              np.zeros((b.shape[1], b.shape[1]))))
+
+        em = np.vstack((em_upper, em_lower))
+        ms = linalg.expm(dt * em)
+
+        # Dispose of the lower rows
+        ms = ms[:a.shape[0], :]
+
+        ad = ms[:, 0:a.shape[1]]
+        bd = ms[:, a.shape[1]:]
+
+        cd = c
+        dd = d
+
+    elif method == 'foh':
+        # Size parameters for convenience
+        n = a.shape[0]
+        m = b.shape[1]
+
+        # Build an exponential matrix similar to 'zoh' method
+        em_upper = linalg.block_diag(np.block([a, b]) * dt, np.eye(m))
+        em_lower = zeros((m, n + 2 * m))
+        em = np.block([[em_upper], [em_lower]])
+
+        ms = linalg.expm(em)
+
+        # Get the three blocks from upper rows
+        ms11 = ms[:n, 0:n]
+        ms12 = ms[:n, n:n + m]
+        ms13 = ms[:n, n + m:]
+
+        ad = ms11
+        bd = ms12 - ms13 + ms11 @ ms13
+        cd = c
+        dd = d + c @ ms13
+
+    elif method == 'impulse':
+        if not np.allclose(d, 0):
+            raise ValueError("Impulse method is only applicable "
+                             "to strictly proper systems")
+
+        ad = linalg.expm(a * dt)
+        bd = ad @ b * dt
+        cd = c
+        dd = c @ b * dt
+
+    else:
+        raise ValueError(f"Unknown transformation method '{method}'")
+
+    return ad, bd, cd, dd, dt

venv/lib/python3.13/site-packages/scipy/signal/_max_len_seq.py

@@ -0,0 +1,139 @@
+# Author: Eric Larson
+# 2014
+
+"""Tools for MLS generation"""
+
+import numpy as np
+
+from ._max_len_seq_inner import _max_len_seq_inner
+
+__all__ = ['max_len_seq']
+
+
+# These are definitions of linear shift register taps for use in max_len_seq()
+_mls_taps = {2: [1], 3: [2], 4: [3], 5: [3], 6: [5], 7: [6], 8: [7, 6, 1],
+             9: [5], 10: [7], 11: [9], 12: [11, 10, 4], 13: [12, 11, 8],
+             14: [13, 12, 2], 15: [14], 16: [15, 13, 4], 17: [14],
+             18: [11], 19: [18, 17, 14], 20: [17], 21: [19], 22: [21],
+             23: [18], 24: [23, 22, 17], 25: [22], 26: [25, 24, 20],
+             27: [26, 25, 22], 28: [25], 29: [27], 30: [29, 28, 7],
+             31: [28], 32: [31, 30, 10]}
+
+def max_len_seq(nbits, state=None, length=None, taps=None):
+    """
+    Maximum length sequence (MLS) generator.
+
+    Parameters
+    ----------
+    nbits : int
+        Number of bits to use. Length of the resulting sequence will
+        be ``(2**nbits) - 1``. Note that generating long sequences
+        (e.g., greater than ``nbits == 16``) can take a long time.
+    state : array_like, optional
+        If array, must be of length ``nbits``, and will be cast to binary
+        (bool) representation. If None, a seed of ones will be used,
+        producing a repeatable representation. If ``state`` is all
+        zeros, an error is raised as this is invalid. Default: None.
+    length : int, optional
+        Number of samples to compute. If None, the entire length
+        ``(2**nbits) - 1`` is computed.
+    taps : array_like, optional
+        Polynomial taps to use (e.g., ``[7, 6, 1]`` for an 8-bit sequence).
+        If None, taps will be automatically selected (for up to
+        ``nbits == 32``).
+
+    Returns
+    -------
+    seq : array
+        Resulting MLS sequence of 0's and 1's.
+    state : array
+        The final state of the shift register.
+
+    Notes
+    -----
+    The algorithm for MLS generation is generically described in:
+
+        https://en.wikipedia.org/wiki/Maximum_length_sequence
+
+    The default values for taps are specifically taken from the first
+    option listed for each value of ``nbits`` in:
+
+        https://web.archive.org/web/20181001062252/http://www.newwaveinstruments.com/resources/articles/m_sequence_linear_feedback_shift_register_lfsr.htm
+
+    .. versionadded:: 0.15.0
+
+    Examples
+    --------
+    MLS uses binary convention:
+
+    >>> from scipy.signal import max_len_seq
+    >>> max_len_seq(4)[0]
+    array([1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0], dtype=int8)
+
+    MLS has a white spectrum (except for DC):
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from numpy.fft import fft, ifft, fftshift, fftfreq
+    >>> seq = max_len_seq(6)[0]*2-1  # +1 and -1
+    >>> spec = fft(seq)
+    >>> N = len(seq)
+    >>> plt.plot(fftshift(fftfreq(N)), fftshift(np.abs(spec)), '.-')
+    >>> plt.margins(0.1, 0.1)
+    >>> plt.grid(True)
+    >>> plt.show()
+
+    Circular autocorrelation of MLS is an impulse:
+
+    >>> acorrcirc = ifft(spec * np.conj(spec)).real
+    >>> plt.figure()
+    >>> plt.plot(np.arange(-N/2+1, N/2+1), fftshift(acorrcirc), '.-')
+    >>> plt.margins(0.1, 0.1)
+    >>> plt.grid(True)
+    >>> plt.show()
+
+    Linear autocorrelation of MLS is approximately an impulse:
+
+    >>> acorr = np.correlate(seq, seq, 'full')
+    >>> plt.figure()
+    >>> plt.plot(np.arange(-N+1, N), acorr, '.-')
+    >>> plt.margins(0.1, 0.1)
+    >>> plt.grid(True)
+    >>> plt.show()
+
+    """
+    taps_dtype = np.int32 if np.intp().itemsize == 4 else np.int64
+    if taps is None:
+        if nbits not in _mls_taps:
+            known_taps = np.array(list(_mls_taps.keys()))
+            raise ValueError(f'nbits must be between {known_taps.min()} and '
+                             f'{known_taps.max()} if taps is None')
+        taps = np.array(_mls_taps[nbits], taps_dtype)
+    else:
+        taps = np.unique(np.array(taps, taps_dtype))[::-1]
+        if np.any(taps < 0) or np.any(taps > nbits) or taps.size < 1:
+            raise ValueError('taps must be non-empty with values between '
+                             'zero and nbits (inclusive)')
+        taps = np.array(taps)  # needed for Cython and Pythran
+    n_max = (2**nbits) - 1
+    if length is None:
+        length = n_max
+    else:
+        length = int(length)
+        if length < 0:
+            raise ValueError('length must be greater than or equal to 0')
+    # We use int8 instead of bool here because NumPy arrays of bools
+    # don't seem to work nicely with Cython
+    if state is None:
+        state = np.ones(nbits, dtype=np.int8, order='c')
+    else:
+        # makes a copy if need be, ensuring it's 0's and 1's
+        state = np.array(state, dtype=bool, order='c').astype(np.int8)
+    if state.ndim != 1 or state.size != nbits:
+        raise ValueError('state must be a 1-D array of size nbits')
+    if np.all(state == 0):
+        raise ValueError('state must not be all zeros')
+
+    seq = np.empty(length, dtype=np.int8, order='c')
+    state = _max_len_seq_inner(taps, state, nbits, length, seq)
+    return seq, state

venv/lib/python3.13/site-packages/scipy/signal/_polyutils.py

@@ -0,0 +1,172 @@
+"""Partial replacements for numpy polynomial routines, with Array API compatibility.
+
+This module contains both "old-style", np.poly1d, routines from the main numpy
+namespace, and "new-style", np.polynomial.polynomial, routines.
+
+To distinguish the two sets, the "new-style" routine names start with `npp_`
+"""
+import scipy._lib.array_api_extra as xpx
+from scipy._lib._array_api import xp_promote, xp_default_dtype
+
+
+def _sort_cmplx(arr, xp):
+    # xp.sort is undefined for complex dtypes. Here we only need some
+    # consistent way to sort a complex array, including equal magnitude elements.
+    arr = xp.asarray(arr)
+    if xp.isdtype(arr.dtype, 'complex floating'):
+        sorter = abs(arr) + xp.real(arr) + xp.imag(arr)**3
+    else:
+        sorter = arr
+
+    idxs = xp.argsort(sorter)
+    return arr[idxs]
+
+
+def polyroots(coef, *, xp):
+    """numpy.roots, best-effor replacement
+    """
+    if coef.shape[0] < 2:
+        return xp.asarray([], dtype=coef.dtype)
+
+    root_func = getattr(xp, 'roots', None)
+    if root_func:
+        # NB: cupy.roots is broken in CuPy 13.x, but CuPy is handled via delegation
+        # so we never hit this code path with xp being cupy
+        return root_func(coef)
+
+    # companion matrix
+    n = coef.shape[0]
+    a = xp.eye(n - 1, n - 1, k=-1, dtype=coef.dtype)
+    a[:, -1] = -xp.flip(coef[1:]) / coef[0]
+
+    # non-symmetric eigenvalue problem is not in the spec but is available on e.g. torch
+    if hasattr(xp.linalg, 'eigvals'):
+        return xp.linalg.eigvals(a)
+    else:
+        import numpy as np
+        return xp.asarray(np.linalg.eigvals(np.asarray(a)))
+
+
+# https://github.com/numpy/numpy/blob/v2.1.0/numpy/lib/_function_base_impl.py#L1874-L1925
+def _trim_zeros(filt, trim='fb'):
+    first = 0
+    trim = trim.upper()
+    if 'F' in trim:
+        for i in filt:
+            if i != 0.:
+                break
+            else:
+                first = first + 1
+    last = filt.shape[0]
+    if 'B' in trim:
+        for i in filt[::-1]:
+            if i != 0.:
+                break
+            else:
+                last = last - 1
+    return filt[first:last]
+
+
+# ### Old-style routines ###
+
+
+# https://github.com/numpy/numpy/blob/v2.2.0/numpy/lib/_polynomial_impl.py#L1232
+def _poly1d(c_or_r, *, xp):
+    """ Constructor of np.poly1d object from an array of coefficients (r=False)
+    """
+    c_or_r = xpx.atleast_nd(c_or_r, ndim=1, xp=xp)
+    if c_or_r.ndim > 1:
+        raise ValueError("Polynomial must be 1d only.")
+    c_or_r = _trim_zeros(c_or_r, trim='f')
+    if c_or_r.shape[0] == 0:
+        c_or_r = xp.asarray([0], dtype=c_or_r.dtype)
+    return c_or_r
+
+
+# https://github.com/numpy/numpy/blob/v2.2.0/numpy/lib/_polynomial_impl.py#L702-L779
+def polyval(p, x, *, xp):
+    """ Old-style polynomial, `np.polyval`
+    """
+    y = xp.zeros_like(x)
+
+    for pv in p:
+        y = y * x + pv
+    return y
+
+
+# https://github.com/numpy/numpy/blob/v2.2.0/numpy/lib/_polynomial_impl.py#L34-L157
+def poly(seq_of_zeros, *, xp):
+    # Only reproduce the 1D variant of np.poly
+    seq_of_zeros = xp.asarray(seq_of_zeros)
+    seq_of_zeros = xpx.atleast_nd(seq_of_zeros, ndim=1, xp=xp)
+
+    if seq_of_zeros.shape[0] == 0:
+        return 1.0
+
+    # prefer np.convolve etc, if available
+    convolve_func = getattr(xp, 'convolve', None)
+    if convolve_func is None:
+        from scipy.signal import convolve as convolve_func
+
+    dt = seq_of_zeros.dtype
+    a = xp.ones((1,), dtype=dt)
+    one = xp.ones_like(seq_of_zeros[0])
+    for zero in seq_of_zeros:
+        a = convolve_func(a, xp.stack((one, -zero)), mode='full')
+
+    if xp.isdtype(a.dtype, 'complex floating'):
+        # if complex roots are all complex conjugates, the roots are real.
+        roots = xp.asarray(seq_of_zeros, dtype=xp.complex128)
+        if xp.all(xp.sort(xp.imag(roots)) == xp.sort(xp.imag(xp.conj(roots)))):
+            a = xp.asarray(xp.real(a), copy=True)
+
+    return a
+
+
+# https://github.com/numpy/numpy/blob/v2.2.0/numpy/lib/_polynomial_impl.py#L912
+def polymul(a1, a2, *, xp):
+    a1, a2 = _poly1d(a1, xp=xp), _poly1d(a2, xp=xp)
+
+    # prefer np.convolve etc, if available
+    convolve_func = getattr(xp, 'convolve', None)
+    if convolve_func is None:
+        from scipy.signal import convolve as convolve_func
+
+    val = convolve_func(a1, a2)
+    return val
+
+
+# ### New-style routines ###
+
+
+# https://github.com/numpy/numpy/blob/v2.2.0/numpy/polynomial/polynomial.py#L663
+def npp_polyval(x, c, *, xp, tensor=True):
+    if xp.isdtype(c.dtype, 'integral'):
+        c = xp.astype(c, xp_default_dtype(xp))
+
+    c = xpx.atleast_nd(c, ndim=1, xp=xp)
+    if isinstance(x, tuple | list):
+        x = xp.asarray(x)
+    if tensor:
+        c = xp.reshape(c, (c.shape + (1,)*x.ndim))
+
+    c0, _ = xp_promote(c[-1, ...], x, broadcast=True, xp=xp)
+    for i in range(2, c.shape[0] + 1):
+        c0 = c[-i, ...] + c0*x
+    return c0
+
+
+# https://github.com/numpy/numpy/blob/v2.2.0/numpy/polynomial/polynomial.py#L758-L842
+def npp_polyvalfromroots(x, r, *, xp, tensor=True):
+    r = xpx.atleast_nd(r, ndim=1, xp=xp)
+    # if r.dtype.char in '?bBhHiIlLqQpP':
+    #    r = r.astype(np.double)
+
+    if isinstance(x, tuple | list):
+        x = xp.asarray(x)
+
+    if tensor:
+        r = xp.reshape(r, r.shape + (1,) * x.ndim)
+    elif x.ndim >= r.ndim:
+        raise ValueError("x.ndim must be < r.ndim when tensor == False")
+    return xp.prod(x - r, axis=0)

venv/lib/python3.13/site-packages/scipy/signal/_savitzky_golay.py

@@ -0,0 +1,357 @@
+import numpy as np
+from scipy.linalg import lstsq
+from scipy._lib._util import float_factorial
+from scipy.ndimage import convolve1d  # type: ignore[attr-defined]
+from ._arraytools import axis_slice
+
+
+def savgol_coeffs(window_length, polyorder, deriv=0, delta=1.0, pos=None,
+                  use="conv"):
+    """Compute the coefficients for a 1-D Savitzky-Golay FIR filter.
+
+    Parameters
+    ----------
+    window_length : int
+        The length of the filter window (i.e., the number of coefficients).
+    polyorder : int
+        The order of the polynomial used to fit the samples.
+        `polyorder` must be less than `window_length`.
+    deriv : int, optional
+        The order of the derivative to compute. This must be a
+        nonnegative integer. The default is 0, which means to filter
+        the data without differentiating.
+    delta : float, optional
+        The spacing of the samples to which the filter will be applied.
+        This is only used if deriv > 0.
+    pos : int or None, optional
+        If pos is not None, it specifies evaluation position within the
+        window. The default is the middle of the window.
+    use : str, optional
+        Either 'conv' or 'dot'. This argument chooses the order of the
+        coefficients. The default is 'conv', which means that the
+        coefficients are ordered to be used in a convolution. With
+        use='dot', the order is reversed, so the filter is applied by
+        dotting the coefficients with the data set.
+
+    Returns
+    -------
+    coeffs : 1-D ndarray
+        The filter coefficients.
+
+    See Also
+    --------
+    savgol_filter
+
+    Notes
+    -----
+    .. versionadded:: 0.14.0
+
+    References
+    ----------
+    A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of Data by
+    Simplified Least Squares Procedures. Analytical Chemistry, 1964, 36 (8),
+    pp 1627-1639.
+    Jianwen Luo, Kui Ying, and Jing Bai. 2005. Savitzky-Golay smoothing and
+    differentiation filter for even number data. Signal Process.
+    85, 7 (July 2005), 1429-1434.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import savgol_coeffs
+    >>> savgol_coeffs(5, 2)
+    array([-0.08571429,  0.34285714,  0.48571429,  0.34285714, -0.08571429])
+    >>> savgol_coeffs(5, 2, deriv=1)
+    array([ 2.00000000e-01,  1.00000000e-01,  2.07548111e-16, -1.00000000e-01,
+           -2.00000000e-01])
+
+    Note that use='dot' simply reverses the coefficients.
+
+    >>> savgol_coeffs(5, 2, pos=3)
+    array([ 0.25714286,  0.37142857,  0.34285714,  0.17142857, -0.14285714])
+    >>> savgol_coeffs(5, 2, pos=3, use='dot')
+    array([-0.14285714,  0.17142857,  0.34285714,  0.37142857,  0.25714286])
+    >>> savgol_coeffs(4, 2, pos=3, deriv=1, use='dot')
+    array([0.45,  -0.85,  -0.65,  1.05])
+
+    `x` contains data from the parabola x = t**2, sampled at
+    t = -1, 0, 1, 2, 3.  `c` holds the coefficients that will compute the
+    derivative at the last position.  When dotted with `x` the result should
+    be 6.
+
+    >>> x = np.array([1, 0, 1, 4, 9])
+    >>> c = savgol_coeffs(5, 2, pos=4, deriv=1, use='dot')
+    >>> c.dot(x)
+    6.0
+    """
+
+    # An alternative method for finding the coefficients when deriv=0 is
+    #    t = np.arange(window_length)
+    #    unit = (t == pos).astype(int)
+    #    coeffs = np.polyval(np.polyfit(t, unit, polyorder), t)
+    # The method implemented here is faster.
+
+    # To recreate the table of sample coefficients shown in the chapter on
+    # the Savitzy-Golay filter in the Numerical Recipes book, use
+    #    window_length = nL + nR + 1
+    #    pos = nL + 1
+    #    c = savgol_coeffs(window_length, M, pos=pos, use='dot')
+
+    if polyorder >= window_length:
+        raise ValueError("polyorder must be less than window_length.")
+
+    halflen, rem = divmod(window_length, 2)
+
+    if pos is None:
+        if rem == 0:
+            pos = halflen - 0.5
+        else:
+            pos = halflen
+
+    if not (0 <= pos < window_length):
+        raise ValueError("pos must be nonnegative and less than "
+                         "window_length.")
+
+    if use not in ['conv', 'dot']:
+        raise ValueError("`use` must be 'conv' or 'dot'")
+
+    if deriv > polyorder:
+        coeffs = np.zeros(window_length)
+        return coeffs
+
+    # Form the design matrix A. The columns of A are powers of the integers
+    # from -pos to window_length - pos - 1. The powers (i.e., rows) range
+    # from 0 to polyorder. (That is, A is a vandermonde matrix, but not
+    # necessarily square.)
+    x = np.arange(-pos, window_length - pos, dtype=float)
+
+    if use == "conv":
+        # Reverse so that result can be used in a convolution.
+        x = x[::-1]
+
+    order = np.arange(polyorder + 1).reshape(-1, 1)
+    A = x ** order
+
+    # y determines which order derivative is returned.
+    y = np.zeros(polyorder + 1)
+    # The coefficient assigned to y[deriv] scales the result to take into
+    # account the order of the derivative and the sample spacing.
+    y[deriv] = float_factorial(deriv) / (delta ** deriv)
+
+    # Find the least-squares solution of A*c = y
+    coeffs, _, _, _ = lstsq(A, y)
+
+    return coeffs
+
+
+def _polyder(p, m):
+    """Differentiate polynomials represented with coefficients.
+
+    p must be a 1-D or 2-D array.  In the 2-D case, each column gives
+    the coefficients of a polynomial; the first row holds the coefficients
+    associated with the highest power. m must be a nonnegative integer.
+    (numpy.polyder doesn't handle the 2-D case.)
+    """
+
+    if m == 0:
+        result = p
+    else:
+        n = len(p)
+        if n <= m:
+            result = np.zeros_like(p[:1, ...])
+        else:
+            dp = p[:-m].copy()
+            for k in range(m):
+                rng = np.arange(n - k - 1, m - k - 1, -1)
+                dp *= rng.reshape((n - m,) + (1,) * (p.ndim - 1))
+            result = dp
+    return result
+
+
+def _fit_edge(x, window_start, window_stop, interp_start, interp_stop,
+              axis, polyorder, deriv, delta, y):
+    """
+    Given an N-d array `x` and the specification of a slice of `x` from
+    `window_start` to `window_stop` along `axis`, create an interpolating
+    polynomial of each 1-D slice, and evaluate that polynomial in the slice
+    from `interp_start` to `interp_stop`. Put the result into the
+    corresponding slice of `y`.
+    """
+
+    # Get the edge into a (window_length, -1) array.
+    x_edge = axis_slice(x, start=window_start, stop=window_stop, axis=axis)
+    if axis == 0 or axis == -x.ndim:
+        xx_edge = x_edge
+        swapped = False
+    else:
+        xx_edge = x_edge.swapaxes(axis, 0)
+        swapped = True
+    xx_edge = xx_edge.reshape(xx_edge.shape[0], -1)
+
+    # Fit the edges.  poly_coeffs has shape (polyorder + 1, -1),
+    # where '-1' is the same as in xx_edge.
+    poly_coeffs = np.polyfit(np.arange(0, window_stop - window_start),
+                             xx_edge, polyorder)
+
+    if deriv > 0:
+        poly_coeffs = _polyder(poly_coeffs, deriv)
+
+    # Compute the interpolated values for the edge.
+    i = np.arange(interp_start - window_start, interp_stop - window_start)
+    values = np.polyval(poly_coeffs, i.reshape(-1, 1)) / (delta ** deriv)
+
+    # Now put the values into the appropriate slice of y.
+    # First reshape values to match y.
+    shp = list(y.shape)
+    shp[0], shp[axis] = shp[axis], shp[0]
+    values = values.reshape(interp_stop - interp_start, *shp[1:])
+    if swapped:
+        values = values.swapaxes(0, axis)
+    # Get a view of the data to be replaced by values.
+    y_edge = axis_slice(y, start=interp_start, stop=interp_stop, axis=axis)
+    y_edge[...] = values
+
+
+def _fit_edges_polyfit(x, window_length, polyorder, deriv, delta, axis, y):
+    """
+    Use polynomial interpolation of x at the low and high ends of the axis
+    to fill in the halflen values in y.
+
+    This function just calls _fit_edge twice, once for each end of the axis.
+    """
+    halflen = window_length // 2
+    _fit_edge(x, 0, window_length, 0, halflen, axis,
+              polyorder, deriv, delta, y)
+    n = x.shape[axis]
+    _fit_edge(x, n - window_length, n, n - halflen, n, axis,
+              polyorder, deriv, delta, y)
+
+
+def savgol_filter(x, window_length, polyorder, deriv=0, delta=1.0,
+                  axis=-1, mode='interp', cval=0.0):
+    """ Apply a Savitzky-Golay filter to an array.
+
+    This is a 1-D filter. If `x`  has dimension greater than 1, `axis`
+    determines the axis along which the filter is applied.
+
+    Parameters
+    ----------
+    x : array_like
+        The data to be filtered. If `x` is not a single or double precision
+        floating point array, it will be converted to type ``numpy.float64``
+        before filtering.
+    window_length : int
+        The length of the filter window (i.e., the number of coefficients).
+        If `mode` is 'interp', `window_length` must be less than or equal
+        to the size of `x`.
+    polyorder : int
+        The order of the polynomial used to fit the samples.
+        `polyorder` must be less than `window_length`.
+    deriv : int, optional
+        The order of the derivative to compute. This must be a
+        nonnegative integer. The default is 0, which means to filter
+        the data without differentiating.
+    delta : float, optional
+        The spacing of the samples to which the filter will be applied.
+        This is only used if deriv > 0. Default is 1.0.
+    axis : int, optional
+        The axis of the array `x` along which the filter is to be applied.
+        Default is -1.
+    mode : str, optional
+        Must be 'mirror', 'constant', 'nearest', 'wrap' or 'interp'. This
+        determines the type of extension to use for the padded signal to
+        which the filter is applied.  When `mode` is 'constant', the padding
+        value is given by `cval`.  See the Notes for more details on 'mirror',
+        'constant', 'wrap', and 'nearest'.
+        When the 'interp' mode is selected (the default), no extension
+        is used.  Instead, a degree `polyorder` polynomial is fit to the
+        last `window_length` values of the edges, and this polynomial is
+        used to evaluate the last `window_length // 2` output values.
+    cval : scalar, optional
+        Value to fill past the edges of the input if `mode` is 'constant'.
+        Default is 0.0.
+
+    Returns
+    -------
+    y : ndarray, same shape as `x`
+        The filtered data.
+
+    See Also
+    --------
+    savgol_coeffs
+
+    Notes
+    -----
+    Details on the `mode` options:
+
+        'mirror':
+            Repeats the values at the edges in reverse order. The value
+            closest to the edge is not included.
+        'nearest':
+            The extension contains the nearest input value.
+        'constant':
+            The extension contains the value given by the `cval` argument.
+        'wrap':
+            The extension contains the values from the other end of the array.
+
+    For example, if the input is [1, 2, 3, 4, 5, 6, 7, 8], and
+    `window_length` is 7, the following shows the extended data for
+    the various `mode` options (assuming `cval` is 0)::
+
+        mode       |   Ext   |         Input          |   Ext
+        -----------+---------+------------------------+---------
+        'mirror'   | 4  3  2 | 1  2  3  4  5  6  7  8 | 7  6  5
+        'nearest'  | 1  1  1 | 1  2  3  4  5  6  7  8 | 8  8  8
+        'constant' | 0  0  0 | 1  2  3  4  5  6  7  8 | 0  0  0
+        'wrap'     | 6  7  8 | 1  2  3  4  5  6  7  8 | 1  2  3
+
+    .. versionadded:: 0.14.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import savgol_filter
+    >>> np.set_printoptions(precision=2)  # For compact display.
+    >>> x = np.array([2, 2, 5, 2, 1, 0, 1, 4, 9])
+
+    Filter with a window length of 5 and a degree 2 polynomial.  Use
+    the defaults for all other parameters.
+
+    >>> savgol_filter(x, 5, 2)
+    array([1.66, 3.17, 3.54, 2.86, 0.66, 0.17, 1.  , 4.  , 9.  ])
+
+    Note that the last five values in x are samples of a parabola, so
+    when mode='interp' (the default) is used with polyorder=2, the last
+    three values are unchanged. Compare that to, for example,
+    `mode='nearest'`:
+
+    >>> savgol_filter(x, 5, 2, mode='nearest')
+    array([1.74, 3.03, 3.54, 2.86, 0.66, 0.17, 1.  , 4.6 , 7.97])
+
+    """
+    if mode not in ["mirror", "constant", "nearest", "interp", "wrap"]:
+        raise ValueError("mode must be 'mirror', 'constant', 'nearest' "
+                         "'wrap' or 'interp'.")
+
+    x = np.asarray(x)
+    # Ensure that x is either single or double precision floating point.
+    if x.dtype != np.float64 and x.dtype != np.float32:
+        x = x.astype(np.float64)
+
+    coeffs = savgol_coeffs(window_length, polyorder, deriv=deriv, delta=delta)
+
+    if mode == "interp":
+        if window_length > x.shape[axis]:
+            raise ValueError("If mode is 'interp', window_length must be less "
+                             "than or equal to the size of x.")
+
+        # Do not pad. Instead, for the elements within `window_length // 2`
+        # of the ends of the sequence, use the polynomial that is fitted to
+        # the last `window_length` elements.
+        y = convolve1d(x, coeffs, axis=axis, mode="constant")
+        _fit_edges_polyfit(x, window_length, polyorder, deriv, delta, axis, y)
+    else:
+        # Any mode other than 'interp' is passed on to ndimage.convolve1d.
+        y = convolve1d(x, coeffs, axis=axis, mode=mode, cval=cval)
+
+    return y

venv/lib/python3.13/site-packages/scipy/signal/_signal_api.py

@@ -0,0 +1,30 @@
+"""This is the 'bare' scipy.signal API.
+
+This --- private! --- module only collects implementations of public  API
+for _support_alternative_backends.
+The latter --- also private! --- module adds delegation to CuPy etc and
+re-exports decorated names to __init__.py
+"""
+
+from . import _sigtools, windows         # noqa: F401
+from ._waveforms import *        # noqa: F403
+from ._max_len_seq import max_len_seq       # noqa: F401
+from ._upfirdn import upfirdn         # noqa: F401
+
+from ._spline import sepfir2d          # noqa: F401
+
+from ._spline_filters import *         # noqa: F403
+from ._filter_design import *         # noqa: F403
+from ._fir_filter_design import *         # noqa: F403
+from ._ltisys import *         # noqa: F403
+from ._lti_conversion import *         # noqa: F403
+from ._signaltools import *         # noqa: F403
+from ._savitzky_golay import savgol_coeffs, savgol_filter  # noqa: F401
+from ._spectral_py import *         # noqa: F403
+from ._short_time_fft import *         # noqa: F403
+from ._peak_finding import *         # noqa: F403
+from ._czt import *         # noqa: F403
+from .windows import get_window  # keep this one in signal namespace  # noqa: F401
+
+
+__all__ = [s for s in dir() if not s.startswith('_')]

venv/lib/python3.13/site-packages/scipy/signal/_spline.pyi

@@ -0,0 +1,34 @@
+
+import numpy as np
+from numpy.typing import NDArray
+
+FloatingArray = NDArray[np.float32] | NDArray[np.float64]
+ComplexArray = NDArray[np.complex64] | NDArray[np.complex128]
+FloatingComplexArray = FloatingArray | ComplexArray
+
+
+def symiirorder1_ic(signal: FloatingComplexArray,
+                    c0: float,
+                    z1: float,
+                    precision: float) -> FloatingComplexArray:
+    ...
+
+
+def symiirorder2_ic_fwd(signal: FloatingArray,
+                        r: float,
+                        omega: float,
+                        precision: float) -> FloatingArray:
+    ...
+
+
+def symiirorder2_ic_bwd(signal: FloatingArray,
+                        r: float,
+                        omega: float,
+                        precision: float) -> FloatingArray:
+    ...
+
+
+def sepfir2d(input: FloatingComplexArray,
+             hrow: FloatingComplexArray,
+             hcol: FloatingComplexArray) -> FloatingComplexArray:
+    ...

venv/lib/python3.13/site-packages/scipy/signal/_spline_filters.py

@@ -0,0 +1,848 @@
+import math
+
+from numpy import (zeros_like, array, tan, arange, floor,
+                   r_, atleast_1d, greater, cos, add, sin,
+                   moveaxis, abs, complex64, float32)
+import numpy as np
+
+from scipy._lib._array_api import array_namespace, xp_promote
+
+from scipy._lib._util import normalize_axis_index
+
+# From splinemodule.c
+from ._spline import sepfir2d, symiirorder1_ic, symiirorder2_ic_fwd, symiirorder2_ic_bwd
+from ._signaltools import lfilter, sosfilt, lfiltic
+from ._arraytools import axis_slice, axis_reverse
+
+from scipy.interpolate import BSpline
+
+
+__all__ = ['spline_filter', 'gauss_spline',
+           'cspline1d', 'qspline1d', 'qspline2d', 'cspline2d',
+           'cspline1d_eval', 'qspline1d_eval', 'symiirorder1', 'symiirorder2']
+
+
+def spline_filter(Iin, lmbda=5.0):
+    """Smoothing spline (cubic) filtering of a rank-2 array.
+
+    Filter an input data set, `Iin`, using a (cubic) smoothing spline of
+    fall-off `lmbda`.
+
+    Parameters
+    ----------
+    Iin : array_like
+        input data set
+    lmbda : float, optional
+        spline smoothing fall-off value, default is `5.0`.
+
+    Returns
+    -------
+    res : ndarray
+        filtered input data
+
+    Examples
+    --------
+    We can filter an multi dimensional signal (ex: 2D image) using cubic
+    B-spline filter:
+
+    >>> import numpy as np
+    >>> from scipy.signal import spline_filter
+    >>> import matplotlib.pyplot as plt
+    >>> orig_img = np.eye(20)  # create an image
+    >>> orig_img[10, :] = 1.0
+    >>> sp_filter = spline_filter(orig_img, lmbda=0.1)
+    >>> f, ax = plt.subplots(1, 2, sharex=True)
+    >>> for ind, data in enumerate([[orig_img, "original image"],
+    ...                             [sp_filter, "spline filter"]]):
+    ...     ax[ind].imshow(data[0], cmap='gray_r')
+    ...     ax[ind].set_title(data[1])
+    >>> plt.tight_layout()
+    >>> plt.show()
+
+    """
+    xp = array_namespace(Iin)
+    Iin = np.asarray(Iin)
+
+    if Iin.dtype not in [np.float32, np.float64, np.complex64, np.complex128]:
+        raise TypeError(f"Invalid data type for Iin: {Iin.dtype = }")
+
+    # XXX: note that complex-valued computations are done in single precision
+    # this is historic, and the root reason is unclear,
+    # see https://github.com/scipy/scipy/issues/9209
+    # Attempting to work in complex double precision leads to symiirorder1
+    # failing to converge for the boundary conditions.
+    intype = Iin.dtype
+    hcol = array([1.0, 4.0, 1.0], np.float32) / 6.0
+    if intype == np.complex128:
+        Iin = Iin.astype(np.complex64)
+
+    ck = cspline2d(Iin, lmbda)
+    out = sepfir2d(ck, hcol, hcol)
+    out = out.astype(intype)
+    return xp.asarray(out)
+
+
+_splinefunc_cache = {}
+
+
+def gauss_spline(x, n):
+    r"""Gaussian approximation to B-spline basis function of order n.
+
+    Parameters
+    ----------
+    x : array_like
+        a knot vector
+    n : int
+        The order of the spline. Must be non-negative, i.e., n >= 0
+
+    Returns
+    -------
+    res : ndarray
+        B-spline basis function values approximated by a zero-mean Gaussian
+        function.
+
+    Notes
+    -----
+    The B-spline basis function can be approximated well by a zero-mean
+    Gaussian function with standard-deviation equal to :math:`\sigma=(n+1)/12`
+    for large `n` :
+
+    .. math::  \frac{1}{\sqrt {2\pi\sigma^2}}exp(-\frac{x^2}{2\sigma})
+
+    References
+    ----------
+    .. [1] Bouma H., Vilanova A., Bescos J.O., ter Haar Romeny B.M., Gerritsen
+       F.A. (2007) Fast and Accurate Gaussian Derivatives Based on B-Splines. In:
+       Sgallari F., Murli A., Paragios N. (eds) Scale Space and Variational
+       Methods in Computer Vision. SSVM 2007. Lecture Notes in Computer
+       Science, vol 4485. Springer, Berlin, Heidelberg
+    .. [2] http://folk.uio.no/inf3330/scripting/doc/python/SciPy/tutorial/old/node24.html
+
+    Examples
+    --------
+    We can calculate B-Spline basis functions approximated by a gaussian
+    distribution:
+
+    >>> import numpy as np
+    >>> from scipy.signal import gauss_spline
+    >>> knots = np.array([-1.0, 0.0, -1.0])
+    >>> gauss_spline(knots, 3)
+    array([0.15418033, 0.6909883, 0.15418033])  # may vary
+
+    """
+    xp = array_namespace(x)
+    x = xp.asarray(x)
+    signsq = (n + 1) / 12.0
+    return 1 / math.sqrt(2 * math.pi * signsq) * xp.exp(-x ** 2 / 2 / signsq)
+
+
+def _cubic(x):
+    xp = array_namespace(x)
+
+    x = np.asarray(x, dtype=float)
+    b = BSpline.basis_element([-2, -1, 0, 1, 2], extrapolate=False)
+    out = b(x)
+    out[(x < -2) | (x > 2)] = 0
+    return xp.asarray(out)
+
+
+def _quadratic(x):
+    xp = array_namespace(x)
+
+    x = abs(np.asarray(x, dtype=float))
+    b = BSpline.basis_element([-1.5, -0.5, 0.5, 1.5], extrapolate=False)
+    out = b(x)
+    out[(x < -1.5) | (x > 1.5)] = 0
+    return xp.asarray(out)
+
+
+def _coeff_smooth(lam):
+    xi = 1 - 96 * lam + 24 * lam * math.sqrt(3 + 144 * lam)
+    omeg = math.atan2(math.sqrt(144 * lam - 1), math.sqrt(xi))
+    rho = (24 * lam - 1 - math.sqrt(xi)) / (24 * lam)
+    rho = rho * math.sqrt((48 * lam + 24 * lam * math.sqrt(3 + 144 * lam)) / xi)
+    return rho, omeg
+
+
+def _hc(k, cs, rho, omega):
+    return (cs / sin(omega) * (rho ** k) * sin(omega * (k + 1)) *
+            greater(k, -1))
+
+
+def _hs(k, cs, rho, omega):
+    c0 = (cs * cs * (1 + rho * rho) / (1 - rho * rho) /
+          (1 - 2 * rho * rho * cos(2 * omega) + rho ** 4))
+    gamma = (1 - rho * rho) / (1 + rho * rho) / tan(omega)
+    ak = abs(k)
+    return c0 * rho ** ak * (cos(omega * ak) + gamma * sin(omega * ak))
+
+
+def _cubic_smooth_coeff(signal, lamb):
+    signal = np.asarray(signal)
+
+    rho, omega = _coeff_smooth(lamb)
+    cs = 1 - 2 * rho * cos(omega) + rho * rho
+    K = len(signal)
+    k = arange(K)
+
+    zi_2 = (_hc(0, cs, rho, omega) * signal[0] +
+            add.reduce(_hc(k + 1, cs, rho, omega) * signal))
+    zi_1 = (_hc(0, cs, rho, omega) * signal[0] +
+            _hc(1, cs, rho, omega) * signal[1] +
+            add.reduce(_hc(k + 2, cs, rho, omega) * signal))
+
+    # Forward filter:
+    # for n in range(2, K):
+    #     yp[n] = (cs * signal[n] + 2 * rho * cos(omega) * yp[n - 1] -
+    #              rho * rho * yp[n - 2])
+    zi = lfiltic(cs, r_[1, -2 * rho * cos(omega), rho * rho], r_[zi_1, zi_2])
+    zi = zi.reshape(1, -1)
+
+    sos = r_[cs, 0, 0, 1, -2 * rho * cos(omega), rho * rho]
+    sos = sos.reshape(1, -1)
+
+    yp, _ = sosfilt(sos, signal[2:], zi=zi)
+    yp = r_[zi_2, zi_1, yp]
+
+    # Reverse filter:
+    # for n in range(K - 3, -1, -1):
+    #     y[n] = (cs * yp[n] + 2 * rho * cos(omega) * y[n + 1] -
+    #             rho * rho * y[n + 2])
+
+    zi_2 = add.reduce((_hs(k, cs, rho, omega) +
+                       _hs(k + 1, cs, rho, omega)) * signal[::-1])
+    zi_1 = add.reduce((_hs(k - 1, cs, rho, omega) +
+                       _hs(k + 2, cs, rho, omega)) * signal[::-1])
+
+    zi = lfiltic(cs, r_[1, -2 * rho * cos(omega), rho * rho], r_[zi_1, zi_2])
+    zi = zi.reshape(1, -1)
+    y, _ = sosfilt(sos, yp[-3::-1], zi=zi)
+    y = r_[y[::-1], zi_1, zi_2]
+    return y
+
+
+def _cubic_coeff(signal):
+    signal = np.asarray(signal)
+
+    zi = -2 + math.sqrt(3)
+    K = len(signal)
+    powers = zi ** np.arange(K)
+
+    if K == 1:
+        yplus = signal[0] + zi * add.reduce(powers * signal)
+        output = zi / (zi - 1) * yplus
+        return atleast_1d(output)
+
+    # Forward filter:
+    # yplus[0] = signal[0] + zi * add.reduce(powers * signal)
+    # for k in range(1, K):
+    #     yplus[k] = signal[k] + zi * yplus[k - 1]
+
+    state = lfiltic(1, np.r_[1, -zi], np.atleast_1d(add.reduce(powers * signal)))
+
+    b = np.ones(1)
+    a = np.r_[1, -zi]
+    yplus, _ = lfilter(b, a, signal, zi=state)
+
+    # Reverse filter:
+    # output[K - 1] = zi / (zi - 1) * yplus[K - 1]
+    # for k in range(K - 2, -1, -1):
+    #     output[k] = zi * (output[k + 1] - yplus[k])
+    out_last = zi / (zi - 1) * yplus[K - 1]
+    state = lfiltic(-zi, r_[1, -zi], np.atleast_1d(out_last))
+
+    b = np.asarray([-zi])
+    output, _ = lfilter(b, a, yplus[-2::-1], zi=state)
+    output = np.r_[output[::-1], out_last]
+    return output * 6.0
+
+
+def _quadratic_coeff(signal):
+    signal = np.asarray(signal)
+
+    zi = -3 + 2 * math.sqrt(2.0)
+    K = len(signal)
+    powers = zi ** np.arange(K)
+
+    if K == 1:
+        yplus = signal[0] + zi * np.add.reduce(powers * signal)
+        output = zi / (zi - 1) * yplus
+        return atleast_1d(output)
+
+    # Forward filter:
+    # yplus[0] = signal[0] + zi * add.reduce(powers * signal)
+    # for k in range(1, K):
+    #     yplus[k] = signal[k] + zi * yplus[k - 1]
+
+    state = lfiltic(1, np.r_[1, -zi], np.atleast_1d(np.add.reduce(powers * signal)))
+
+    b = np.ones(1)
+    a = np.r_[1, -zi]
+    yplus, _ = lfilter(b, a, signal, zi=state)
+
+    # Reverse filter:
+    # output[K - 1] = zi / (zi - 1) * yplus[K - 1]
+    # for k in range(K - 2, -1, -1):
+    #     output[k] = zi * (output[k + 1] - yplus[k])
+    out_last = zi / (zi - 1) * yplus[K - 1]
+    state = lfiltic(-zi, r_[1, -zi], np.atleast_1d(out_last))
+
+    b = np.asarray([-zi])
+    output, _ = lfilter(b, a, yplus[-2::-1], zi=state)
+    output = np.r_[output[::-1], out_last]
+    return output * 8.0
+
+
+def compute_root_from_lambda(lamb):
+    tmp = math.sqrt(3 + 144 * lamb)
+    xi = 1 - 96 * lamb + 24 * lamb * tmp
+    omega = math.atan(math.sqrt((144 * lamb - 1.0) / xi))
+    tmp2 = math.sqrt(xi)
+    r = ((24 * lamb - 1 - tmp2) / (24 * lamb) *
+         math.sqrt(48*lamb + 24 * lamb * tmp) / tmp2)
+    return r, omega
+
+
+def cspline1d(signal, lamb=0.0):
+    """
+    Compute cubic spline coefficients for rank-1 array.
+
+    Find the cubic spline coefficients for a 1-D signal assuming
+    mirror-symmetric boundary conditions. To obtain the signal back from the
+    spline representation mirror-symmetric-convolve these coefficients with a
+    length 3 FIR window [1.0, 4.0, 1.0]/ 6.0 .
+
+    Parameters
+    ----------
+    signal : ndarray
+        A rank-1 array representing samples of a signal.
+    lamb : float, optional
+        Smoothing coefficient, default is 0.0.
+
+    Returns
+    -------
+    c : ndarray
+        Cubic spline coefficients.
+
+    See Also
+    --------
+    cspline1d_eval : Evaluate a cubic spline at the new set of points.
+
+    Examples
+    --------
+    We can filter a signal to reduce and smooth out high-frequency noise with
+    a cubic spline:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.signal import cspline1d, cspline1d_eval
+    >>> rng = np.random.default_rng()
+    >>> sig = np.repeat([0., 1., 0.], 100)
+    >>> sig += rng.standard_normal(len(sig))*0.05  # add noise
+    >>> time = np.linspace(0, len(sig))
+    >>> filtered = cspline1d_eval(cspline1d(sig), time)
+    >>> plt.plot(sig, label="signal")
+    >>> plt.plot(time, filtered, label="filtered")
+    >>> plt.legend()
+    >>> plt.show()
+
+    """
+    xp = array_namespace(signal)
+
+    if lamb != 0.0:
+        ret = _cubic_smooth_coeff(signal, lamb)
+    else:
+        ret = _cubic_coeff(signal)
+    return xp.asarray(ret)
+
+
+def qspline1d(signal, lamb=0.0):
+    """Compute quadratic spline coefficients for rank-1 array.
+
+    Parameters
+    ----------
+    signal : ndarray
+        A rank-1 array representing samples of a signal.
+    lamb : float, optional
+        Smoothing coefficient (must be zero for now).
+
+    Returns
+    -------
+    c : ndarray
+        Quadratic spline coefficients.
+
+    See Also
+    --------
+    qspline1d_eval : Evaluate a quadratic spline at the new set of points.
+
+    Notes
+    -----
+    Find the quadratic spline coefficients for a 1-D signal assuming
+    mirror-symmetric boundary conditions. To obtain the signal back from the
+    spline representation mirror-symmetric-convolve these coefficients with a
+    length 3 FIR window [1.0, 6.0, 1.0]/ 8.0 .
+
+    Examples
+    --------
+    We can filter a signal to reduce and smooth out high-frequency noise with
+    a quadratic spline:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.signal import qspline1d, qspline1d_eval
+    >>> rng = np.random.default_rng()
+    >>> sig = np.repeat([0., 1., 0.], 100)
+    >>> sig += rng.standard_normal(len(sig))*0.05  # add noise
+    >>> time = np.linspace(0, len(sig))
+    >>> filtered = qspline1d_eval(qspline1d(sig), time)
+    >>> plt.plot(sig, label="signal")
+    >>> plt.plot(time, filtered, label="filtered")
+    >>> plt.legend()
+    >>> plt.show()
+
+    """
+    xp = array_namespace(signal)
+
+    if lamb != 0.0:
+        raise ValueError("Smoothing quadratic splines not supported yet.")
+    else:
+        return xp.asarray(_quadratic_coeff(signal))
+
+
+def collapse_2d(x, axis):
+    x = moveaxis(x, axis, -1)
+    x_shape = x.shape
+    x = x.reshape(-1, x.shape[-1])
+    if not x.flags.c_contiguous:
+        x = x.copy()
+    return x, x_shape
+
+
+def symiirorder_nd(func, input, *args, axis=-1, **kwargs):
+    axis = normalize_axis_index(axis, input.ndim)
+    input_shape = input.shape
+    input_ndim = input.ndim
+    if input.ndim > 1:
+        input, input_shape = collapse_2d(input, axis)
+
+    out = func(input, *args, **kwargs)
+
+    if input_ndim > 1:
+        out = out.reshape(input_shape)
+        out = moveaxis(out, -1, axis)
+        if not out.flags.c_contiguous:
+            out = out.copy()
+    return out
+
+
+def qspline2d(signal, lamb=0.0, precision=-1.0):
+    """
+    Coefficients for 2-D quadratic (2nd order) B-spline.
+
+    Return the second-order B-spline coefficients over a regularly spaced
+    input grid for the two-dimensional input image.
+
+    Parameters
+    ----------
+    input : ndarray
+        The input signal.
+    lamb : float
+        Specifies the amount of smoothing in the transfer function.
+    precision : float
+        Specifies the precision for computing the infinite sum needed to apply
+        mirror-symmetric boundary conditions.
+
+    Returns
+    -------
+    output : ndarray
+        The filtered signal.
+    """
+    if precision < 0.0 or precision >= 1.0:
+        if signal.dtype in [float32, complex64]:
+            precision = 1e-3
+        else:
+            precision = 1e-6
+
+    if lamb > 0:
+        raise ValueError('lambda must be negative or zero')
+
+    # normal quadratic spline
+    r = -3 + 2 * math.sqrt(2.0)
+    c0 = -r * 8.0
+    z1 = r
+
+    out = symiirorder_nd(symiirorder1, signal, c0, z1, precision, axis=-1)
+    out = symiirorder_nd(symiirorder1, out, c0, z1, precision, axis=0)
+    return out
+
+
+def cspline2d(signal, lamb=0.0, precision=-1.0):
+    """
+    Coefficients for 2-D cubic (3rd order) B-spline.
+
+    Return the third-order B-spline coefficients over a regularly spaced
+    input grid for the two-dimensional input image.
+
+    Parameters
+    ----------
+    input : ndarray
+        The input signal.
+    lamb : float
+        Specifies the amount of smoothing in the transfer function.
+    precision : float
+        Specifies the precision for computing the infinite sum needed to apply
+        mirror-symmetric boundary conditions.
+
+    Returns
+    -------
+    output : ndarray
+        The filtered signal.
+    """
+    xp = array_namespace(signal)
+    signal = np.asarray(signal)
+
+    if precision < 0.0 or precision >= 1.0:
+        if signal.dtype in [np.float32, np.complex64]:
+            precision = 1e-3
+        else:
+            precision = 1e-6
+
+    if lamb <= 1 / 144.0:
+        # Normal cubic spline
+        r = -2 + math.sqrt(3.0)
+        out = symiirorder_nd(
+            symiirorder1, signal, -r * 6.0, r, precision=precision, axis=-1)
+        out = symiirorder_nd(
+            symiirorder1, out, -r * 6.0, r, precision=precision, axis=0)
+        return out
+
+    r, omega = compute_root_from_lambda(lamb)
+    out = symiirorder_nd(symiirorder2, signal, r, omega,
+                         precision=precision, axis=-1)
+    out = symiirorder_nd(symiirorder2, out, r, omega,
+                         precision=precision, axis=0)
+    return xp.asarray(out)
+
+
+def cspline1d_eval(cj, newx, dx=1.0, x0=0):
+    """Evaluate a cubic spline at the new set of points.
+
+    `dx` is the old sample-spacing while `x0` was the old origin. In
+    other-words the old-sample points (knot-points) for which the `cj`
+    represent spline coefficients were at equally-spaced points of:
+
+      oldx = x0 + j*dx  j=0...N-1, with N=len(cj)
+
+    Edges are handled using mirror-symmetric boundary conditions.
+
+    Parameters
+    ----------
+    cj : ndarray
+        cublic spline coefficients
+    newx : ndarray
+        New set of points.
+    dx : float, optional
+        Old sample-spacing, the default value is 1.0.
+    x0 : int, optional
+        Old origin, the default value is 0.
+
+    Returns
+    -------
+    res : ndarray
+        Evaluated a cubic spline points.
+
+    See Also
+    --------
+    cspline1d : Compute cubic spline coefficients for rank-1 array.
+
+    Examples
+    --------
+    We can filter a signal to reduce and smooth out high-frequency noise with
+    a cubic spline:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.signal import cspline1d, cspline1d_eval
+    >>> rng = np.random.default_rng()
+    >>> sig = np.repeat([0., 1., 0.], 100)
+    >>> sig += rng.standard_normal(len(sig))*0.05  # add noise
+    >>> time = np.linspace(0, len(sig))
+    >>> filtered = cspline1d_eval(cspline1d(sig), time)
+    >>> plt.plot(sig, label="signal")
+    >>> plt.plot(time, filtered, label="filtered")
+    >>> plt.legend()
+    >>> plt.show()
+
+    """
+    xp = array_namespace(cj, newx)
+
+    newx = (np.asarray(newx) - x0) / float(dx)
+    cj = np.asarray(cj)
+
+    if cj.size == 0:
+        raise ValueError("Spline coefficients 'cj' must not be empty.")
+
+    res = zeros_like(newx, dtype=cj.dtype)
+    if res.size == 0:
+        return xp.asarray(res)
+    N = len(cj)
+    cond1 = newx < 0
+    cond2 = newx > (N - 1)
+    cond3 = ~(cond1 | cond2)
+    # handle general mirror-symmetry
+    res[cond1] = cspline1d_eval(cj, -newx[cond1])
+    res[cond2] = cspline1d_eval(cj, 2 * (N - 1) - newx[cond2])
+    newx = newx[cond3]
+    if newx.size == 0:
+        return xp.asarray(res)
+    result = zeros_like(newx, dtype=cj.dtype)
+    jlower = floor(newx - 2).astype(int) + 1
+    for i in range(4):
+        thisj = jlower + i
+        indj = thisj.clip(0, N - 1)  # handle edge cases
+        result += cj[indj] * _cubic(newx - thisj)
+    res[cond3] = result
+    return xp.asarray(res)
+
+
+def qspline1d_eval(cj, newx, dx=1.0, x0=0):
+    """Evaluate a quadratic spline at the new set of points.
+
+    Parameters
+    ----------
+    cj : ndarray
+        Quadratic spline coefficients
+    newx : ndarray
+        New set of points.
+    dx : float, optional
+        Old sample-spacing, the default value is 1.0.
+    x0 : int, optional
+        Old origin, the default value is 0.
+
+    Returns
+    -------
+    res : ndarray
+        Evaluated a quadratic spline points.
+
+    See Also
+    --------
+    qspline1d : Compute quadratic spline coefficients for rank-1 array.
+
+    Notes
+    -----
+    `dx` is the old sample-spacing while `x0` was the old origin. In
+    other-words the old-sample points (knot-points) for which the `cj`
+    represent spline coefficients were at equally-spaced points of::
+
+      oldx = x0 + j*dx  j=0...N-1, with N=len(cj)
+
+    Edges are handled using mirror-symmetric boundary conditions.
+
+    Examples
+    --------
+    We can filter a signal to reduce and smooth out high-frequency noise with
+    a quadratic spline:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.signal import qspline1d, qspline1d_eval
+    >>> rng = np.random.default_rng()
+    >>> sig = np.repeat([0., 1., 0.], 100)
+    >>> sig += rng.standard_normal(len(sig))*0.05  # add noise
+    >>> time = np.linspace(0, len(sig))
+    >>> filtered = qspline1d_eval(qspline1d(sig), time)
+    >>> plt.plot(sig, label="signal")
+    >>> plt.plot(time, filtered, label="filtered")
+    >>> plt.legend()
+    >>> plt.show()
+
+    """
+    xp = array_namespace(newx, cj)
+
+    newx = (np.asarray(newx) - x0) / dx
+    res = np.zeros_like(newx)
+    if res.size == 0:
+        return xp.asarray(res)
+
+    cj = np.asarray(cj)
+    if cj.size == 0:
+        raise ValueError("Spline coefficients 'cj' must not be empty.")
+    
+    N = len(cj)
+    cond1 = newx < 0
+    cond2 = newx > (N - 1)
+    cond3 = ~(cond1 | cond2)
+    # handle general mirror-symmetry
+    res[cond1] = qspline1d_eval(cj, -newx[cond1])
+    res[cond2] = qspline1d_eval(cj, 2 * (N - 1) - newx[cond2])
+    newx = newx[cond3]
+    if newx.size == 0:
+        return xp.asarray(res)
+    result = zeros_like(newx)
+    jlower = floor(newx - 1.5).astype(int) + 1
+    for i in range(3):
+        thisj = jlower + i
+        indj = thisj.clip(0, N - 1)  # handle edge cases
+        result += cj[indj] * _quadratic(newx - thisj)
+    res[cond3] = result
+    return xp.asarray(res)
+
+
+def symiirorder1(signal, c0, z1, precision=-1.0):
+    """
+    Implement a smoothing IIR filter with mirror-symmetric boundary conditions
+    using a cascade of first-order sections.
+
+    The second section uses a reversed sequence.  This implements a system with
+    the following transfer function and mirror-symmetric boundary conditions::
+
+                           c0
+           H(z) = ---------------------
+                   (1-z1/z) (1 - z1 z)
+
+    The resulting signal will have mirror symmetric boundary conditions
+    as well.
+
+    Parameters
+    ----------
+    signal : ndarray
+        The input signal. If 2D, then the filter will be applied in a batched
+        fashion across the last axis.
+    c0, z1 : scalar
+        Parameters in the transfer function.
+    precision :
+        Specifies the precision for calculating initial conditions
+        of the recursive filter based on mirror-symmetric input.
+
+    Returns
+    -------
+    output : ndarray
+        The filtered signal.
+    """
+    xp = array_namespace(signal)
+    signal = xp_promote(signal, force_floating=True, xp=xp)
+    # This function uses C internals
+    signal = np.asarray(signal)
+
+    if abs(z1) >= 1:
+        raise ValueError('|z1| must be less than 1.0')
+
+    if signal.ndim > 2:
+        raise ValueError('Input must be 1D or 2D')
+
+    squeeze_dim = False
+    if signal.ndim == 1:
+        signal = signal[None, :]
+        squeeze_dim = True
+
+    y0 = symiirorder1_ic(signal, z1, precision)
+
+    # Apply first the system 1 / (1 - z1 * z^-1)
+    b = np.ones(1, dtype=signal.dtype)
+    a = np.r_[1, -z1]
+    a = a.astype(signal.dtype)
+
+    # Compute the initial state for lfilter.
+    zii = y0 * z1
+
+    y1, _ = lfilter(b, a, axis_slice(signal, 1), zi=zii)
+    y1 = np.c_[y0, y1]
+
+    # Compute backward symmetric condition and apply the system
+    # c0 / (1 - z1 * z)
+    b = np.asarray([c0], dtype=signal.dtype)
+    out_last = -c0 / (z1 - 1.0) * axis_slice(y1, -1)
+
+    # Compute the initial state for lfilter.
+    zii = out_last * z1
+
+    # Apply the system c0 / (1 - z1 * z) by reversing the output of the previous stage
+    out, _ = lfilter(b, a, axis_slice(y1, -2, step=-1), zi=zii)
+    out = np.c_[axis_reverse(out), out_last]
+
+    if squeeze_dim:
+        out = out[0]
+
+    return xp.asarray(out)
+
+
+def symiirorder2(input, r, omega, precision=-1.0):
+    """
+    Implement a smoothing IIR filter with mirror-symmetric boundary conditions
+    using a cascade of second-order sections.
+
+    The second section uses a reversed sequence.  This implements the following
+    transfer function::
+
+                                  cs^2
+         H(z) = ---------------------------------------
+                (1 - a2/z - a3/z^2) (1 - a2 z - a3 z^2 )
+
+    where::
+
+          a2 = 2 * r * cos(omega)
+          a3 = - r ** 2
+          cs = 1 - 2 * r * cos(omega) + r ** 2
+
+    Parameters
+    ----------
+    input : ndarray
+        The input signal.
+    r, omega : float
+        Parameters in the transfer function.
+    precision : float
+        Specifies the precision for calculating initial conditions
+        of the recursive filter based on mirror-symmetric input.
+
+    Returns
+    -------
+    output : ndarray
+        The filtered signal.
+    """
+    xp = array_namespace(input)
+    input = xp_promote(input, force_floating=True, xp=xp)
+    # This function uses C internals
+    input = np.ascontiguousarray(input)
+
+    if r >= 1.0:
+        raise ValueError('r must be less than 1.0')
+
+    if input.ndim > 2:
+        raise ValueError('Input must be 1D or 2D')
+
+    squeeze_dim = False
+    if input.ndim == 1:
+        input = input[None, :]
+        squeeze_dim = True
+
+    rsq = r * r
+    a2 = 2 * r * math.cos(omega)
+    a3 = -rsq
+    cs = 1 - 2 * r * math.cos(omega) + rsq
+    sos = np.asarray([cs, 0, 0, 1, -a2, -a3], dtype=input.dtype)
+
+    # Find the starting (forward) conditions.
+    ic_fwd = symiirorder2_ic_fwd(input, r, omega, precision)
+
+    # Apply first the system cs / (1 - a2 * z^-1 - a3 * z^-2)
+    # Compute the initial conditions in the form expected by sosfilt
+    # coef = np.asarray([[a3, a2], [0, a3]], dtype=input.dtype)
+    coef = np.asarray([[a3, a2], [0, a3]], dtype=input.dtype)
+    zi = np.matmul(coef, ic_fwd[:, :, None])[:, :, 0]
+
+    y_fwd, _ = sosfilt(sos, axis_slice(input, 2), zi=zi[None])
+    y_fwd = np.c_[ic_fwd, y_fwd]
+
+    # Then compute the symmetric backward starting conditions
+    ic_bwd = symiirorder2_ic_bwd(input, r, omega, precision)
+
+    # Apply the system cs / (1 - a2 * z^1 - a3 * z^2)
+    # Compute the initial conditions in the form expected by sosfilt
+    zi = np.matmul(coef, ic_bwd[:, :, None])[:, :, 0]
+    y, _ = sosfilt(sos, axis_slice(y_fwd, -3, step=-1), zi=zi[None])
+    out = np.c_[axis_reverse(y), axis_reverse(ic_bwd)]
+
+    if squeeze_dim:
+        out = out[0]
+
+    return xp.asarray(out)

venv/lib/python3.13/site-packages/scipy/signal/_support_alternative_backends.py

@@ -0,0 +1,73 @@
+import functools
+from scipy._lib._array_api import (
+    is_cupy, is_jax, scipy_namespace_for, SCIPY_ARRAY_API
+)
+
+from ._signal_api import *   # noqa: F403
+from . import _signal_api
+from . import _delegators
+__all__ = _signal_api.__all__
+
+
+MODULE_NAME = 'signal'
+
+# jax.scipy.signal has only partial coverage of scipy.signal, so we keep the list
+# of functions we can delegate to JAX
+# https://jax.readthedocs.io/en/latest/jax.scipy.html
+JAX_SIGNAL_FUNCS = [
+    'fftconvolve', 'convolve', 'convolve2d', 'correlate', 'correlate2d',
+    'csd', 'detrend', 'istft', 'welch'
+]
+
+# some cupyx.scipy.signal functions are incompatible with their scipy counterparts
+CUPY_BLACKLIST = ['lfilter_zi', 'sosfilt_zi', 'get_window', 'envelope', 'remez']
+
+# freqz_sos is a sosfreqz rename, and cupy does not have the new name yet (in v13.x)
+CUPY_RENAMES = {'freqz_sos': 'sosfreqz'}
+
+
+def delegate_xp(delegator, module_name):
+    def inner(func):
+        @functools.wraps(func)
+        def wrapper(*args, **kwds):
+            try:
+                xp = delegator(*args, **kwds)
+            except TypeError:
+                # object arrays
+                import numpy as np
+                xp = np
+
+            # try delegating to a cupyx/jax namesake
+            if is_cupy(xp) and func.__name__ not in CUPY_BLACKLIST:
+                func_name = CUPY_RENAMES.get(func.__name__, func.__name__)
+
+                # https://github.com/cupy/cupy/issues/8336
+                import importlib
+                cupyx_module = importlib.import_module(f"cupyx.scipy.{module_name}")
+                cupyx_func = getattr(cupyx_module, func_name)
+                return cupyx_func(*args, **kwds)
+            elif is_jax(xp) and func.__name__ in JAX_SIGNAL_FUNCS:
+                spx = scipy_namespace_for(xp)
+                jax_module = getattr(spx, module_name)
+                jax_func = getattr(jax_module, func.__name__)
+                return jax_func(*args, **kwds)
+            else:
+                # the original function
+                return func(*args, **kwds)
+        return wrapper
+    return inner
+
+
+
+# ### decorate ###
+for obj_name in _signal_api.__all__:
+    bare_obj = getattr(_signal_api, obj_name)
+    delegator = getattr(_delegators, obj_name + "_signature", None)
+
+    if SCIPY_ARRAY_API and delegator is not None:
+        f = delegate_xp(delegator, MODULE_NAME)(bare_obj)
+    else:
+        f = bare_obj
+
+    # add the decorated function to the namespace, to be imported in __init__.py
+    vars()[obj_name] = f

venv/lib/python3.13/site-packages/scipy/signal/_upfirdn.py

@@ -0,0 +1,219 @@
+# Code adapted from "upfirdn" python library with permission:
+#
+# Copyright (c) 2009, Motorola, Inc
+#
+# All Rights Reserved.
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions are
+# met:
+#
+# * Redistributions of source code must retain the above copyright notice,
+# this list of conditions and the following disclaimer.
+#
+# * Redistributions in binary form must reproduce the above copyright
+# notice, this list of conditions and the following disclaimer in the
+# documentation and/or other materials provided with the distribution.
+#
+# * Neither the name of Motorola nor the names of its contributors may be
+# used to endorse or promote products derived from this software without
+# specific prior written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+# IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
+# THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+# PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR
+# CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+# EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+# PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+# PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+# LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+import numpy as np
+
+from scipy._lib._array_api import array_namespace
+from ._upfirdn_apply import _output_len, _apply, mode_enum
+
+__all__ = ['upfirdn', '_output_len']
+
+_upfirdn_modes = [
+    'constant', 'wrap', 'edge', 'smooth', 'symmetric', 'reflect',
+    'antisymmetric', 'antireflect', 'line',
+]
+
+
+def _pad_h(h, up):
+    """Store coefficients in a transposed, flipped arrangement.
+
+    For example, suppose upRate is 3, and the
+    input number of coefficients is 10, represented as h[0], ..., h[9].
+
+    Then the internal buffer will look like this::
+
+       h[9], h[6], h[3], h[0],   // flipped phase 0 coefs
+       0,    h[7], h[4], h[1],   // flipped phase 1 coefs (zero-padded)
+       0,    h[8], h[5], h[2],   // flipped phase 2 coefs (zero-padded)
+
+    """
+    h_padlen = len(h) + (-len(h) % up)
+    h_full = np.zeros(h_padlen, h.dtype)
+    h_full[:len(h)] = h
+    h_full = h_full.reshape(-1, up).T[:, ::-1].ravel()
+    return h_full
+
+
+def _check_mode(mode):
+    mode = mode.lower()
+    enum = mode_enum(mode)
+    return enum
+
+
+class _UpFIRDn:
+    """Helper for resampling."""
+
+    def __init__(self, h, x_dtype, up, down):
+        h = np.asarray(h)
+        if h.ndim != 1 or h.size == 0:
+            raise ValueError('h must be 1-D with non-zero length')
+        self._output_type = np.result_type(h.dtype, x_dtype, np.float32)
+        h = np.asarray(h, self._output_type)
+        self._up = int(up)
+        self._down = int(down)
+        if self._up < 1 or self._down < 1:
+            raise ValueError('Both up and down must be >= 1')
+        # This both transposes, and "flips" each phase for filtering
+        self._h_trans_flip = _pad_h(h, self._up)
+        self._h_trans_flip = np.ascontiguousarray(self._h_trans_flip)
+        self._h_len_orig = len(h)
+
+    def apply_filter(self, x, axis=-1, mode='constant', cval=0):
+        """Apply the prepared filter to the specified axis of N-D signal x."""
+        output_len = _output_len(self._h_len_orig, x.shape[axis],
+                                 self._up, self._down)
+        # Explicit use of np.int64 for output_shape dtype avoids OverflowError
+        # when allocating large array on platforms where intp is 32 bits.
+        output_shape = np.asarray(x.shape, dtype=np.int64)
+        output_shape[axis] = output_len
+        out = np.zeros(output_shape, dtype=self._output_type, order='C')
+        axis = axis % x.ndim
+        mode = _check_mode(mode)
+        _apply(np.asarray(x, self._output_type),
+               self._h_trans_flip, out,
+               self._up, self._down, axis, mode, cval)
+        return out
+
+
+def upfirdn(h, x, up=1, down=1, axis=-1, mode='constant', cval=0):
+    """Upsample, FIR filter, and downsample.
+
+    Parameters
+    ----------
+    h : array_like
+        1-D FIR (finite-impulse response) filter coefficients.
+    x : array_like
+        Input signal array.
+    up : int, optional
+        Upsampling rate. Default is 1.
+    down : int, optional
+        Downsampling rate. Default is 1.
+    axis : int, optional
+        The axis of the input data array along which to apply the
+        linear filter. The filter is applied to each subarray along
+        this axis. Default is -1.
+    mode : str, optional
+        The signal extension mode to use. The set
+        ``{"constant", "symmetric", "reflect", "edge", "wrap"}`` correspond to
+        modes provided by `numpy.pad`. ``"smooth"`` implements a smooth
+        extension by extending based on the slope of the last 2 points at each
+        end of the array. ``"antireflect"`` and ``"antisymmetric"`` are
+        anti-symmetric versions of ``"reflect"`` and ``"symmetric"``. The mode
+        `"line"` extends the signal based on a linear trend defined by the
+        first and last points along the ``axis``.
+
+        .. versionadded:: 1.4.0
+    cval : float, optional
+        The constant value to use when ``mode == "constant"``.
+
+        .. versionadded:: 1.4.0
+
+    Returns
+    -------
+    y : ndarray
+        The output signal array. Dimensions will be the same as `x` except
+        for along `axis`, which will change size according to the `h`,
+        `up`,  and `down` parameters.
+
+    Notes
+    -----
+    The algorithm is an implementation of the block diagram shown on page 129
+    of the Vaidyanathan text [1]_ (Figure 4.3-8d).
+
+    The direct approach of upsampling by factor of P with zero insertion,
+    FIR filtering of length ``N``, and downsampling by factor of Q is
+    O(N*Q) per output sample. The polyphase implementation used here is
+    O(N/P).
+
+    .. versionadded:: 0.18
+
+    References
+    ----------
+    .. [1] P. P. Vaidyanathan, Multirate Systems and Filter Banks,
+           Prentice Hall, 1993.
+
+    Examples
+    --------
+    Simple operations:
+
+    >>> import numpy as np
+    >>> from scipy.signal import upfirdn
+    >>> upfirdn([1, 1, 1], [1, 1, 1])   # FIR filter
+    array([ 1.,  2.,  3.,  2.,  1.])
+    >>> upfirdn([1], [1, 2, 3], 3)  # upsampling with zeros insertion
+    array([ 1.,  0.,  0.,  2.,  0.,  0.,  3.])
+    >>> upfirdn([1, 1, 1], [1, 2, 3], 3)  # upsampling with sample-and-hold
+    array([ 1.,  1.,  1.,  2.,  2.,  2.,  3.,  3.,  3.])
+    >>> upfirdn([.5, 1, .5], [1, 1, 1], 2)  # linear interpolation
+    array([ 0.5,  1. ,  1. ,  1. ,  1. ,  1. ,  0.5])
+    >>> upfirdn([1], np.arange(10), 1, 3)  # decimation by 3
+    array([ 0.,  3.,  6.,  9.])
+    >>> upfirdn([.5, 1, .5], np.arange(10), 2, 3)  # linear interp, rate 2/3
+    array([ 0. ,  1. ,  2.5,  4. ,  5.5,  7. ,  8.5])
+
+    Apply a single filter to multiple signals:
+
+    >>> x = np.reshape(np.arange(8), (4, 2))
+    >>> x
+    array([[0, 1],
+           [2, 3],
+           [4, 5],
+           [6, 7]])
+
+    Apply along the last dimension of ``x``:
+
+    >>> h = [1, 1]
+    >>> upfirdn(h, x, 2)
+    array([[ 0.,  0.,  1.,  1.],
+           [ 2.,  2.,  3.,  3.],
+           [ 4.,  4.,  5.,  5.],
+           [ 6.,  6.,  7.,  7.]])
+
+    Apply along the 0th dimension of ``x``:
+
+    >>> upfirdn(h, x, 2, axis=0)
+    array([[ 0.,  1.],
+           [ 0.,  1.],
+           [ 2.,  3.],
+           [ 2.,  3.],
+           [ 4.,  5.],
+           [ 4.,  5.],
+           [ 6.,  7.],
+           [ 6.,  7.]])
+    """
+    xp = array_namespace(h, x)
+
+    x = np.asarray(x)
+    ufd = _UpFIRDn(h, x.dtype, up, down)
+    # This is equivalent to (but faster than) using np.apply_along_axis
+    return xp.asarray(ufd.apply_filter(x, axis, mode, cval))

venv/lib/python3.13/site-packages/scipy/signal/_waveforms.py

@@ -0,0 +1,687 @@
+# Author: Travis Oliphant
+# 2003
+#
+# Feb. 2010: Updated by Warren Weckesser:
+#   Rewrote much of chirp()
+#   Added sweep_poly()
+import numpy as np
+from numpy import asarray, zeros, place, nan, mod, pi, extract, log, sqrt, \
+    exp, cos, sin, polyval, polyint
+
+
+__all__ = ['sawtooth', 'square', 'gausspulse', 'chirp', 'sweep_poly',
+           'unit_impulse']
+
+
+def sawtooth(t, width=1):
+    """
+    Return a periodic sawtooth or triangle waveform.
+
+    The sawtooth waveform has a period ``2*pi``, rises from -1 to 1 on the
+    interval 0 to ``width*2*pi``, then drops from 1 to -1 on the interval
+    ``width*2*pi`` to ``2*pi``. `width` must be in the interval [0, 1].
+
+    Note that this is not band-limited.  It produces an infinite number
+    of harmonics, which are aliased back and forth across the frequency
+    spectrum.
+
+    Parameters
+    ----------
+    t : array_like
+        Time.
+    width : array_like, optional
+        Width of the rising ramp as a proportion of the total cycle.
+        Default is 1, producing a rising ramp, while 0 produces a falling
+        ramp.  `width` = 0.5 produces a triangle wave.
+        If an array, causes wave shape to change over time, and must be the
+        same length as t.
+
+    Returns
+    -------
+    y : ndarray
+        Output array containing the sawtooth waveform.
+
+    Examples
+    --------
+    A 5 Hz waveform sampled at 500 Hz for 1 second:
+
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+    >>> t = np.linspace(0, 1, 500)
+    >>> plt.plot(t, signal.sawtooth(2 * np.pi * 5 * t))
+
+    """
+    t, w = asarray(t), asarray(width)
+    w = asarray(w + (t - t))
+    t = asarray(t + (w - w))
+    y = zeros(t.shape, dtype="d")
+
+    # width must be between 0 and 1 inclusive
+    mask1 = (w > 1) | (w < 0)
+    place(y, mask1, nan)
+
+    # take t modulo 2*pi
+    tmod = mod(t, 2 * pi)
+
+    # on the interval 0 to width*2*pi function is
+    #  tmod / (pi*w) - 1
+    mask2 = (1 - mask1) & (tmod < w * 2 * pi)
+    tsub = extract(mask2, tmod)
+    wsub = extract(mask2, w)
+    place(y, mask2, tsub / (pi * wsub) - 1)
+
+    # on the interval width*2*pi to 2*pi function is
+    #  (pi*(w+1)-tmod) / (pi*(1-w))
+
+    mask3 = (1 - mask1) & (1 - mask2)
+    tsub = extract(mask3, tmod)
+    wsub = extract(mask3, w)
+    place(y, mask3, (pi * (wsub + 1) - tsub) / (pi * (1 - wsub)))
+    return y
+
+
+def square(t, duty=0.5):
+    """
+    Return a periodic square-wave waveform.
+
+    The square wave has a period ``2*pi``, has value +1 from 0 to
+    ``2*pi*duty`` and -1 from ``2*pi*duty`` to ``2*pi``. `duty` must be in
+    the interval [0,1].
+
+    Note that this is not band-limited.  It produces an infinite number
+    of harmonics, which are aliased back and forth across the frequency
+    spectrum.
+
+    Parameters
+    ----------
+    t : array_like
+        The input time array.
+    duty : array_like, optional
+        Duty cycle.  Default is 0.5 (50% duty cycle).
+        If an array, causes wave shape to change over time, and must be the
+        same length as t.
+
+    Returns
+    -------
+    y : ndarray
+        Output array containing the square waveform.
+
+    Examples
+    --------
+    A 5 Hz waveform sampled at 500 Hz for 1 second:
+
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+    >>> t = np.linspace(0, 1, 500, endpoint=False)
+    >>> plt.plot(t, signal.square(2 * np.pi * 5 * t))
+    >>> plt.ylim(-2, 2)
+
+    A pulse-width modulated sine wave:
+
+    >>> plt.figure()
+    >>> sig = np.sin(2 * np.pi * t)
+    >>> pwm = signal.square(2 * np.pi * 30 * t, duty=(sig + 1)/2)
+    >>> plt.subplot(2, 1, 1)
+    >>> plt.plot(t, sig)
+    >>> plt.subplot(2, 1, 2)
+    >>> plt.plot(t, pwm)
+    >>> plt.ylim(-1.5, 1.5)
+
+    """
+    t, w = asarray(t), asarray(duty)
+    w = asarray(w + (t - t))
+    t = asarray(t + (w - w))
+    y = zeros(t.shape, dtype="d")
+
+    # width must be between 0 and 1 inclusive
+    mask1 = (w > 1) | (w < 0)
+    place(y, mask1, nan)
+
+    # on the interval 0 to duty*2*pi function is 1
+    tmod = mod(t, 2 * pi)
+    mask2 = (1 - mask1) & (tmod < w * 2 * pi)
+    place(y, mask2, 1)
+
+    # on the interval duty*2*pi to 2*pi function is
+    #  (pi*(w+1)-tmod) / (pi*(1-w))
+    mask3 = (1 - mask1) & (1 - mask2)
+    place(y, mask3, -1)
+    return y
+
+
+def gausspulse(t, fc=1000, bw=0.5, bwr=-6, tpr=-60, retquad=False,
+               retenv=False):
+    """
+    Return a Gaussian modulated sinusoid:
+
+        ``exp(-a t^2) exp(1j*2*pi*fc*t).``
+
+    If `retquad` is True, then return the real and imaginary parts
+    (in-phase and quadrature).
+    If `retenv` is True, then return the envelope (unmodulated signal).
+    Otherwise, return the real part of the modulated sinusoid.
+
+    Parameters
+    ----------
+    t : ndarray or the string 'cutoff'
+        Input array.
+    fc : float, optional
+        Center frequency (e.g. Hz).  Default is 1000.
+    bw : float, optional
+        Fractional bandwidth in frequency domain of pulse (e.g. Hz).
+        Default is 0.5.
+    bwr : float, optional
+        Reference level at which fractional bandwidth is calculated (dB).
+        Default is -6.
+    tpr : float, optional
+        If `t` is 'cutoff', then the function returns the cutoff
+        time for when the pulse amplitude falls below `tpr` (in dB).
+        Default is -60.
+    retquad : bool, optional
+        If True, return the quadrature (imaginary) as well as the real part
+        of the signal.  Default is False.
+    retenv : bool, optional
+        If True, return the envelope of the signal.  Default is False.
+
+    Returns
+    -------
+    yI : ndarray
+        Real part of signal.  Always returned.
+    yQ : ndarray
+        Imaginary part of signal.  Only returned if `retquad` is True.
+    yenv : ndarray
+        Envelope of signal.  Only returned if `retenv` is True.
+
+    Examples
+    --------
+    Plot real component, imaginary component, and envelope for a 5 Hz pulse,
+    sampled at 100 Hz for 2 seconds:
+
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+    >>> t = np.linspace(-1, 1, 2 * 100, endpoint=False)
+    >>> i, q, e = signal.gausspulse(t, fc=5, retquad=True, retenv=True)
+    >>> plt.plot(t, i, t, q, t, e, '--')
+
+    """
+    if fc < 0:
+        raise ValueError(f"Center frequency (fc={fc:.2f}) must be >=0.")
+    if bw <= 0:
+        raise ValueError(f"Fractional bandwidth (bw={bw:.2f}) must be > 0.")
+    if bwr >= 0:
+        raise ValueError(f"Reference level for bandwidth (bwr={bwr:.2f}) "
+                         "must be < 0 dB")
+
+    # exp(-a t^2) <->  sqrt(pi/a) exp(-pi^2/a * f^2)  = g(f)
+
+    ref = pow(10.0, bwr / 20.0)
+    # fdel = fc*bw/2:  g(fdel) = ref --- solve this for a
+    #
+    # pi^2/a * fc^2 * bw^2 /4=-log(ref)
+    a = -(pi * fc * bw) ** 2 / (4.0 * log(ref))
+
+    if isinstance(t, str):
+        if t == 'cutoff':  # compute cut_off point
+            #  Solve exp(-a tc**2) = tref  for tc
+            #   tc = sqrt(-log(tref) / a) where tref = 10^(tpr/20)
+            if tpr >= 0:
+                raise ValueError("Reference level for time cutoff must "
+                                 "be < 0 dB")
+            tref = pow(10.0, tpr / 20.0)
+            return sqrt(-log(tref) / a)
+        else:
+            raise ValueError("If `t` is a string, it must be 'cutoff'")
+
+    yenv = exp(-a * t * t)
+    yI = yenv * cos(2 * pi * fc * t)
+    yQ = yenv * sin(2 * pi * fc * t)
+    if not retquad and not retenv:
+        return yI
+    if not retquad and retenv:
+        return yI, yenv
+    if retquad and not retenv:
+        return yI, yQ
+    if retquad and retenv:
+        return yI, yQ, yenv
+
+
+def chirp(t, f0, t1, f1, method='linear', phi=0, vertex_zero=True, *,
+          complex=False):
+    r"""Frequency-swept cosine generator.
+
+    In the following, 'Hz' should be interpreted as 'cycles per unit';
+    there is no requirement here that the unit is one second.  The
+    important distinction is that the units of rotation are cycles, not
+    radians. Likewise, `t` could be a measurement of space instead of time.
+
+    Parameters
+    ----------
+    t : array_like
+        Times at which to evaluate the waveform.
+    f0 : float
+        Frequency (e.g. Hz) at time t=0.
+    t1 : float
+        Time at which `f1` is specified.
+    f1 : float
+        Frequency (e.g. Hz) of the waveform at time `t1`.
+    method : {'linear', 'quadratic', 'logarithmic', 'hyperbolic'}, optional
+        Kind of frequency sweep.  If not given, `linear` is assumed.  See
+        Notes below for more details.
+    phi : float, optional
+        Phase offset, in degrees. Default is 0.
+    vertex_zero : bool, optional
+        This parameter is only used when `method` is 'quadratic'.
+        It determines whether the vertex of the parabola that is the graph
+        of the frequency is at t=0 or t=t1.
+    complex : bool, optional
+        This parameter creates a complex-valued analytic signal instead of a
+        real-valued signal. It allows the use of complex baseband (in communications
+        domain). Default is False.
+
+        .. versionadded:: 1.15.0
+
+    Returns
+    -------
+    y : ndarray
+        A numpy array containing the signal evaluated at `t` with the requested
+        time-varying frequency.  More precisely, the function returns
+        ``exp(1j*phase + 1j*(pi/180)*phi) if complex else cos(phase + (pi/180)*phi)``
+        where `phase` is the integral (from 0 to `t`) of ``2*pi*f(t)``.
+        The instantaneous frequency ``f(t)`` is defined below.
+
+    See Also
+    --------
+    sweep_poly
+
+    Notes
+    -----
+    There are four possible options for the parameter `method`, which have a (long)
+    standard form and some allowed abbreviations. The formulas for the instantaneous
+    frequency :math:`f(t)` of the generated signal are as follows:
+
+    1. Parameter `method` in ``('linear', 'lin', 'li')``:
+
+       .. math::
+           f(t) = f_0 + \beta\, t           \quad\text{with}\quad
+           \beta = \frac{f_1 - f_0}{t_1}
+
+       Frequency :math:`f(t)` varies linearly over time with a constant rate
+       :math:`\beta`.
+
+    2. Parameter `method` in ``('quadratic', 'quad', 'q')``:
+
+       .. math::
+            f(t) =
+            \begin{cases}
+              f_0 + \beta\, t^2          & \text{if vertex_zero is True,}\\
+              f_1 + \beta\, (t_1 - t)^2  & \text{otherwise,}
+            \end{cases}
+            \quad\text{with}\quad
+            \beta = \frac{f_1 - f_0}{t_1^2}
+
+       The graph of the frequency f(t) is a parabola through :math:`(0, f_0)` and
+       :math:`(t_1, f_1)`.  By default, the vertex of the parabola is at
+       :math:`(0, f_0)`. If `vertex_zero` is ``False``, then the vertex is at
+       :math:`(t_1, f_1)`.
+       To use a more general quadratic function, or an arbitrary
+       polynomial, use the function `scipy.signal.sweep_poly`.
+
+    3. Parameter `method` in ``('logarithmic', 'log', 'lo')``:
+
+       .. math::
+            f(t) = f_0  \left(\frac{f_1}{f_0}\right)^{t/t_1}
+
+       :math:`f_0` and :math:`f_1` must be nonzero and have the same sign.
+       This signal is also known as a geometric or exponential chirp.
+
+    4. Parameter `method` in ``('hyperbolic', 'hyp')``:
+
+       .. math::
+              f(t) = \frac{\alpha}{\beta\, t + \gamma} \quad\text{with}\quad
+              \alpha = f_0 f_1 t_1, \ \beta = f_0 - f_1, \ \gamma = f_1 t_1
+
+       :math:`f_0` and :math:`f_1` must be nonzero.
+
+
+    Examples
+    --------
+    For the first example, a linear chirp ranging from 6 Hz to 1 Hz over 10 seconds is
+    plotted:
+
+    >>> import numpy as np
+    >>> from matplotlib.pyplot import tight_layout
+    >>> from scipy.signal import chirp, square, ShortTimeFFT
+    >>> from scipy.signal.windows import gaussian
+    >>> import matplotlib.pyplot as plt
+    ...
+    >>> N, T = 1000, 0.01  # number of samples and sampling interval for 10 s signal
+    >>> t = np.arange(N) * T  # timestamps
+    ...
+    >>> x_lin = chirp(t, f0=6, f1=1, t1=10, method='linear')
+    ...
+    >>> fg0, ax0 = plt.subplots()
+    >>> ax0.set_title(r"Linear Chirp from $f(0)=6\,$Hz to $f(10)=1\,$Hz")
+    >>> ax0.set(xlabel="Time $t$ in Seconds", ylabel=r"Amplitude $x_\text{lin}(t)$")
+    >>> ax0.plot(t, x_lin)
+    >>> plt.show()
+
+    The following four plots each show the short-time Fourier transform of a chirp
+    ranging from 45 Hz to 5 Hz with different values for the parameter `method`
+    (and `vertex_zero`):
+
+    >>> x_qu0 = chirp(t, f0=45, f1=5, t1=N*T, method='quadratic', vertex_zero=True)
+    >>> x_qu1 = chirp(t, f0=45, f1=5, t1=N*T, method='quadratic', vertex_zero=False)
+    >>> x_log = chirp(t, f0=45, f1=5, t1=N*T, method='logarithmic')
+    >>> x_hyp = chirp(t, f0=45, f1=5, t1=N*T, method='hyperbolic')
+    ...
+    >>> win = gaussian(50, std=12, sym=True)
+    >>> SFT = ShortTimeFFT(win, hop=2, fs=1/T, mfft=800, scale_to='magnitude')
+    >>> ts = ("'quadratic', vertex_zero=True", "'quadratic', vertex_zero=False",
+    ...       "'logarithmic'", "'hyperbolic'")
+    >>> fg1, ax1s = plt.subplots(2, 2, sharex='all', sharey='all',
+    ...                          figsize=(6, 5),  layout="constrained")
+    >>> for x_, ax_, t_ in zip([x_qu0, x_qu1, x_log, x_hyp], ax1s.ravel(), ts):
+    ...     aSx = abs(SFT.stft(x_))
+    ...     im_ = ax_.imshow(aSx, origin='lower', aspect='auto', extent=SFT.extent(N),
+    ...                      cmap='plasma')
+    ...     ax_.set_title(t_)
+    ...     if t_ == "'hyperbolic'":
+    ...         fg1.colorbar(im_, ax=ax1s, label='Magnitude $|S_z(t,f)|$')
+    >>> _ = fg1.supxlabel("Time $t$ in Seconds")  # `_ =` is needed to pass doctests
+    >>> _ = fg1.supylabel("Frequency $f$ in Hertz")
+    >>> plt.show()
+
+    Finally, the short-time Fourier transform of a complex-valued linear chirp
+    ranging from -30 Hz to 30 Hz is depicted:
+
+    >>> z_lin = chirp(t, f0=-30, f1=30, t1=N*T, method="linear", complex=True)
+    >>> SFT.fft_mode = 'centered'  # needed to work with complex signals
+    >>> aSz = abs(SFT.stft(z_lin))
+    ...
+    >>> fg2, ax2 = plt.subplots()
+    >>> ax2.set_title(r"Linear Chirp from $-30\,$Hz to $30\,$Hz")
+    >>> ax2.set(xlabel="Time $t$ in Seconds", ylabel="Frequency $f$ in Hertz")
+    >>> im2 = ax2.imshow(aSz, origin='lower', aspect='auto',
+    ...                  extent=SFT.extent(N), cmap='viridis')
+    >>> fg2.colorbar(im2, label='Magnitude $|S_z(t,f)|$')
+    >>> plt.show()
+
+    Note that using negative frequencies makes only sense with complex-valued signals.
+    Furthermore, the magnitude of the complex exponential function is one whereas the
+    magnitude of the real-valued cosine function is only 1/2.
+    """
+    # 'phase' is computed in _chirp_phase, to make testing easier.
+    phase = _chirp_phase(t, f0, t1, f1, method, vertex_zero) + np.deg2rad(phi)
+    return np.exp(1j*phase) if complex else np.cos(phase)
+
+
+def _chirp_phase(t, f0, t1, f1, method='linear', vertex_zero=True):
+    """
+    Calculate the phase used by `chirp` to generate its output.
+
+    See `chirp` for a description of the arguments.
+
+    """
+    t = asarray(t)
+    f0 = float(f0)
+    t1 = float(t1)
+    f1 = float(f1)
+    if method in ['linear', 'lin', 'li']:
+        beta = (f1 - f0) / t1
+        phase = 2 * pi * (f0 * t + 0.5 * beta * t * t)
+
+    elif method in ['quadratic', 'quad', 'q']:
+        beta = (f1 - f0) / (t1 ** 2)
+        if vertex_zero:
+            phase = 2 * pi * (f0 * t + beta * t ** 3 / 3)
+        else:
+            phase = 2 * pi * (f1 * t + beta * ((t1 - t) ** 3 - t1 ** 3) / 3)
+
+    elif method in ['logarithmic', 'log', 'lo']:
+        if f0 * f1 <= 0.0:
+            raise ValueError("For a logarithmic chirp, f0 and f1 must be "
+                             "nonzero and have the same sign.")
+        if f0 == f1:
+            phase = 2 * pi * f0 * t
+        else:
+            beta = t1 / log(f1 / f0)
+            phase = 2 * pi * beta * f0 * (pow(f1 / f0, t / t1) - 1.0)
+
+    elif method in ['hyperbolic', 'hyp']:
+        if f0 == 0 or f1 == 0:
+            raise ValueError("For a hyperbolic chirp, f0 and f1 must be "
+                             "nonzero.")
+        if f0 == f1:
+            # Degenerate case: constant frequency.
+            phase = 2 * pi * f0 * t
+        else:
+            # Singular point: the instantaneous frequency blows up
+            # when t == sing.
+            sing = -f1 * t1 / (f0 - f1)
+            phase = 2 * pi * (-sing * f0) * log(np.abs(1 - t/sing))
+
+    else:
+        raise ValueError("method must be 'linear', 'quadratic', 'logarithmic', "
+                         f"or 'hyperbolic', but a value of {method!r} was given.")
+
+    return phase
+
+
+def sweep_poly(t, poly, phi=0):
+    """
+    Frequency-swept cosine generator, with a time-dependent frequency.
+
+    This function generates a sinusoidal function whose instantaneous
+    frequency varies with time.  The frequency at time `t` is given by
+    the polynomial `poly`.
+
+    Parameters
+    ----------
+    t : ndarray
+        Times at which to evaluate the waveform.
+    poly : 1-D array_like or instance of numpy.poly1d
+        The desired frequency expressed as a polynomial.  If `poly` is
+        a list or ndarray of length n, then the elements of `poly` are
+        the coefficients of the polynomial, and the instantaneous
+        frequency is
+
+          ``f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ... + poly[n-1]``
+
+        If `poly` is an instance of numpy.poly1d, then the
+        instantaneous frequency is
+
+          ``f(t) = poly(t)``
+
+    phi : float, optional
+        Phase offset, in degrees, Default: 0.
+
+    Returns
+    -------
+    sweep_poly : ndarray
+        A numpy array containing the signal evaluated at `t` with the
+        requested time-varying frequency.  More precisely, the function
+        returns ``cos(phase + (pi/180)*phi)``, where `phase` is the integral
+        (from 0 to t) of ``2 * pi * f(t)``; ``f(t)`` is defined above.
+
+    See Also
+    --------
+    chirp
+
+    Notes
+    -----
+    .. versionadded:: 0.8.0
+
+    If `poly` is a list or ndarray of length `n`, then the elements of
+    `poly` are the coefficients of the polynomial, and the instantaneous
+    frequency is:
+
+        ``f(t) = poly[0]*t**(n-1) + poly[1]*t**(n-2) + ... + poly[n-1]``
+
+    If `poly` is an instance of `numpy.poly1d`, then the instantaneous
+    frequency is:
+
+          ``f(t) = poly(t)``
+
+    Finally, the output `s` is:
+
+        ``cos(phase + (pi/180)*phi)``
+
+    where `phase` is the integral from 0 to `t` of ``2 * pi * f(t)``,
+    ``f(t)`` as defined above.
+
+    Examples
+    --------
+    Compute the waveform with instantaneous frequency::
+
+        f(t) = 0.025*t**3 - 0.36*t**2 + 1.25*t + 2
+
+    over the interval 0 <= t <= 10.
+
+    >>> import numpy as np
+    >>> from scipy.signal import sweep_poly
+    >>> p = np.poly1d([0.025, -0.36, 1.25, 2.0])
+    >>> t = np.linspace(0, 10, 5001)
+    >>> w = sweep_poly(t, p)
+
+    Plot it:
+
+    >>> import matplotlib.pyplot as plt
+    >>> plt.subplot(2, 1, 1)
+    >>> plt.plot(t, w)
+    >>> plt.title("Sweep Poly\\nwith frequency " +
+    ...           "$f(t) = 0.025t^3 - 0.36t^2 + 1.25t + 2$")
+    >>> plt.subplot(2, 1, 2)
+    >>> plt.plot(t, p(t), 'r', label='f(t)')
+    >>> plt.legend()
+    >>> plt.xlabel('t')
+    >>> plt.tight_layout()
+    >>> plt.show()
+
+    """
+    # 'phase' is computed in _sweep_poly_phase, to make testing easier.
+    phase = _sweep_poly_phase(t, poly)
+    # Convert to radians.
+    phi *= pi / 180
+    return cos(phase + phi)
+
+
+def _sweep_poly_phase(t, poly):
+    """
+    Calculate the phase used by sweep_poly to generate its output.
+
+    See `sweep_poly` for a description of the arguments.
+
+    """
+    # polyint handles lists, ndarrays and instances of poly1d automatically.
+    intpoly = polyint(poly)
+    phase = 2 * pi * polyval(intpoly, t)
+    return phase
+
+
+def unit_impulse(shape, idx=None, dtype=float):
+    r"""
+    Unit impulse signal (discrete delta function) or unit basis vector.
+
+    Parameters
+    ----------
+    shape : int or tuple of int
+        Number of samples in the output (1-D), or a tuple that represents the
+        shape of the output (N-D).
+    idx : None or int or tuple of int or 'mid', optional
+        Index at which the value is 1.  If None, defaults to the 0th element.
+        If ``idx='mid'``, the impulse will be centered at ``shape // 2`` in
+        all dimensions.  If an int, the impulse will be at `idx` in all
+        dimensions.
+    dtype : data-type, optional
+        The desired data-type for the array, e.g., ``numpy.int8``.  Default is
+        ``numpy.float64``.
+
+    Returns
+    -------
+    y : ndarray
+        Output array containing an impulse signal.
+
+    Notes
+    -----
+    In digital signal processing literature the unit impulse signal is often
+    represented by the Kronecker delta. [1]_ I.e., a signal :math:`u_k[n]`,
+    which is zero everywhere except being one at the :math:`k`-th sample,
+    can be expressed as
+
+    .. math::
+
+        u_k[n] = \delta[n-k] \equiv \delta_{n,k}\ .
+
+    Furthermore, the unit impulse is frequently interpreted as the discrete-time
+    version of the continuous-time Dirac distribution. [2]_
+
+    References
+    ----------
+    .. [1] "Kronecker delta", *Wikipedia*,
+           https://en.wikipedia.org/wiki/Kronecker_delta#Digital_signal_processing
+    .. [2] "Dirac delta function" *Wikipedia*,
+           https://en.wikipedia.org/wiki/Dirac_delta_function#Relationship_to_the_Kronecker_delta
+
+    .. versionadded:: 0.19.0
+
+    Examples
+    --------
+    An impulse at the 0th element (:math:`\\delta[n]`):
+
+    >>> from scipy import signal
+    >>> signal.unit_impulse(8)
+    array([ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])
+
+    Impulse offset by 2 samples (:math:`\\delta[n-2]`):
+
+    >>> signal.unit_impulse(7, 2)
+    array([ 0.,  0.,  1.,  0.,  0.,  0.,  0.])
+
+    2-dimensional impulse, centered:
+
+    >>> signal.unit_impulse((3, 3), 'mid')
+    array([[ 0.,  0.,  0.],
+           [ 0.,  1.,  0.],
+           [ 0.,  0.,  0.]])
+
+    Impulse at (2, 2), using broadcasting:
+
+    >>> signal.unit_impulse((4, 4), 2)
+    array([[ 0.,  0.,  0.,  0.],
+           [ 0.,  0.,  0.,  0.],
+           [ 0.,  0.,  1.,  0.],
+           [ 0.,  0.,  0.,  0.]])
+
+    Plot the impulse response of a 4th-order Butterworth lowpass filter:
+
+    >>> imp = signal.unit_impulse(100, 'mid')
+    >>> b, a = signal.butter(4, 0.2)
+    >>> response = signal.lfilter(b, a, imp)
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(np.arange(-50, 50), imp)
+    >>> plt.plot(np.arange(-50, 50), response)
+    >>> plt.margins(0.1, 0.1)
+    >>> plt.xlabel('Time [samples]')
+    >>> plt.ylabel('Amplitude')
+    >>> plt.grid(True)
+    >>> plt.show()
+
+    """
+    out = zeros(shape, dtype)
+
+    shape = np.atleast_1d(shape)
+
+    if idx is None:
+        idx = (0,) * len(shape)
+    elif idx == 'mid':
+        idx = tuple(shape // 2)
+    elif not hasattr(idx, "__iter__"):
+        idx = (idx,) * len(shape)
+
+    out[idx] = 1
+    return out

venv/lib/python3.13/site-packages/scipy/signal/_wavelets.py

@@ -0,0 +1,29 @@
+import numpy as np
+from scipy.signal._signaltools import convolve
+
+
+def _ricker(points, a):
+    A = 2 / (np.sqrt(3 * a) * (np.pi**0.25))
+    wsq = a**2
+    vec = np.arange(0, points) - (points - 1.0) / 2
+    xsq = vec**2
+    mod = (1 - xsq / wsq)
+    gauss = np.exp(-xsq / (2 * wsq))
+    total = A * mod * gauss
+    return total
+
+
+def _cwt(data, wavelet, widths, dtype=None, **kwargs):
+    # Determine output type
+    if dtype is None:
+        if np.asarray(wavelet(1, widths[0], **kwargs)).dtype.char in 'FDG':
+            dtype = np.complex128
+        else:
+            dtype = np.float64
+
+    output = np.empty((len(widths), len(data)), dtype=dtype)
+    for ind, width in enumerate(widths):
+        N = np.min([10 * width, len(data)])
+        wavelet_data = np.conj(wavelet(N, width, **kwargs)[::-1])
+        output[ind] = convolve(data, wavelet_data, mode='same')
+    return output

venv/lib/python3.13/site-packages/scipy/signal/bsplines.py

@@ -0,0 +1,21 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'spline_filter', 'gauss_spline',
+    'cspline1d', 'qspline1d', 'cspline1d_eval', 'qspline1d_eval',
+    'cspline2d', 'sepfir2d'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="bsplines",
+                                   private_modules=["_spline_filters"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/filter_design.py

@@ -0,0 +1,28 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'findfreqs', 'freqs', 'freqz', 'tf2zpk', 'zpk2tf', 'normalize',
+    'lp2lp', 'lp2hp', 'lp2bp', 'lp2bs', 'bilinear', 'iirdesign',
+    'iirfilter', 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel',
+    'band_stop_obj', 'buttord', 'cheb1ord', 'cheb2ord', 'ellipord',
+    'buttap', 'cheb1ap', 'cheb2ap', 'ellipap', 'besselap',
+    'BadCoefficients', 'freqs_zpk', 'freqz_zpk',
+    'tf2sos', 'sos2tf', 'zpk2sos', 'sos2zpk', 'group_delay',
+    'sosfreqz', 'freqz_sos', 'iirnotch', 'iirpeak', 'bilinear_zpk',
+    'lp2lp_zpk', 'lp2hp_zpk', 'lp2bp_zpk', 'lp2bs_zpk',
+    'gammatone', 'iircomb',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="filter_design",
+                                   private_modules=["_filter_design"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/fir_filter_design.py

@@ -0,0 +1,21 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'kaiser_beta', 'kaiser_atten', 'kaiserord',
+    'firwin', 'firwin2', 'remez', 'firls', 'minimum_phase',
+    'firwin_2d',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="fir_filter_design",
+                                   private_modules=["_fir_filter_design"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/lti_conversion.py

@@ -0,0 +1,20 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk',
+    'cont2discrete', 'tf2zpk', 'zpk2tf', 'normalize'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="lti_conversion",
+                                   private_modules=["_lti_conversion"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/ltisys.py

@@ -0,0 +1,25 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'lti', 'dlti', 'TransferFunction', 'ZerosPolesGain', 'StateSpace',
+    'lsim', 'impulse', 'step', 'bode',
+    'freqresp', 'place_poles', 'dlsim', 'dstep', 'dimpulse',
+    'dfreqresp', 'dbode',
+    'tf2zpk', 'zpk2tf', 'normalize', 'freqs',
+    'freqz', 'freqs_zpk', 'freqz_zpk', 'tf2ss', 'abcd_normalize',
+    'ss2tf', 'zpk2ss', 'ss2zpk', 'cont2discrete',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="ltisys",
+                                   private_modules=["_ltisys"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/signaltools.py

@@ -0,0 +1,27 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'correlate', 'correlation_lags', 'correlate2d',
+    'convolve', 'convolve2d', 'fftconvolve', 'oaconvolve',
+    'order_filter', 'medfilt', 'medfilt2d', 'wiener', 'lfilter',
+    'lfiltic', 'sosfilt', 'deconvolve', 'hilbert', 'hilbert2',
+    'unique_roots', 'invres', 'invresz', 'residue',
+    'residuez', 'resample', 'resample_poly', 'detrend',
+    'lfilter_zi', 'sosfilt_zi', 'sosfiltfilt', 'choose_conv_method',
+    'filtfilt', 'decimate', 'vectorstrength',
+    'dlti', 'upfirdn', 'get_window', 'cheby1', 'firwin'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="signaltools",
+                                   private_modules=["_signaltools"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/spectral.py

@@ -0,0 +1,21 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.signal` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'periodogram', 'welch', 'lombscargle', 'csd', 'coherence',
+    'spectrogram', 'stft', 'istft', 'check_COLA', 'check_NOLA',
+    'get_window',
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="spectral",
+                                   private_modules=["_spectral_py"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/spline.py

@@ -0,0 +1,18 @@
+# This file is not meant for public use and will be removed in the future
+# versions of SciPy. Use the `scipy.signal` namespace for importing the
+# functions included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+
+__all__ = ['sepfir2d']  # noqa: F822
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="spline",
+                                   private_modules=["_spline"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/signal/tests/_scipy_spectral_test_shim.py

@@ -0,0 +1,311 @@
+"""Helpers to utilize existing stft / istft tests for testing `ShortTimeFFT`.
+
+This module provides the functions stft_compare() and istft_compare(), which,
+compares the output between the existing (i)stft() and the shortTimeFFT based
+_(i)stft_wrapper() implementations in this module.
+
+For testing add the following imports to the file ``tests/test_spectral.py``::
+
+    from ._scipy_spectral_test_shim import stft_compare as stft
+    from ._scipy_spectral_test_shim import istft_compare as istft
+
+and remove the existing imports of stft and istft.
+
+The idea of these wrappers is not to provide a backward-compatible interface
+but to demonstrate that the ShortTimeFFT implementation is at least as capable
+as the existing one and delivers comparable results. Furthermore, the
+wrappers highlight the different philosophies of the implementations,
+especially in the border handling.
+"""
+import platform
+from typing import cast, Literal
+
+import numpy as np
+from numpy.testing import assert_allclose
+
+from scipy.signal import ShortTimeFFT
+from scipy.signal import get_window, stft, istft
+from scipy.signal._arraytools import const_ext, even_ext, odd_ext, zero_ext
+from scipy.signal._short_time_fft import FFT_MODE_TYPE
+from scipy.signal._spectral_py import _triage_segments
+
+
+def _stft_wrapper(x, fs=1.0, window='hann', nperseg=256, noverlap=None,
+                  nfft=None, detrend=False, return_onesided=True,
+                  boundary='zeros', padded=True, axis=-1, scaling='spectrum'):
+    """Wrapper for the SciPy `stft()` function based on `ShortTimeFFT` for
+    unit testing.
+
+    Handling the boundary and padding is where `ShortTimeFFT` and `stft()`
+    differ in behavior. Parts of `_spectral_helper()` were copied to mimic
+    the` stft()` behavior.
+
+    This function is meant to be solely used by `stft_compare()`.
+    """
+    if scaling not in ('psd', 'spectrum'):  # same errors as in original stft:
+        raise ValueError(f"Parameter {scaling=} not in ['spectrum', 'psd']!")
+
+    # The following lines are taken from the original _spectral_helper():
+    boundary_funcs = {'even': even_ext,
+                      'odd': odd_ext,
+                      'constant': const_ext,
+                      'zeros': zero_ext,
+                      None: None}
+
+    if boundary not in boundary_funcs:
+        raise ValueError(f"Unknown boundary option '{boundary}', must be one" +
+                         f" of: {list(boundary_funcs.keys())}")
+    if x.size == 0:
+        return np.empty(x.shape), np.empty(x.shape), np.empty(x.shape)
+
+    if nperseg is not None:  # if specified by user
+        nperseg = int(nperseg)
+        if nperseg < 1:
+            raise ValueError('nperseg must be a positive integer')
+
+    # parse window; if array like, then set nperseg = win.shape
+    win, nperseg = _triage_segments(window, nperseg,
+                                    input_length=x.shape[axis])
+
+    if nfft is None:
+        nfft = nperseg
+    elif nfft < nperseg:
+        raise ValueError('nfft must be greater than or equal to nperseg.')
+    else:
+        nfft = int(nfft)
+
+    if noverlap is None:
+        noverlap = nperseg//2
+    else:
+        noverlap = int(noverlap)
+    if noverlap >= nperseg:
+        raise ValueError('noverlap must be less than nperseg.')
+    nstep = nperseg - noverlap
+    n = x.shape[axis]
+
+    # Padding occurs after boundary extension, so that the extended signal ends
+    # in zeros, instead of introducing an impulse at the end.
+    # I.e. if x = [..., 3, 2]
+    # extend then pad -> [..., 3, 2, 2, 3, 0, 0, 0]
+    # pad then extend -> [..., 3, 2, 0, 0, 0, 2, 3]
+
+    if boundary is not None:
+        ext_func = boundary_funcs[boundary]
+        # Extend by nperseg//2 in front and back:
+        x = ext_func(x, nperseg//2, axis=axis)
+
+    if padded:
+        # Pad to integer number of windowed segments
+        # I.e make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg
+        x = np.moveaxis(x, axis, -1)
+
+        # This is an edge case where shortTimeFFT returns one more time slice
+        # than the Scipy stft() shorten to remove last time slice:
+        if n % 2 == 1 and nperseg % 2 == 1 and noverlap % 2 == 1:
+            x = x[..., : -1]
+
+        nadd = (-(x.shape[-1]-nperseg) % nstep) % nperseg
+        zeros_shape = list(x.shape[:-1]) + [nadd]
+        x = np.concatenate((x, np.zeros(zeros_shape)), axis=-1)
+        x = np.moveaxis(x, -1, axis)
+
+    #  ... end original _spectral_helper() code.
+    scale_to = {'spectrum': 'magnitude', 'psd': 'psd'}[scaling]
+
+    if np.iscomplexobj(x) and return_onesided:
+        return_onesided = False
+    # using cast() to make mypy happy:
+    fft_mode = cast(FFT_MODE_TYPE, 'onesided' if return_onesided else 'twosided')
+
+    ST = ShortTimeFFT(win, nstep, fs, fft_mode=fft_mode, mfft=nfft,
+                      scale_to=scale_to, phase_shift=None)
+
+    k_off = nperseg // 2
+    p0 = 0  # ST.lower_border_end[1] + 1
+    nn = x.shape[axis] if padded else n+k_off+1
+    # number of frames akin to legacy stft computation
+    p1 = (x.shape[axis] - nperseg) // nstep + 1 
+
+    detr = None if detrend is False else detrend
+    Sxx = ST.stft_detrend(x, detr, p0, p1, k_offset=k_off, axis=axis)
+    t = ST.t(nn, 0, p1 - p0, k_offset=0 if boundary is not None else k_off)
+    if x.dtype in (np.float32, np.complex64):
+        Sxx = Sxx.astype(np.complex64)
+
+    return ST.f, t, Sxx
+
+
+def _istft_wrapper(Zxx, fs=1.0, window='hann', nperseg=None, noverlap=None,
+                   nfft=None, input_onesided=True, boundary=True, time_axis=-1,
+                   freq_axis=-2, scaling='spectrum') -> \
+        tuple[np.ndarray, np.ndarray, tuple[int, int]]:
+    """Wrapper for the SciPy `istft()` function based on `ShortTimeFFT` for
+        unit testing.
+
+    Note that only option handling is implemented as far as to handle the unit
+    tests. E.g., the case ``nperseg=None`` is not handled.
+
+    This function is meant to be solely used by `istft_compare()`.
+    """
+    # *** Lines are taken from _spectral_py.istft() ***:
+    if Zxx.ndim < 2:
+        raise ValueError('Input stft must be at least 2d!')
+
+    if freq_axis == time_axis:
+        raise ValueError('Must specify differing time and frequency axes!')
+
+    nseg = Zxx.shape[time_axis]
+
+    if input_onesided:
+        # Assume even segment length
+        n_default = 2*(Zxx.shape[freq_axis] - 1)
+    else:
+        n_default = Zxx.shape[freq_axis]
+
+    # Check windowing parameters
+    if nperseg is None:
+        nperseg = n_default
+    else:
+        nperseg = int(nperseg)
+        if nperseg < 1:
+            raise ValueError('nperseg must be a positive integer')
+
+    if nfft is None:
+        if input_onesided and (nperseg == n_default + 1):
+            # Odd nperseg, no FFT padding
+            nfft = nperseg
+        else:
+            nfft = n_default
+    elif nfft < nperseg:
+        raise ValueError('nfft must be greater than or equal to nperseg.')
+    else:
+        nfft = int(nfft)
+
+    if noverlap is None:
+        noverlap = nperseg//2
+    else:
+        noverlap = int(noverlap)
+    if noverlap >= nperseg:
+        raise ValueError('noverlap must be less than nperseg.')
+    nstep = nperseg - noverlap
+
+    # Get window as array
+    if isinstance(window, str) or type(window) is tuple:
+        win = get_window(window, nperseg)
+    else:
+        win = np.asarray(window)
+        if len(win.shape) != 1:
+            raise ValueError('window must be 1-D')
+        if win.shape[0] != nperseg:
+            raise ValueError(f'window must have length of {nperseg}')
+
+    outputlength = nperseg + (nseg-1)*nstep
+    # *** End block of: Taken from _spectral_py.istft() ***
+
+    # Using cast() to make mypy happy:
+    fft_mode = cast(FFT_MODE_TYPE, 'onesided' if input_onesided else 'twosided')
+    scale_to = cast(Literal['magnitude', 'psd'],
+                    {'spectrum': 'magnitude', 'psd': 'psd'}[scaling])
+
+    ST = ShortTimeFFT(win, nstep, fs, fft_mode=fft_mode, mfft=nfft,
+                      scale_to=scale_to, phase_shift=None)
+
+    if boundary:
+        j = nperseg if nperseg % 2 == 0 else nperseg - 1
+        k0 = ST.k_min + nperseg // 2
+        k1 = outputlength - j + k0
+    else:
+        raise NotImplementedError("boundary=False does not make sense with" +
+                                  "ShortTimeFFT.istft()!")
+
+    x = ST.istft(Zxx, k0=k0, k1=k1, f_axis=freq_axis, t_axis=time_axis)
+    t = np.arange(k1 - k0) * ST.T
+    k_hi = ST.upper_border_begin(k1 - k0)[0]
+    # using cast() to make mypy happy:
+    return t, x, (ST.lower_border_end[0], k_hi)
+
+
+def stft_compare(x, fs=1.0, window='hann', nperseg=256, noverlap=None,
+                 nfft=None, detrend=False, return_onesided=True,
+                 boundary='zeros', padded=True, axis=-1, scaling='spectrum'):
+    """Assert that the results from the existing `stft()` and `_stft_wrapper()`
+    are close to each other.
+
+    For comparing the STFT values an absolute tolerance of the floating point
+    resolution was added to circumvent problems with the following tests:
+    * For float32 the tolerances are much higher in
+      TestSTFT.test_roundtrip_float32()).
+    * The TestSTFT.test_roundtrip_scaling() has a high relative deviation.
+      Interestingly this did not appear in Scipy 1.9.1 but only in the current
+      development version.
+    """
+    kw = dict(x=x, fs=fs, window=window, nperseg=nperseg, noverlap=noverlap,
+              nfft=nfft, detrend=detrend, return_onesided=return_onesided,
+              boundary=boundary, padded=padded, axis=axis, scaling=scaling)
+    f, t, Zxx = stft(**kw)
+    f_wrapper, t_wrapper, Zxx_wrapper = _stft_wrapper(**kw)
+
+    e_msg_part = " of `stft_wrapper()` differ from `stft()`."
+    assert_allclose(f_wrapper, f, err_msg=f"Frequencies {e_msg_part}")
+    assert_allclose(t_wrapper, t, err_msg=f"Time slices {e_msg_part}")
+
+    # Adapted tolerances to account for:
+    atol = np.finfo(Zxx.dtype).resolution * 2
+    assert_allclose(Zxx_wrapper, Zxx, atol=atol,
+                    err_msg=f"STFT values {e_msg_part}")
+    return f, t, Zxx
+
+
+def istft_compare(Zxx, fs=1.0, window='hann', nperseg=None, noverlap=None,
+                  nfft=None, input_onesided=True, boundary=True, time_axis=-1,
+                  freq_axis=-2, scaling='spectrum'):
+    """Assert that the results from the existing `istft()` and
+    `_istft_wrapper()` are close to each other.
+
+    Quirks:
+    * If ``boundary=False`` the comparison is skipped, since it does not
+      make sense with ShortTimeFFT.istft(). Only used in test
+      TestSTFT.test_roundtrip_boundary_extension().
+    * If ShortTimeFFT.istft() decides the STFT is not invertible, the
+      comparison is skipped, since istft() only emits a warning and does not
+      return a correct result. Only used in
+      ShortTimeFFT.test_roundtrip_not_nola().
+    * For comparing the signals an absolute tolerance of the floating point
+      resolution was added to account for the low accuracy of float32 (Occurs
+      only in TestSTFT.test_roundtrip_float32()).
+    """
+    kw = dict(Zxx=Zxx, fs=fs, window=window, nperseg=nperseg,
+              noverlap=noverlap, nfft=nfft, input_onesided=input_onesided,
+              boundary=boundary, time_axis=time_axis, freq_axis=freq_axis,
+              scaling=scaling)
+
+    t, x = istft(**kw)
+    if not boundary:  # skip test_roundtrip_boundary_extension():
+        return t, x  # _istft_wrapper does() not implement this case
+    try:  # if inversion fails, istft() only emits a warning:
+        t_wrapper, x_wrapper, (k_lo, k_hi) = _istft_wrapper(**kw)
+    except ValueError as v:  # Do nothing if inversion fails:
+        if v.args[0] == "Short-time Fourier Transform not invertible!":
+            return t, x
+        raise v
+
+    e_msg_part = " of `istft_wrapper()` differ from `istft()`"
+    assert_allclose(t, t_wrapper, err_msg=f"Sample times {e_msg_part}")
+
+    # Adapted tolerances to account for resolution loss:
+    atol = np.finfo(x.dtype).resolution*2  # instead of default atol = 0
+    rtol = 1e-7  # default for np.allclose()
+
+    # Relax atol on 32-Bit platforms a bit to pass CI tests.
+    #  - Not clear why there are discrepancies (in the FFT maybe?)
+    #  - Not sure what changed on 'i686' since earlier on those test passed
+    if x.dtype == np.float32 and platform.machine() == 'i686':
+        # float32 gets only used by TestSTFT.test_roundtrip_float32() so
+        # we are using the tolerances from there to circumvent CI problems
+        atol, rtol = 1e-4, 1e-5
+    elif platform.machine() in ('aarch64', 'i386', 'i686'):
+        atol = max(atol, 1e-12)  # 2e-15 seems too tight for 32-Bit platforms
+
+    assert_allclose(x_wrapper[k_lo:k_hi], x[k_lo:k_hi], atol=atol, rtol=rtol,
+                    err_msg=f"Signal values {e_msg_part}")
+    return t, x

venv/lib/python3.13/site-packages/scipy/signal/tests/mpsig.py

@@ -0,0 +1,122 @@
+"""
+Some signal functions implemented using mpmath.
+"""
+
+try:
+    import mpmath
+except ImportError:
+    mpmath = None
+
+
+def _prod(seq):
+    """Returns the product of the elements in the sequence `seq`."""
+    p = 1
+    for elem in seq:
+        p *= elem
+    return p
+
+
+def _relative_degree(z, p):
+    """
+    Return relative degree of transfer function from zeros and poles.
+
+    This is simply len(p) - len(z), which must be nonnegative.
+    A ValueError is raised if len(p) < len(z).
+    """
+    degree = len(p) - len(z)
+    if degree < 0:
+        raise ValueError("Improper transfer function. "
+                         "Must have at least as many poles as zeros.")
+    return degree
+
+
+def _zpkbilinear(z, p, k, fs):
+    """Bilinear transformation to convert a filter from analog to digital."""
+
+    degree = _relative_degree(z, p)
+
+    fs2 = 2*fs
+
+    # Bilinear transform the poles and zeros
+    z_z = [(fs2 + z1) / (fs2 - z1) for z1 in z]
+    p_z = [(fs2 + p1) / (fs2 - p1) for p1 in p]
+
+    # Any zeros that were at infinity get moved to the Nyquist frequency
+    z_z.extend([-1] * degree)
+
+    # Compensate for gain change
+    numer = _prod(fs2 - z1 for z1 in z)
+    denom = _prod(fs2 - p1 for p1 in p)
+    k_z = k * numer / denom
+
+    return z_z, p_z, k_z.real
+
+
+def _zpklp2lp(z, p, k, wo=1):
+    """Transform a lowpass filter to a different cutoff frequency."""
+
+    degree = _relative_degree(z, p)
+
+    # Scale all points radially from origin to shift cutoff frequency
+    z_lp = [wo * z1 for z1 in z]
+    p_lp = [wo * p1 for p1 in p]
+
+    # Each shifted pole decreases gain by wo, each shifted zero increases it.
+    # Cancel out the net change to keep overall gain the same
+    k_lp = k * wo**degree
+
+    return z_lp, p_lp, k_lp
+
+
+def _butter_analog_poles(n):
+    """
+    Poles of an analog Butterworth lowpass filter.
+
+    This is the same calculation as scipy.signal.buttap(n) or
+    scipy.signal.butter(n, 1, analog=True, output='zpk'), but mpmath is used,
+    and only the poles are returned.
+    """
+    poles = [-mpmath.exp(1j*mpmath.pi*k/(2*n)) for k in range(-n+1, n, 2)]
+    return poles
+
+
+def butter_lp(n, Wn):
+    """
+    Lowpass Butterworth digital filter design.
+
+    This computes the same result as scipy.signal.butter(n, Wn, output='zpk'),
+    but it uses mpmath, and the results are returned in lists instead of NumPy
+    arrays.
+    """
+    zeros = []
+    poles = _butter_analog_poles(n)
+    k = 1
+    fs = 2
+    warped = 2 * fs * mpmath.tan(mpmath.pi * Wn / fs)
+    z, p, k = _zpklp2lp(zeros, poles, k, wo=warped)
+    z, p, k = _zpkbilinear(z, p, k, fs=fs)
+    return z, p, k
+
+
+def zpkfreqz(z, p, k, worN=None):
+    """
+    Frequency response of a filter in zpk format, using mpmath.
+
+    This is the same calculation as scipy.signal.freqz, but the input is in
+    zpk format, the calculation is performed using mpath, and the results are
+    returned in lists instead of NumPy arrays.
+    """
+    if worN is None or isinstance(worN, int):
+        N = worN or 512
+        ws = [mpmath.pi * mpmath.mpf(j) / N for j in range(N)]
+    else:
+        ws = worN
+
+    h = []
+    for wk in ws:
+        zm1 = mpmath.exp(1j * wk)
+        numer = _prod([zm1 - t for t in z])
+        denom = _prod([zm1 - t for t in p])
+        hk = k * numer / denom
+        h.append(hk)
+    return ws, h

venv/lib/python3.13/site-packages/scipy/signal/tests/test_array_tools.py

@@ -0,0 +1,111 @@
+import numpy as np
+
+from scipy._lib._array_api import xp_assert_equal
+from pytest import raises as assert_raises
+
+from scipy.signal._arraytools import (axis_slice, axis_reverse,
+     odd_ext, even_ext, const_ext, zero_ext)
+
+
+class TestArrayTools:
+
+    def test_axis_slice(self):
+        a = np.arange(12).reshape(3, 4)
+
+        s = axis_slice(a, start=0, stop=1, axis=0)
+        xp_assert_equal(s, a[0:1, :])
+
+        s = axis_slice(a, start=-1, axis=0)
+        xp_assert_equal(s, a[-1:, :])
+
+        s = axis_slice(a, start=0, stop=1, axis=1)
+        xp_assert_equal(s, a[:, 0:1])
+
+        s = axis_slice(a, start=-1, axis=1)
+        xp_assert_equal(s, a[:, -1:])
+
+        s = axis_slice(a, start=0, step=2, axis=0)
+        xp_assert_equal(s, a[::2, :])
+
+        s = axis_slice(a, start=0, step=2, axis=1)
+        xp_assert_equal(s, a[:, ::2])
+
+    def test_axis_reverse(self):
+        a = np.arange(12).reshape(3, 4)
+
+        r = axis_reverse(a, axis=0)
+        xp_assert_equal(r, a[::-1, :])
+
+        r = axis_reverse(a, axis=1)
+        xp_assert_equal(r, a[:, ::-1])
+
+    def test_odd_ext(self):
+        a = np.array([[1, 2, 3, 4, 5],
+                      [9, 8, 7, 6, 5]])
+
+        odd = odd_ext(a, 2, axis=1)
+        expected = np.array([[-1, 0, 1, 2, 3, 4, 5, 6, 7],
+                             [11, 10, 9, 8, 7, 6, 5, 4, 3]])
+        xp_assert_equal(odd, expected)
+
+        odd = odd_ext(a, 1, axis=0)
+        expected = np.array([[-7, -4, -1, 2, 5],
+                             [1, 2, 3, 4, 5],
+                             [9, 8, 7, 6, 5],
+                             [17, 14, 11, 8, 5]])
+        xp_assert_equal(odd, expected)
+
+        assert_raises(ValueError, odd_ext, a, 2, axis=0)
+        assert_raises(ValueError, odd_ext, a, 5, axis=1)
+
+    def test_even_ext(self):
+        a = np.array([[1, 2, 3, 4, 5],
+                      [9, 8, 7, 6, 5]])
+
+        even = even_ext(a, 2, axis=1)
+        expected = np.array([[3, 2, 1, 2, 3, 4, 5, 4, 3],
+                             [7, 8, 9, 8, 7, 6, 5, 6, 7]])
+        xp_assert_equal(even, expected)
+
+        even = even_ext(a, 1, axis=0)
+        expected = np.array([[9, 8, 7, 6, 5],
+                             [1, 2, 3, 4, 5],
+                             [9, 8, 7, 6, 5],
+                             [1, 2, 3, 4, 5]])
+        xp_assert_equal(even, expected)
+
+        assert_raises(ValueError, even_ext, a, 2, axis=0)
+        assert_raises(ValueError, even_ext, a, 5, axis=1)
+
+    def test_const_ext(self):
+        a = np.array([[1, 2, 3, 4, 5],
+                      [9, 8, 7, 6, 5]])
+
+        const = const_ext(a, 2, axis=1)
+        expected = np.array([[1, 1, 1, 2, 3, 4, 5, 5, 5],
+                             [9, 9, 9, 8, 7, 6, 5, 5, 5]])
+        xp_assert_equal(const, expected)
+
+        const = const_ext(a, 1, axis=0)
+        expected = np.array([[1, 2, 3, 4, 5],
+                             [1, 2, 3, 4, 5],
+                             [9, 8, 7, 6, 5],
+                             [9, 8, 7, 6, 5]])
+        xp_assert_equal(const, expected)
+
+    def test_zero_ext(self):
+        a = np.array([[1, 2, 3, 4, 5],
+                      [9, 8, 7, 6, 5]])
+
+        zero = zero_ext(a, 2, axis=1)
+        expected = np.array([[0, 0, 1, 2, 3, 4, 5, 0, 0],
+                             [0, 0, 9, 8, 7, 6, 5, 0, 0]])
+        xp_assert_equal(zero, expected)
+
+        zero = zero_ext(a, 1, axis=0)
+        expected = np.array([[0, 0, 0, 0, 0],
+                             [1, 2, 3, 4, 5],
+                             [9, 8, 7, 6, 5],
+                             [0, 0, 0, 0, 0]])
+        xp_assert_equal(zero, expected)
+

venv/lib/python3.13/site-packages/scipy/signal/tests/test_bsplines.py

@@ -0,0 +1,365 @@
+# pylint: disable=missing-docstring
+import math
+import numpy as np
+
+from scipy._lib._array_api import (
+    assert_almost_equal, xp_assert_close, xp_assert_equal
+)
+import pytest
+from pytest import raises
+
+from scipy import signal
+
+skip_xp_backends = pytest.mark.skip_xp_backends
+xfail_xp_backends = pytest.mark.xfail_xp_backends
+
+
+class TestBSplines:
+    """Test behaviors of B-splines. Some of the values tested against were
+    returned as of SciPy 1.1.0 and are included for regression testing
+    purposes. Others (at integer points) are compared to theoretical
+    expressions (cf. Unser, Aldroubi, Eden, IEEE TSP 1993, Table 1)."""
+
+    @skip_xp_backends(cpu_only=True, exceptions=["cupy"])
+    def test_spline_filter(self, xp):
+        rng = np.random.RandomState(12457)
+        # Test the type-error branch
+        raises(TypeError, signal.spline_filter, xp.asarray([0]), 0)
+        # Test the real branch
+        data_array_real = rng.rand(12, 12)
+        # make the magnitude exceed 1, and make some negative
+        data_array_real = 10*(1-2*data_array_real)
+        data_array_real = xp.asarray(data_array_real)
+        result_array_real = xp.asarray(
+            [[-.463312621, 8.33391222, .697290949, 5.28390836,
+              5.92066474, 6.59452137, 9.84406950, -8.78324188,
+              7.20675750, -8.17222994, -4.38633345, 9.89917069],
+             [2.67755154, 6.24192170, -3.15730578, 9.87658581,
+              -9.96930425, 3.17194115, -4.50919947, 5.75423446,
+              9.65979824, -8.29066885, .971416087, -2.38331897],
+             [-7.08868346, 4.89887705, -1.37062289, 7.70705838,
+              2.51526461, 3.65885497, 5.16786604, -8.77715342e-03,
+              4.10533325, 9.04761993, -.577960351, 9.86382519],
+             [-4.71444301, -1.68038985, 2.84695116, 1.14315938,
+              -3.17127091, 1.91830461, 7.13779687, -5.35737482,
+              -9.66586425, -9.87717456, 9.93160672, 4.71948144],
+             [9.49551194, -1.92958436, 6.25427993, -9.05582911,
+              3.97562282, 7.68232426, -1.04514824, -5.86021443,
+              -8.43007451, 5.47528997, 2.06330736, -8.65968112],
+             [-8.91720100, 8.87065356, 3.76879937, 2.56222894,
+              -.828387146, 8.72288903, 6.42474741, -6.84576083,
+              9.94724115, 6.90665380, -6.61084494, -9.44907391],
+             [9.25196790, -.774032030, 7.05371046, -2.73505725,
+              2.53953305, -1.82889155, 2.95454824, -1.66362046,
+              5.72478916, -3.10287679, 1.54017123, -7.87759020],
+             [-3.98464539, -2.44316992, -1.12708657, 1.01725672,
+              -8.89294671, -5.42145629, -6.16370321, 2.91775492,
+              9.64132208, .702499998, -2.02622392, 1.56308431],
+             [-2.22050773, 7.89951554, 5.98970713, -7.35861835,
+              5.45459283, -7.76427957, 3.67280490, -4.05521315,
+              4.51967507, -3.22738749, -3.65080177, 3.05630155],
+             [-6.21240584, -.296796126, -8.34800163, 9.21564563,
+              -3.61958784, -4.77120006, -3.99454057, 1.05021988e-03,
+              -6.95982829, 6.04380797, 8.43181250, -2.71653339],
+             [1.19638037, 6.99718842e-02, 6.72020394, -2.13963198,
+              3.75309875, -5.70076744, 5.92143551, -7.22150575,
+              -3.77114594, -1.11903194, -5.39151466, 3.06620093],
+             [9.86326886, 1.05134482, -7.75950607, -3.64429655,
+              7.81848957, -9.02270373, 3.73399754, -4.71962549,
+              -7.71144306, 3.78263161, 6.46034818, -4.43444731]], dtype=xp.float64)
+        xp_assert_close(signal.spline_filter(data_array_real, 0),
+                        result_array_real)
+
+    @skip_xp_backends(cpu_only=True, exceptions=["cupy"])
+    def test_spline_filter_complex(self, xp):
+        rng = np.random.RandomState(12457)
+        data_array_complex = rng.rand(7, 7) + rng.rand(7, 7)*1j
+        # make the magnitude exceed 1, and make some negative
+        data_array_complex = 10*(1+1j-2*data_array_complex)
+        data_array_complex = xp.asarray(data_array_complex)
+
+        result_array_complex = xp.asarray(
+            [[-4.61489230e-01-1.92994022j, 8.33332443+6.25519943j,
+              6.96300745e-01-9.05576038j, 5.28294849+3.97541356j,
+              5.92165565+7.68240595j, 6.59493160-1.04542804j,
+              9.84503460-5.85946894j],
+             [-8.78262329-8.4295969j, 7.20675516+5.47528982j,
+              -8.17223072+2.06330729j, -4.38633347-8.65968037j,
+              9.89916801-8.91720295j, 2.67755103+8.8706522j,
+              6.24192142+3.76879835j],
+             [-3.15627527+2.56303072j, 9.87658501-0.82838702j,
+              -9.96930313+8.72288895j, 3.17193985+6.42474651j,
+              -4.50919819-6.84576082j, 5.75423431+9.94723988j,
+              9.65979767+6.90665293j],
+             [-8.28993416-6.61064005j, 9.71416473e-01-9.44907284j,
+              -2.38331890+9.25196648j, -7.08868170-0.77403212j,
+              4.89887714+7.05371094j, -1.37062311-2.73505688j,
+              7.70705748+2.5395329j],
+             [2.51528406-1.82964492j, 3.65885472+2.95454836j,
+              5.16786575-1.66362023j, -8.77737999e-03+5.72478867j,
+              4.10533333-3.10287571j, 9.04761887+1.54017115j,
+              -5.77960968e-01-7.87758923j],
+             [9.86398506-3.98528528j, -4.71444130-2.44316983j,
+              -1.68038976-1.12708664j, 2.84695053+1.01725709j,
+              1.14315915-8.89294529j, -3.17127085-5.42145538j,
+              1.91830420-6.16370344j],
+             [7.13875294+2.91851187j, -5.35737514+9.64132309j,
+              -9.66586399+0.70250005j, -9.87717438-2.0262239j,
+              9.93160629+1.5630846j, 4.71948051-2.22050714j,
+              9.49550819+7.8995142j]], dtype=xp.complex128)
+        # FIXME: for complex types, the computations are done in
+        # single precision (reason unclear). When this is changed,
+        # this test needs updating.
+        xp_assert_close(signal.spline_filter(data_array_complex, 0),
+                        result_array_complex, rtol=1e-6)
+
+    def test_gauss_spline(self, xp):
+        assert math.isclose(signal.gauss_spline(0, 0), 1.381976597885342)
+
+        xp_assert_close(signal.gauss_spline(xp.asarray([1.]), 1),
+                        xp.asarray([0.04865217]), atol=1e-9
+        )
+
+    @skip_xp_backends(np_only=True, reason="deliberate: array-likes are accepted")
+    def test_gauss_spline_list(self, xp):
+        # regression test for gh-12152 (accept array_like)
+        knots = [-1.0, 0.0, -1.0]
+        assert_almost_equal(signal.gauss_spline(knots, 3),
+                            np.asarray([0.15418033, 0.6909883, 0.15418033])
+        )
+
+    @skip_xp_backends(cpu_only=True)
+    def test_cspline1d(self, xp):
+        xp_assert_equal(signal.cspline1d(xp.asarray([0])),
+                        xp.asarray([0.], dtype=xp.float64))
+        c1d = xp.asarray([1.21037185, 1.86293902, 2.98834059, 4.11660378,
+                          4.78893826], dtype=xp.float64)
+        # test lamda != 0
+        xp_assert_close(signal.cspline1d(xp.asarray([1., 2, 3, 4, 5]), 1), c1d)
+        c1d0 = xp.asarray([0.78683946, 2.05333735, 2.99981113, 3.94741812,
+                           5.21051638], dtype=xp.float64)
+        xp_assert_close(signal.cspline1d(xp.asarray([1., 2, 3, 4, 5])), c1d0)
+
+    @skip_xp_backends(cpu_only=True)
+    def test_qspline1d(self, xp):
+        xp_assert_equal(signal.qspline1d(xp.asarray([0])),
+                        xp.asarray([0.], dtype=xp.float64))
+        # test lamda != 0
+        raises(ValueError, signal.qspline1d, xp.asarray([1., 2, 3, 4, 5]), 1.)
+        raises(ValueError, signal.qspline1d, xp.asarray([1., 2, 3, 4, 5]), -1.)
+        q1d0 = xp.asarray([0.85350007, 2.02441743, 2.99999534, 3.97561055,
+                           5.14634135], dtype=xp.float64)
+        xp_assert_close(
+            signal.qspline1d(xp.asarray([1., 2, 3, 4, 5], dtype=xp.float64)), q1d0
+        )
+
+    @skip_xp_backends(cpu_only=True)
+    def test_cspline1d_eval(self, xp):
+        r = signal.cspline1d_eval(xp.asarray([0., 0], dtype=xp.float64),
+                               xp.asarray([0.], dtype=xp.float64))
+        xp_assert_close(r, xp.asarray([0.], dtype=xp.float64))
+
+        r = signal.cspline1d_eval(xp.asarray([1., 0, 1], dtype=xp.float64),
+                               xp.asarray([], dtype=xp.float64))
+        xp_assert_equal(r, xp.asarray([], dtype=xp.float64))
+        x = [-3, -2, -1, 0, 1, 2, 3, 4, 5, 6]
+        dx = x[1] - x[0]
+        newx = [-6., -5.5, -5., -4.5, -4., -3.5, -3., -2.5, -2., -1.5, -1.,
+                -0.5, 0., 0.5, 1., 1.5, 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5, 6.,
+                6.5, 7., 7.5, 8., 8.5, 9., 9.5, 10., 10.5, 11., 11.5, 12.,
+                12.5]
+        y = xp.asarray([4.216, 6.864, 3.514, 6.203, 6.759, 7.433, 7.874, 5.879,
+                        1.396, 4.094])
+        cj = signal.cspline1d(y)
+        newy = xp.asarray([6.203, 4.41570658, 3.514, 5.16924703, 6.864, 6.04643068,
+                           4.21600281, 6.04643068, 6.864, 5.16924703, 3.514,
+                           4.41570658, 6.203, 6.80717667, 6.759, 6.98971173, 7.433,
+                           7.79560142, 7.874, 7.41525761, 5.879, 3.18686814, 1.396,
+                           2.24889482, 4.094, 2.24889482, 1.396, 3.18686814, 5.879,
+                           7.41525761, 7.874, 7.79560142, 7.433, 6.98971173, 6.759,
+                           6.80717667, 6.203, 4.41570658], dtype=xp.float64)
+        xp_assert_close(
+            signal.cspline1d_eval(cj, xp.asarray(newx), dx=dx, x0=x[0]), newy
+        )
+
+        with pytest.raises(ValueError,
+                            match="Spline coefficients 'cj' must not be empty."):
+            signal.cspline1d_eval(xp.asarray([], dtype=xp.float64), 
+                                  xp.asarray([0.0], dtype=xp.float64))
+
+    @skip_xp_backends(cpu_only=True)
+    def test_qspline1d_eval(self, xp):
+        xp_assert_close(signal.qspline1d_eval(xp.asarray([0., 0]), xp.asarray([0.])),
+                        xp.asarray([0.])
+        )
+        xp_assert_equal(signal.qspline1d_eval(xp.asarray([1., 0, 1]), xp.asarray([])),
+                        xp.asarray([])
+        )
+        x = [-3, -2, -1, 0, 1, 2, 3, 4, 5, 6]
+        dx = x[1] - x[0]
+        newx = [-6., -5.5, -5., -4.5, -4., -3.5, -3., -2.5, -2., -1.5, -1.,
+                -0.5, 0., 0.5, 1., 1.5, 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5, 6.,
+                6.5, 7., 7.5, 8., 8.5, 9., 9.5, 10., 10.5, 11., 11.5, 12.,
+                12.5]
+        y = xp.asarray([4.216, 6.864, 3.514, 6.203, 6.759, 7.433, 7.874, 5.879,
+                        1.396, 4.094])
+        cj = signal.qspline1d(y)
+        newy = xp.asarray([6.203, 4.49418159, 3.514, 5.18390821, 6.864, 5.91436915,
+                           4.21600002, 5.91436915, 6.864, 5.18390821, 3.514,
+                           4.49418159, 6.203, 6.71900226, 6.759, 7.03980488, 7.433,
+                           7.81016848, 7.874, 7.32718426, 5.879, 3.23872593, 1.396,
+                           2.34046013, 4.094, 2.34046013, 1.396, 3.23872593, 5.879,
+                           7.32718426, 7.874, 7.81016848, 7.433, 7.03980488, 6.759,
+                           6.71900226, 6.203, 4.49418159], dtype=xp.float64)
+        r = signal.qspline1d_eval(
+            cj, xp.asarray(newx, dtype=xp.float64), dx=dx, x0=x[0]
+        )
+        xp_assert_close(r, newy)
+
+        with pytest.raises(ValueError, 
+                           match="Spline coefficients 'cj' must not be empty."):
+            signal.qspline1d_eval(xp.asarray([], dtype=xp.float64), 
+                                  xp.asarray([0.0], dtype=xp.float64))
+
+
+# i/o dtypes with scipy 1.9.1, likely fixed by backwards compat
+sepfir_dtype_map = {np.uint8: np.float32, int: np.float64,
+                    np.float32: np.float32, float: float,
+                    np.complex64: np.complex64, complex: complex}
+
+
+@skip_xp_backends(np_only=True)
+class TestSepfir2d:
+    def test_sepfir2d_invalid_filter(self, xp):
+        filt = xp.asarray([1.0, 2.0, 4.0, 2.0, 1.0])
+        image = np.random.rand(7, 9)
+        image = xp.asarray(image)
+        # No error for odd lengths
+        signal.sepfir2d(image, filt, filt[2:])
+
+        # Row or column filter must be odd
+        with pytest.raises(ValueError, match="odd length"):
+            signal.sepfir2d(image, filt, filt[1:])
+        with pytest.raises(ValueError, match="odd length"):
+            signal.sepfir2d(image, filt[1:], filt)
+
+        # Filters must be 1-dimensional
+        with pytest.raises(ValueError, match="object too deep"):
+            signal.sepfir2d(image, xp.reshape(filt, (1, -1)), filt)
+        with pytest.raises(ValueError, match="object too deep"):
+            signal.sepfir2d(image, filt, xp.reshape(filt, (1, -1)))
+
+    def test_sepfir2d_invalid_image(self, xp):
+        filt = xp.asarray([1.0, 2.0, 4.0, 2.0, 1.0])
+        image = np.random.rand(8, 8)
+        image = xp.asarray(image)
+
+        # Image must be 2 dimensional
+        with pytest.raises(ValueError, match="object too deep"):
+            signal.sepfir2d(xp.reshape(image, (4, 4, 4)), filt, filt)
+
+        with pytest.raises(ValueError, match="object of too small depth"):
+            signal.sepfir2d(image[0, :], filt, filt)
+
+    @pytest.mark.parametrize('dtyp',
+        [np.uint8, int, np.float32, float, np.complex64, complex]
+    )
+    def test_simple(self, dtyp, xp):
+        # test values on a paper-and-pencil example
+        a = np.array([[1, 2, 3, 3, 2, 1],
+                      [1, 2, 3, 3, 2, 1],
+                      [1, 2, 3, 3, 2, 1],
+                      [1, 2, 3, 3, 2, 1]], dtype=dtyp)
+        h1 = [0.5, 1, 0.5]
+        h2 = [1]
+        result = signal.sepfir2d(a, h1, h2)
+        dt = sepfir_dtype_map[dtyp]
+        expected = np.asarray([[2.5, 4. , 5.5, 5.5, 4. , 2.5],
+                               [2.5, 4. , 5.5, 5.5, 4. , 2.5],
+                               [2.5, 4. , 5.5, 5.5, 4. , 2.5],
+                               [2.5, 4. , 5.5, 5.5, 4. , 2.5]], dtype=dt)
+        xp_assert_close(result, expected, atol=1e-16)
+
+        result = signal.sepfir2d(a, h2, h1)
+        expected = np.asarray([[2., 4., 6., 6., 4., 2.],
+                               [2., 4., 6., 6., 4., 2.],
+                               [2., 4., 6., 6., 4., 2.],
+                               [2., 4., 6., 6., 4., 2.]], dtype=dt)
+        xp_assert_close(result, expected, atol=1e-16)
+
+    @skip_xp_backends(np_only=True, reason="TODO: convert this test")
+    @pytest.mark.parametrize('dtyp',
+        [np.uint8, int, np.float32, float, np.complex64, complex]
+    )
+    def test_strided(self, dtyp, xp):
+        a = np.array([[1, 2, 3, 3, 2, 1, 1, 2, 3],
+                     [1, 2, 3, 3, 2, 1, 1, 2, 3],
+                     [1, 2, 3, 3, 2, 1, 1, 2, 3],
+                     [1, 2, 3, 3, 2, 1, 1, 2, 3]])
+        h1, h2 = [0.5, 1, 0.5], [1]
+        result_strided = signal.sepfir2d(a[:, ::2], h1, h2)
+        result_contig = signal.sepfir2d(a[:, ::2].copy(), h1, h2)
+        xp_assert_close(result_strided, result_contig, atol=1e-15)
+        assert result_strided.dtype == result_contig.dtype
+
+    @skip_xp_backends(np_only=True, reason="TODO: convert this test")
+    @pytest.mark.xfail(reason="XXX: filt.size > image.shape: flaky")
+    def test_sepfir2d_strided_2(self, xp):
+        # XXX: this test is flaky: fails on some reruns, with
+        # result[0, 1] and result[1, 1] being ~1e+224.
+        filt = np.array([1.0, 2.0, 4.0, 2.0, 1.0, 3.0, 2.0])
+        image = np.random.rand(4, 4)
+
+        expected = np.asarray([[36.018162, 30.239061, 38.71187 , 43.878183],
+                                [38.180999, 35.824583, 43.525247, 43.874945],
+                                [43.269533, 40.834018, 46.757772, 44.276423],
+                                [49.120928, 39.681844, 43.596067, 45.085854]])
+        xp_assert_close(signal.sepfir2d(image, filt, filt[::3]), expected)
+
+    @skip_xp_backends(np_only=True, reason="TODO: convert this test")
+    @pytest.mark.xfail(reason="XXX: flaky. pointers OOB on some platforms")
+    @pytest.mark.parametrize('dtyp',
+        [np.uint8, int, np.float32, float, np.complex64, complex]
+    )
+    def test_sepfir2d_strided_3(self, dtyp, xp):
+        # NB: 'image' and 'filt' dtypes match here. Otherwise we can run into
+        # unsafe casting errors for many combinations. Historically, dtype handling
+        # in `sepfir2d` is a tad baroque; fixing it is an enhancement.
+        filt = np.array([1, 2, 4, 2, 1, 3, 2], dtype=dtyp)
+        image = np.asarray([[0, 3, 0, 1, 2],
+                            [2, 2, 3, 3, 3],
+                            [0, 1, 3, 0, 3],
+                            [2, 3, 0, 1, 3],
+                            [3, 3, 2, 1, 2]], dtype=dtyp)
+
+        expected = [[123., 101.,  91., 136., 127.],
+                    [133., 125., 126., 152., 160.],
+                    [136., 137., 150., 162., 177.],
+                    [133., 124., 132., 148., 147.],
+                    [173., 158., 152., 164., 141.]]
+        expected = np.asarray(expected)
+        result = signal.sepfir2d(image, filt, filt[::3])
+        xp_assert_close(result, expected, atol=1e-15)
+        assert result.dtype == sepfir_dtype_map[dtyp]
+
+        expected = [[22., 35., 41., 31., 47.],
+                    [27., 39., 48., 47., 55.],
+                    [33., 42., 49., 53., 59.],
+                    [39., 44., 41., 36., 48.],
+                    [67., 62., 47., 34., 46.]]
+        expected = np.asarray(expected)
+        result = signal.sepfir2d(image, filt[::3], filt[::3])
+        xp_assert_close(result, expected, atol=1e-15)
+        assert result.dtype == sepfir_dtype_map[dtyp]
+
+
+def test_cspline2d(xp):
+    rng = np.random.RandomState(181819142)
+    image = rng.rand(71, 73)
+    signal.cspline2d(image, 8.0)
+
+
+def test_qspline2d(xp):
+    rng = np.random.RandomState(181819143)
+    image = rng.rand(71, 73)
+    signal.qspline2d(image)

venv/lib/python3.13/site-packages/scipy/signal/tests/test_cont2discrete.py

@@ -0,0 +1,424 @@
+import numpy as np
+from scipy._lib._array_api import (
+    assert_array_almost_equal, assert_almost_equal, xp_assert_close
+)
+
+import pytest
+from scipy.signal import cont2discrete as c2d
+from scipy.signal import dlsim, ss2tf, ss2zpk, lsim, lti
+from scipy.signal import tf2ss, impulse, dimpulse, step, dstep
+
+# Author: Jeffrey Armstrong <jeff@approximatrix.com>
+# March 29, 2011
+
+
+class TestC2D:
+    def test_zoh(self):
+        ac = np.eye(2, dtype=np.float64)
+        bc = np.full((2, 1), 0.5, dtype=np.float64)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [-0.33]])
+
+        ad_truth = 1.648721270700128 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.324360635350064)
+        # c and d in discrete should be equal to their continuous counterparts
+        dt_requested = 0.5
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='zoh')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cc, cd)
+        assert_array_almost_equal(dc, dd)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_foh(self):
+        ac = np.eye(2)
+        bc = np.full((2, 1), 0.5)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [-0.33]])
+
+        # True values are verified with Matlab
+        ad_truth = 1.648721270700128 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.420839287058789)
+        cd_truth = cc
+        dd_truth = np.array([[0.260262223725224],
+                             [0.297442541400256],
+                             [-0.144098411624840]])
+        dt_requested = 0.5
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='foh')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_impulse(self):
+        ac = np.eye(2)
+        bc = np.full((2, 1), 0.5)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [0.0]])
+
+        # True values are verified with Matlab
+        ad_truth = 1.648721270700128 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.412180317675032)
+        cd_truth = cc
+        dd_truth = np.array([[0.4375], [0.5], [0.3125]])
+        dt_requested = 0.5
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
+                                 method='impulse')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_gbt(self):
+        ac = np.eye(2)
+        bc = np.full((2, 1), 0.5)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [-0.33]])
+
+        dt_requested = 0.5
+        alpha = 1.0 / 3.0
+
+        ad_truth = 1.6 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.3)
+        cd_truth = np.array([[0.9, 1.2],
+                             [1.2, 1.2],
+                             [1.2, 0.3]])
+        dd_truth = np.array([[0.175],
+                             [0.2],
+                             [-0.205]])
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
+                                 method='gbt', alpha=alpha)
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+
+    def test_euler(self):
+        ac = np.eye(2)
+        bc = np.full((2, 1), 0.5)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [-0.33]])
+
+        dt_requested = 0.5
+
+        ad_truth = 1.5 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.25)
+        cd_truth = np.array([[0.75, 1.0],
+                             [1.0, 1.0],
+                             [1.0, 0.25]])
+        dd_truth = dc
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
+                                 method='euler')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_backward_diff(self):
+        ac = np.eye(2)
+        bc = np.full((2, 1), 0.5)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [-0.33]])
+
+        dt_requested = 0.5
+
+        ad_truth = 2.0 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.5)
+        cd_truth = np.array([[1.5, 2.0],
+                             [2.0, 2.0],
+                             [2.0, 0.5]])
+        dd_truth = np.array([[0.875],
+                             [1.0],
+                             [0.295]])
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
+                                 method='backward_diff')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+
+    def test_bilinear(self):
+        ac = np.eye(2)
+        bc = np.full((2, 1), 0.5)
+        cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
+        dc = np.array([[0.0], [0.0], [-0.33]])
+
+        dt_requested = 0.5
+
+        ad_truth = (5.0 / 3.0) * np.eye(2)
+        bd_truth = np.full((2, 1), 1.0 / 3.0)
+        cd_truth = np.array([[1.0, 4.0 / 3.0],
+                             [4.0 / 3.0, 4.0 / 3.0],
+                             [4.0 / 3.0, 1.0 / 3.0]])
+        dd_truth = np.array([[0.291666666666667],
+                             [1.0 / 3.0],
+                             [-0.121666666666667]])
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
+                                 method='bilinear')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+        assert_almost_equal(dt_requested, dt)
+
+        # Same continuous system again, but change sampling rate
+
+        ad_truth = 1.4 * np.eye(2)
+        bd_truth = np.full((2, 1), 0.2)
+        cd_truth = np.array([[0.9, 1.2], [1.2, 1.2], [1.2, 0.3]])
+        dd_truth = np.array([[0.175], [0.2], [-0.205]])
+
+        dt_requested = 1.0 / 3.0
+
+        ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
+                                 method='bilinear')
+
+        assert_array_almost_equal(ad_truth, ad)
+        assert_array_almost_equal(bd_truth, bd)
+        assert_array_almost_equal(cd_truth, cd)
+        assert_array_almost_equal(dd_truth, dd)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_transferfunction(self):
+        numc = np.array([0.25, 0.25, 0.5])
+        denc = np.array([0.75, 0.75, 1.0])
+
+        numd = np.array([[1.0 / 3.0, -0.427419169438754, 0.221654141101125]])
+        dend = np.array([1.0, -1.351394049721225, 0.606530659712634])
+
+        dt_requested = 0.5
+
+        num, den, dt = c2d((numc, denc), dt_requested, method='zoh')
+
+        assert_array_almost_equal(numd, num)
+        assert_array_almost_equal(dend, den)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_zerospolesgain(self):
+        zeros_c = np.array([0.5, -0.5])
+        poles_c = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
+        k_c = 1.0
+
+        zeros_d = [1.23371727305860, 0.735356894461267]
+        polls_d = [0.938148335039729 + 0.346233593780536j,
+                   0.938148335039729 - 0.346233593780536j]
+        k_d = 1.0
+
+        dt_requested = 0.5
+
+        zeros, poles, k, dt = c2d((zeros_c, poles_c, k_c), dt_requested,
+                                  method='zoh')
+
+        assert_array_almost_equal(zeros_d, zeros)
+        assert_array_almost_equal(polls_d, poles)
+        assert_almost_equal(k_d, k)
+        assert_almost_equal(dt_requested, dt)
+
+    def test_gbt_with_sio_tf_and_zpk(self):
+        """Test method='gbt' with alpha=0.25 for tf and zpk cases."""
+        # State space coefficients for the continuous SIO system.
+        A = -1.0
+        B = 1.0
+        C = 1.0
+        D = 0.5
+
+        # The continuous transfer function coefficients.
+        cnum, cden = ss2tf(A, B, C, D)
+
+        # Continuous zpk representation
+        cz, cp, ck = ss2zpk(A, B, C, D)
+
+        h = 1.0
+        alpha = 0.25
+
+        # Explicit formulas, in the scalar case.
+        Ad = (1 + (1 - alpha) * h * A) / (1 - alpha * h * A)
+        Bd = h * B / (1 - alpha * h * A)
+        Cd = C / (1 - alpha * h * A)
+        Dd = D + alpha * C * Bd
+
+        # Convert the explicit solution to tf
+        dnum, dden = ss2tf(Ad, Bd, Cd, Dd)
+
+        # Compute the discrete tf using cont2discrete.
+        c2dnum, c2dden, dt = c2d((cnum, cden), h, method='gbt', alpha=alpha)
+
+        xp_assert_close(dnum, c2dnum)
+        xp_assert_close(dden, c2dden)
+
+        # Convert explicit solution to zpk.
+        dz, dp, dk = ss2zpk(Ad, Bd, Cd, Dd)
+
+        # Compute the discrete zpk using cont2discrete.
+        c2dz, c2dp, c2dk, dt = c2d((cz, cp, ck), h, method='gbt', alpha=alpha)
+
+        xp_assert_close(dz, c2dz)
+        xp_assert_close(dp, c2dp)
+        xp_assert_close(dk, c2dk)
+
+    def test_discrete_approx(self):
+        """
+        Test that the solution to the discrete approximation of a continuous
+        system actually approximates the solution to the continuous system.
+        This is an indirect test of the correctness of the implementation
+        of cont2discrete.
+        """
+
+        def u(t):
+            return np.sin(2.5 * t)
+
+        a = np.array([[-0.01]])
+        b = np.array([[1.0]])
+        c = np.array([[1.0]])
+        d = np.array([[0.2]])
+        x0 = 1.0
+
+        t = np.linspace(0, 10.0, 101)
+        dt = t[1] - t[0]
+        u1 = u(t)
+
+        # Use lsim to compute the solution to the continuous system.
+        t, yout, xout = lsim((a, b, c, d), T=t, U=u1, X0=x0)
+
+        # Convert the continuous system to a discrete approximation.
+        dsys = c2d((a, b, c, d), dt, method='bilinear')
+
+        # Use dlsim with the pairwise averaged input to compute the output
+        # of the discrete system.
+        u2 = 0.5 * (u1[:-1] + u1[1:])
+        t2 = t[:-1]
+        td2, yd2, xd2 = dlsim(dsys, u=u2.reshape(-1, 1), t=t2, x0=x0)
+
+        # ymid is the average of consecutive terms of the "exact" output
+        # computed by lsim2.  This is what the discrete approximation
+        # actually approximates.
+        ymid = 0.5 * (yout[:-1] + yout[1:])
+
+        xp_assert_close(yd2.ravel(), ymid, rtol=1e-4)
+
+    def test_simo_tf(self):
+        # See gh-5753
+        tf = ([[1, 0], [1, 1]], [1, 1])
+        num, den, dt = c2d(tf, 0.01)
+
+        assert dt == 0.01  # sanity check
+        xp_assert_close(den, [1, -0.990404983], rtol=1e-3)
+        xp_assert_close(num, [[1, -1], [1, -0.99004983]], rtol=1e-3)
+
+    def test_multioutput(self):
+        ts = 0.01  # time step
+
+        tf = ([[1, -3], [1, 5]], [1, 1])
+        num, den, dt = c2d(tf, ts)
+
+        tf1 = (tf[0][0], tf[1])
+        num1, den1, dt1 = c2d(tf1, ts)
+
+        tf2 = (tf[0][1], tf[1])
+        num2, den2, dt2 = c2d(tf2, ts)
+
+        # Sanity checks
+        assert dt == dt1
+        assert dt == dt2
+
+        # Check that we get the same results
+        xp_assert_close(num, np.vstack((num1, num2)), rtol=1e-13)
+
+        # Single input, so the denominator should
+        # not be multidimensional like the numerator
+        xp_assert_close(den, den1, rtol=1e-13)
+        xp_assert_close(den, den2, rtol=1e-13)
+
+class TestC2dLti:
+    def test_c2d_ss(self):
+        # StateSpace
+        A = np.array([[-0.3, 0.1], [0.2, -0.7]])
+        B = np.array([[0], [1]])
+        C = np.array([[1, 0]])
+        D = 0
+        dt = 0.05
+
+        A_res = np.array([[0.985136404135682, 0.004876671474795],
+                          [0.009753342949590, 0.965629718236502]])
+        B_res = np.array([[0.000122937599964], [0.049135527547844]])
+
+        sys_ssc = lti(A, B, C, D)
+        sys_ssd = sys_ssc.to_discrete(dt=dt)
+
+        xp_assert_close(sys_ssd.A, A_res)
+        xp_assert_close(sys_ssd.B, B_res)
+        xp_assert_close(sys_ssd.C, C)
+        xp_assert_close(sys_ssd.D, np.zeros_like(sys_ssd.D))
+
+        sys_ssd2 = c2d(sys_ssc, dt=dt)
+
+        xp_assert_close(sys_ssd2.A, A_res)
+        xp_assert_close(sys_ssd2.B, B_res)
+        xp_assert_close(sys_ssd2.C, C)
+        xp_assert_close(sys_ssd2.D, np.zeros_like(sys_ssd2.D))
+
+    def test_c2d_tf(self):
+
+        sys = lti([0.5, 0.3], [1.0, 0.4])
+        sys = sys.to_discrete(0.005)
+
+        # Matlab results
+        num_res = np.array([0.5, -0.485149004980066])
+        den_res = np.array([1.0, -0.980198673306755])
+
+        # Somehow a lot of numerical errors
+        xp_assert_close(sys.den, den_res, atol=0.02)
+        xp_assert_close(sys.num, num_res, atol=0.02)
+
+
+class TestC2dInvariants:
+    # Some test cases for checking the invariances.
+    # Array of triplets: (system, sample time, number of samples)
+    cases = [
+        (tf2ss([1, 1], [1, 1.5, 1]), 0.25, 10),
+        (tf2ss([1, 2], [1, 1.5, 3, 1]), 0.5, 10),
+        (tf2ss(0.1, [1, 1, 2, 1]), 0.5, 10),
+    ]
+
+    # Check that systems discretized with the impulse-invariant
+    # method really hold the invariant
+    @pytest.mark.parametrize("sys,sample_time,samples_number", cases)
+    def test_impulse_invariant(self, sys, sample_time, samples_number):
+        time = np.arange(samples_number) * sample_time
+        _, yout_cont = impulse(sys, T=time)
+        _, yout_disc = dimpulse(c2d(sys, sample_time, method='impulse'),
+                                n=len(time))
+        xp_assert_close(sample_time * yout_cont.ravel(), yout_disc[0].ravel())
+
+    # Step invariant should hold for ZOH discretized systems
+    @pytest.mark.parametrize("sys,sample_time,samples_number", cases)
+    def test_step_invariant(self, sys, sample_time, samples_number):
+        time = np.arange(samples_number) * sample_time
+        _, yout_cont = step(sys, T=time)
+        _, yout_disc = dstep(c2d(sys, sample_time, method='zoh'), n=len(time))
+        xp_assert_close(yout_cont.ravel(), yout_disc[0].ravel())
+
+    # Linear invariant should hold for FOH discretized systems
+    @pytest.mark.parametrize("sys,sample_time,samples_number", cases)
+    def test_linear_invariant(self, sys, sample_time, samples_number):
+        time = np.arange(samples_number) * sample_time
+        _, yout_cont, _ = lsim(sys, T=time, U=time)
+        _, yout_disc, _ = dlsim(c2d(sys, sample_time, method='foh'), u=time)
+        xp_assert_close(yout_cont.ravel(), yout_disc.ravel())

venv/lib/python3.13/site-packages/scipy/signal/tests/test_czt.py

@@ -0,0 +1,221 @@
+# This program is public domain
+# Authors: Paul Kienzle, Nadav Horesh
+'''
+A unit test module for czt.py
+'''
+import pytest
+from scipy._lib._array_api import xp_assert_close
+from scipy.fft import fft
+from scipy.signal import (czt, zoom_fft, czt_points, CZT, ZoomFFT)
+import numpy as np
+
+
+def check_czt(x):
+    # Check that czt is the equivalent of normal fft
+    y = fft(x)
+    y1 = czt(x)
+    xp_assert_close(y1, y, rtol=1e-13)
+
+    # Check that interpolated czt is the equivalent of normal fft
+    y = fft(x, 100*len(x))
+    y1 = czt(x, 100*len(x))
+    xp_assert_close(y1, y, rtol=1e-12)
+
+
+def check_zoom_fft(x):
+    # Check that zoom_fft is the equivalent of normal fft
+    y = fft(x)
+    y1 = zoom_fft(x, [0, 2-2./len(y)], endpoint=True)
+    xp_assert_close(y1, y, rtol=1e-11, atol=1e-14)
+    y1 = zoom_fft(x, [0, 2])
+    xp_assert_close(y1, y, rtol=1e-11, atol=1e-14)
+
+    # Test fn scalar
+    y1 = zoom_fft(x, 2-2./len(y), endpoint=True)
+    xp_assert_close(y1, y, rtol=1e-11, atol=1e-14)
+    y1 = zoom_fft(x, 2)
+    xp_assert_close(y1, y, rtol=1e-11, atol=1e-14)
+
+    # Check that zoom_fft with oversampling is equivalent to zero padding
+    over = 10
+    yover = fft(x, over*len(x))
+    y2 = zoom_fft(x, [0, 2-2./len(yover)], m=len(yover), endpoint=True)
+    xp_assert_close(y2, yover, rtol=1e-12, atol=1e-10)
+    y2 = zoom_fft(x, [0, 2], m=len(yover))
+    xp_assert_close(y2, yover, rtol=1e-12, atol=1e-10)
+
+    # Check that zoom_fft works on a subrange
+    w = np.linspace(0, 2-2./len(x), len(x))
+    f1, f2 = w[3], w[6]
+    y3 = zoom_fft(x, [f1, f2], m=3*over+1, endpoint=True)
+    idx3 = slice(3*over, 6*over+1)
+    xp_assert_close(y3, yover[idx3], rtol=1e-13)
+
+
+def test_1D():
+    # Test of 1D version of the transforms
+
+    rng = np.random.RandomState(0)  # Deterministic randomness
+
+    # Random signals
+    lengths = rng.randint(8, 200, 20)
+    np.append(lengths, 1)
+    for length in lengths:
+        x = rng.random(length)
+        check_zoom_fft(x)
+        check_czt(x)
+
+    # Gauss
+    t = np.linspace(-2, 2, 128)
+    x = np.exp(-t**2/0.01)
+    check_zoom_fft(x)
+
+    # Linear
+    x = [1, 2, 3, 4, 5, 6, 7]
+    check_zoom_fft(x)
+
+    # Check near powers of two
+    check_zoom_fft(range(126-31))
+    check_zoom_fft(range(127-31))
+    check_zoom_fft(range(128-31))
+    check_zoom_fft(range(129-31))
+    check_zoom_fft(range(130-31))
+
+    # Check transform on n-D array input
+    x = np.reshape(np.arange(3*2*28), (3, 2, 28))
+    y1 = zoom_fft(x, [0, 2-2./28])
+    y2 = zoom_fft(x[2, 0, :], [0, 2-2./28])
+    xp_assert_close(y1[2, 0], y2, rtol=1e-13, atol=1e-12)
+
+    y1 = zoom_fft(x, [0, 2], endpoint=False)
+    y2 = zoom_fft(x[2, 0, :], [0, 2], endpoint=False)
+    xp_assert_close(y1[2, 0], y2, rtol=1e-13, atol=1e-12)
+
+    # Random (not a test condition)
+    x = rng.rand(101)
+    check_zoom_fft(x)
+
+    # Spikes
+    t = np.linspace(0, 1, 128)
+    x = np.sin(2*np.pi*t*5)+np.sin(2*np.pi*t*13)
+    check_zoom_fft(x)
+
+    # Sines
+    x = np.zeros(100, dtype=complex)
+    x[[1, 5, 21]] = 1
+    check_zoom_fft(x)
+
+    # Sines plus complex component
+    x += 1j*np.linspace(0, 0.5, x.shape[0])
+    check_zoom_fft(x)
+
+
+def test_large_prime_lengths():
+    rng = np.random.RandomState(0)  # Deterministic randomness
+    for N in (101, 1009, 10007):
+        x = rng.rand(N)
+        y = fft(x)
+        y1 = czt(x)
+        xp_assert_close(y, y1, rtol=1e-12)
+
+
+@pytest.mark.slow
+def test_czt_vs_fft():
+    rng = np.random.RandomState(123)  # Deterministic randomness
+    random_lengths = rng.exponential(100000, size=10).astype('int')
+    for n in random_lengths:
+        a = rng.randn(n)
+        xp_assert_close(czt(a), fft(a), rtol=1e-11)
+
+
+def test_empty_input():
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        czt([])
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        zoom_fft([], 0.5)
+
+
+def test_0_rank_input():
+    with pytest.raises(IndexError, match='tuple index out of range'):
+        czt(5)
+    with pytest.raises(IndexError, match='tuple index out of range'):
+        zoom_fft(5, 0.5)
+
+
+@pytest.mark.parametrize('impulse', ([0, 0, 1], [0, 0, 1, 0, 0],
+                                     np.concatenate((np.array([0, 0, 1]),
+                                                     np.zeros(100)))))
+@pytest.mark.parametrize('m', (1, 3, 5, 8, 101, 1021))
+@pytest.mark.parametrize('a', (1, 2, 0.5, 1.1))
+# Step that tests away from the unit circle, but not so far it explodes from
+# numerical error
+@pytest.mark.parametrize('w', (None, 0.98534 + 0.17055j))
+def test_czt_math(impulse, m, w, a):
+    # z-transform of an impulse is 1 everywhere
+    xp_assert_close(czt(impulse[2:], m=m, w=w, a=a),
+                    np.ones(m, dtype=np.complex128), rtol=1e-10)
+
+    # z-transform of a delayed impulse is z**-1
+    xp_assert_close(czt(impulse[1:], m=m, w=w, a=a),
+                    czt_points(m=m, w=w, a=a)**-1, rtol=1e-10)
+
+    # z-transform of a 2-delayed impulse is z**-2
+    xp_assert_close(czt(impulse, m=m, w=w, a=a),
+                    czt_points(m=m, w=w, a=a)**-2, rtol=1e-10)
+
+
+def test_int_args():
+    # Integer argument `a` was producing all 0s
+    xp_assert_close(abs(czt([0, 1], m=10, a=2)), 0.5*np.ones(10), rtol=1e-15)
+    xp_assert_close(czt_points(11, w=2),
+                    1/(2**np.arange(11, dtype=np.complex128)), rtol=1e-30)
+
+
+def test_czt_points():
+    for N in (1, 2, 3, 8, 11, 100, 101, 10007):
+        xp_assert_close(czt_points(N), np.exp(2j*np.pi*np.arange(N)/N),
+                        rtol=1e-30)
+
+    xp_assert_close(czt_points(7, w=1), np.ones(7, dtype=np.complex128), rtol=1e-30)
+    xp_assert_close(czt_points(11, w=2.),
+                    1/(2**np.arange(11, dtype=np.complex128)), rtol=1e-30)
+
+    func = CZT(12, m=11, w=2., a=1)
+    xp_assert_close(func.points(), 1/(2**np.arange(11)), rtol=1e-30)
+
+
+@pytest.mark.parametrize('cls, args', [(CZT, (100,)), (ZoomFFT, (100, 0.2))])
+def test_CZT_size_mismatch(cls, args):
+    # Data size doesn't match function's expected size
+    myfunc = cls(*args)
+    with pytest.raises(ValueError, match='CZT defined for'):
+        myfunc(np.arange(5))
+
+
+def test_invalid_range():
+    with pytest.raises(ValueError, match='2-length sequence'):
+        ZoomFFT(100, [1, 2, 3])
+
+
+@pytest.mark.parametrize('m', [0, -11, 5.5, 4.0])
+def test_czt_points_errors(m):
+    # Invalid number of points
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        czt_points(m)
+
+
+@pytest.mark.parametrize('size', [0, -5, 3.5, 4.0])
+def test_nonsense_size(size):
+    # Numpy and Scipy fft() give ValueError for 0 output size, so we do, too
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        CZT(size, 3)
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        ZoomFFT(size, 0.2, 3)
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        CZT(3, size)
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        ZoomFFT(3, 0.2, size)
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        czt([1, 2, 3], size)
+    with pytest.raises(ValueError, match='Invalid number of CZT'):
+        zoom_fft([1, 2, 3], 0.2, size)

venv/lib/python3.13/site-packages/scipy/signal/tests/test_dltisys.py

@@ -0,0 +1,599 @@
+# Author: Jeffrey Armstrong <jeff@approximatrix.com>
+# April 4, 2011
+
+import numpy as np
+from numpy.testing import suppress_warnings
+from pytest import raises as assert_raises
+from scipy._lib._array_api import (
+    assert_array_almost_equal, assert_almost_equal, xp_assert_close, xp_assert_equal,
+)
+
+from scipy.signal import (dlsim, dstep, dimpulse, tf2zpk, lti, dlti,
+                          StateSpace, TransferFunction, ZerosPolesGain,
+                          dfreqresp, dbode, BadCoefficients)
+
+
+class TestDLTI:
+
+    def test_dlsim(self):
+
+        a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
+        b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
+        c = np.asarray([[0.1, 0.3]])
+        d = np.asarray([[0.0, -0.1, 0.0]])
+        dt = 0.5
+
+        # Create an input matrix with inputs down the columns (3 cols) and its
+        # respective time input vector
+        u = np.hstack((np.linspace(0, 4.0, num=5)[:, np.newaxis],
+                       np.full((5, 1), 0.01),
+                       np.full((5, 1), -0.002)))
+        t_in = np.linspace(0, 2.0, num=5)
+
+        # Define the known result
+        yout_truth = np.array([[-0.001,
+                                -0.00073,
+                                0.039446,
+                                0.0915387,
+                                0.13195948]]).T
+        xout_truth = np.asarray([[0, 0],
+                                 [0.0012, 0.0005],
+                                 [0.40233, 0.00071],
+                                 [1.163368, -0.079327],
+                                 [2.2402985, -0.3035679]])
+
+        tout, yout, xout = dlsim((a, b, c, d, dt), u, t_in)
+
+        assert_array_almost_equal(yout_truth, yout)
+        assert_array_almost_equal(xout_truth, xout)
+        assert_array_almost_equal(t_in, tout)
+
+        # Make sure input with single-dimension doesn't raise error
+        dlsim((1, 2, 3), 4)
+
+        # Interpolated control - inputs should have different time steps
+        # than the discrete model uses internally
+        u_sparse = u[[0, 4], :]
+        t_sparse = np.asarray([0.0, 2.0])
+
+        tout, yout, xout = dlsim((a, b, c, d, dt), u_sparse, t_sparse)
+
+        assert_array_almost_equal(yout_truth, yout)
+        assert_array_almost_equal(xout_truth, xout)
+        assert len(tout) == len(yout)
+
+        # Transfer functions (assume dt = 0.5)
+        num = np.asarray([1.0, -0.1])
+        den = np.asarray([0.3, 1.0, 0.2])
+        yout_truth = np.array([[0.0,
+                                0.0,
+                                3.33333333333333,
+                                -4.77777777777778,
+                                23.0370370370370]]).T
+
+        # Assume use of the first column of the control input built earlier
+        tout, yout = dlsim((num, den, 0.5), u[:, 0], t_in)
+
+        assert_array_almost_equal(yout, yout_truth)
+        assert_array_almost_equal(t_in, tout)
+
+        # Retest the same with a 1-D input vector
+        uflat = np.asarray(u[:, 0])
+        uflat = uflat.reshape((5,))
+        tout, yout = dlsim((num, den, 0.5), uflat, t_in)
+
+        assert_array_almost_equal(yout, yout_truth)
+        assert_array_almost_equal(t_in, tout)
+
+        # zeros-poles-gain representation
+        zd = np.array([0.5, -0.5])
+        pd = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
+        k = 1.0
+        yout_truth = np.array([[0.0, 1.0, 2.0, 2.25, 2.5]]).T
+
+        tout, yout = dlsim((zd, pd, k, 0.5), u[:, 0], t_in)
+
+        assert_array_almost_equal(yout, yout_truth)
+        assert_array_almost_equal(t_in, tout)
+
+        # Raise an error for continuous-time systems
+        system = lti([1], [1, 1])
+        assert_raises(AttributeError, dlsim, system, u)
+
+    def test_dstep(self):
+
+        a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
+        b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
+        c = np.asarray([[0.1, 0.3]])
+        d = np.asarray([[0.0, -0.1, 0.0]])
+        dt = 0.5
+
+        # Because b.shape[1] == 3, dstep should result in a tuple of three
+        # result vectors
+        yout_step_truth = (np.asarray([0.0, 0.04, 0.052, 0.0404, 0.00956,
+                                       -0.036324, -0.093318, -0.15782348,
+                                       -0.226628324, -0.2969374948]),
+                           np.asarray([-0.1, -0.075, -0.058, -0.04815,
+                                       -0.04453, -0.0461895, -0.0521812,
+                                       -0.061588875, -0.073549579,
+                                       -0.08727047595]),
+                           np.asarray([0.0, -0.01, -0.013, -0.0101, -0.00239,
+                                       0.009081, 0.0233295, 0.03945587,
+                                       0.056657081, 0.0742343737]))
+
+        tout, yout = dstep((a, b, c, d, dt), n=10)
+
+        assert len(yout) == 3
+
+        for i in range(0, len(yout)):
+            assert yout[i].shape[0] == 10
+            assert_array_almost_equal(yout[i].flatten(), yout_step_truth[i])
+
+        # Check that the other two inputs (tf, zpk) will work as well
+        tfin = ([1.0], [1.0, 1.0], 0.5)
+        yout_tfstep = np.asarray([0.0, 1.0, 0.0])
+        tout, yout = dstep(tfin, n=3)
+        assert len(yout) == 1
+        assert_array_almost_equal(yout[0].flatten(), yout_tfstep)
+
+        zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,)
+        tout, yout = dstep(zpkin, n=3)
+        assert len(yout) == 1
+        assert_array_almost_equal(yout[0].flatten(), yout_tfstep)
+
+        # Raise an error for continuous-time systems
+        system = lti([1], [1, 1])
+        assert_raises(AttributeError, dstep, system)
+
+    def test_dimpulse(self):
+
+        a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
+        b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
+        c = np.asarray([[0.1, 0.3]])
+        d = np.asarray([[0.0, -0.1, 0.0]])
+        dt = 0.5
+
+        # Because b.shape[1] == 3, dimpulse should result in a tuple of three
+        # result vectors
+        yout_imp_truth = (np.asarray([0.0, 0.04, 0.012, -0.0116, -0.03084,
+                                      -0.045884, -0.056994, -0.06450548,
+                                      -0.068804844, -0.0703091708]),
+                          np.asarray([-0.1, 0.025, 0.017, 0.00985, 0.00362,
+                                      -0.0016595, -0.0059917, -0.009407675,
+                                      -0.011960704, -0.01372089695]),
+                          np.asarray([0.0, -0.01, -0.003, 0.0029, 0.00771,
+                                      0.011471, 0.0142485, 0.01612637,
+                                      0.017201211, 0.0175772927]))
+
+        tout, yout = dimpulse((a, b, c, d, dt), n=10)
+
+        assert len(yout) == 3
+
+        for i in range(0, len(yout)):
+            assert yout[i].shape[0] == 10
+            assert_array_almost_equal(yout[i].flatten(), yout_imp_truth[i])
+
+        # Check that the other two inputs (tf, zpk) will work as well
+        tfin = ([1.0], [1.0, 1.0], 0.5)
+        yout_tfimpulse = np.asarray([0.0, 1.0, -1.0])
+        tout, yout = dimpulse(tfin, n=3)
+        assert len(yout) == 1
+        assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse)
+
+        zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,)
+        tout, yout = dimpulse(zpkin, n=3)
+        assert len(yout) == 1
+        assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse)
+
+        # Raise an error for continuous-time systems
+        system = lti([1], [1, 1])
+        assert_raises(AttributeError, dimpulse, system)
+
+    def test_dlsim_trivial(self):
+        a = np.array([[0.0]])
+        b = np.array([[0.0]])
+        c = np.array([[0.0]])
+        d = np.array([[0.0]])
+        n = 5
+        u = np.zeros(n).reshape(-1, 1)
+        tout, yout, xout = dlsim((a, b, c, d, 1), u)
+        xp_assert_equal(tout, np.arange(float(n)))
+        xp_assert_equal(yout, np.zeros((n, 1)))
+        xp_assert_equal(xout, np.zeros((n, 1)))
+
+    def test_dlsim_simple1d(self):
+        a = np.array([[0.5]])
+        b = np.array([[0.0]])
+        c = np.array([[1.0]])
+        d = np.array([[0.0]])
+        n = 5
+        u = np.zeros(n).reshape(-1, 1)
+        tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1)
+        xp_assert_equal(tout, np.arange(float(n)))
+        expected = (0.5 ** np.arange(float(n))).reshape(-1, 1)
+        xp_assert_equal(yout, expected)
+        xp_assert_equal(xout, expected)
+
+    def test_dlsim_simple2d(self):
+        lambda1 = 0.5
+        lambda2 = 0.25
+        a = np.array([[lambda1, 0.0],
+                      [0.0, lambda2]])
+        b = np.array([[0.0],
+                      [0.0]])
+        c = np.array([[1.0, 0.0],
+                      [0.0, 1.0]])
+        d = np.array([[0.0],
+                      [0.0]])
+        n = 5
+        u = np.zeros(n).reshape(-1, 1)
+        tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1)
+        xp_assert_equal(tout, np.arange(float(n)))
+        # The analytical solution:
+        expected = (np.array([lambda1, lambda2]) **
+                                np.arange(float(n)).reshape(-1, 1))
+        xp_assert_equal(yout, expected)
+        xp_assert_equal(xout, expected)
+
+    def test_more_step_and_impulse(self):
+        lambda1 = 0.5
+        lambda2 = 0.75
+        a = np.array([[lambda1, 0.0],
+                      [0.0, lambda2]])
+        b = np.array([[1.0, 0.0],
+                      [0.0, 1.0]])
+        c = np.array([[1.0, 1.0]])
+        d = np.array([[0.0, 0.0]])
+
+        n = 10
+
+        # Check a step response.
+        ts, ys = dstep((a, b, c, d, 1), n=n)
+
+        # Create the exact step response.
+        stp0 = (1.0 / (1 - lambda1)) * (1.0 - lambda1 ** np.arange(n))
+        stp1 = (1.0 / (1 - lambda2)) * (1.0 - lambda2 ** np.arange(n))
+
+        xp_assert_close(ys[0][:, 0], stp0)
+        xp_assert_close(ys[1][:, 0], stp1)
+
+        # Check an impulse response with an initial condition.
+        x0 = np.array([1.0, 1.0])
+        ti, yi = dimpulse((a, b, c, d, 1), n=n, x0=x0)
+
+        # Create the exact impulse response.
+        imp = (np.array([lambda1, lambda2]) **
+                            np.arange(-1, n + 1).reshape(-1, 1))
+        imp[0, :] = 0.0
+        # Analytical solution to impulse response
+        y0 = imp[:n, 0] + np.dot(imp[1:n + 1, :], x0)
+        y1 = imp[:n, 1] + np.dot(imp[1:n + 1, :], x0)
+
+        xp_assert_close(yi[0][:, 0], y0)
+        xp_assert_close(yi[1][:, 0], y1)
+
+        # Check that dt=0.1, n=3 gives 3 time values.
+        system = ([1.0], [1.0, -0.5], 0.1)
+        t, (y,) = dstep(system, n=3)
+        xp_assert_close(t, [0, 0.1, 0.2])
+        xp_assert_equal(y.T, [[0, 1.0, 1.5]])
+        t, (y,) = dimpulse(system, n=3)
+        xp_assert_close(t, [0, 0.1, 0.2])
+        xp_assert_equal(y.T, [[0, 1, 0.5]])
+
+
+class TestDlti:
+    def test_dlti_instantiation(self):
+        # Test that lti can be instantiated.
+
+        dt = 0.05
+        # TransferFunction
+        s = dlti([1], [-1], dt=dt)
+        assert isinstance(s, TransferFunction)
+        assert isinstance(s, dlti)
+        assert not isinstance(s, lti)
+        assert s.dt == dt
+
+        # ZerosPolesGain
+        s = dlti(np.array([]), np.array([-1]), 1, dt=dt)
+        assert isinstance(s, ZerosPolesGain)
+        assert isinstance(s, dlti)
+        assert not isinstance(s, lti)
+        assert s.dt == dt
+
+        # StateSpace
+        s = dlti([1], [-1], 1, 3, dt=dt)
+        assert isinstance(s, StateSpace)
+        assert isinstance(s, dlti)
+        assert not isinstance(s, lti)
+        assert s.dt == dt
+
+        # Number of inputs
+        assert_raises(ValueError, dlti, 1)
+        assert_raises(ValueError, dlti, 1, 1, 1, 1, 1)
+
+
+class TestStateSpaceDisc:
+    def test_initialization(self):
+        # Check that all initializations work
+        dt = 0.05
+        StateSpace(1, 1, 1, 1, dt=dt)
+        StateSpace([1], [2], [3], [4], dt=dt)
+        StateSpace(np.array([[1, 2], [3, 4]]), np.array([[1], [2]]),
+                   np.array([[1, 0]]), np.array([[0]]), dt=dt)
+        StateSpace(1, 1, 1, 1, dt=True)
+
+    def test_conversion(self):
+        # Check the conversion functions
+        s = StateSpace(1, 2, 3, 4, dt=0.05)
+        assert isinstance(s.to_ss(), StateSpace)
+        assert isinstance(s.to_tf(), TransferFunction)
+        assert isinstance(s.to_zpk(), ZerosPolesGain)
+
+        # Make sure copies work
+        assert StateSpace(s) is not s
+        assert s.to_ss() is not s
+
+    def test_properties(self):
+        # Test setters/getters for cross class properties.
+        # This implicitly tests to_tf() and to_zpk()
+
+        # Getters
+        s = StateSpace(1, 1, 1, 1, dt=0.05)
+        xp_assert_equal(s.poles, [1.])
+        xp_assert_equal(s.zeros, [0.])
+
+
+class TestTransferFunction:
+    def test_initialization(self):
+        # Check that all initializations work
+        dt = 0.05
+        TransferFunction(1, 1, dt=dt)
+        TransferFunction([1], [2], dt=dt)
+        TransferFunction(np.array([1]), np.array([2]), dt=dt)
+        TransferFunction(1, 1, dt=True)
+
+    def test_conversion(self):
+        # Check the conversion functions
+        s = TransferFunction([1, 0], [1, -1], dt=0.05)
+        assert isinstance(s.to_ss(), StateSpace)
+        assert isinstance(s.to_tf(), TransferFunction)
+        assert isinstance(s.to_zpk(), ZerosPolesGain)
+
+        # Make sure copies work
+        assert TransferFunction(s) is not s
+        assert s.to_tf() is not s
+
+    def test_properties(self):
+        # Test setters/getters for cross class properties.
+        # This implicitly tests to_ss() and to_zpk()
+
+        # Getters
+        s = TransferFunction([1, 0], [1, -1], dt=0.05)
+        xp_assert_equal(s.poles, [1.])
+        xp_assert_equal(s.zeros, [0.])
+
+
+class TestZerosPolesGain:
+    def test_initialization(self):
+        # Check that all initializations work
+        dt = 0.05
+        ZerosPolesGain(1, 1, 1, dt=dt)
+        ZerosPolesGain([1], [2], 1, dt=dt)
+        ZerosPolesGain(np.array([1]), np.array([2]), 1, dt=dt)
+        ZerosPolesGain(1, 1, 1, dt=True)
+
+    def test_conversion(self):
+        # Check the conversion functions
+        s = ZerosPolesGain(1, 2, 3, dt=0.05)
+        assert isinstance(s.to_ss(), StateSpace)
+        assert isinstance(s.to_tf(), TransferFunction)
+        assert isinstance(s.to_zpk(), ZerosPolesGain)
+
+        # Make sure copies work
+        assert ZerosPolesGain(s) is not s
+        assert s.to_zpk() is not s
+
+
+class Test_dfreqresp:
+
+    def test_manual(self):
+        # Test dfreqresp() real part calculation (manual sanity check).
+        # 1st order low-pass filter: H(z) = 1 / (z - 0.2),
+        system = TransferFunction(1, [1, -0.2], dt=0.1)
+        w = [0.1, 1, 10]
+        w, H = dfreqresp(system, w=w)
+
+        # test real
+        expected_re = [1.2383, 0.4130, -0.7553]
+        assert_almost_equal(H.real, expected_re, decimal=4)
+
+        # test imag
+        expected_im = [-0.1555, -1.0214, 0.3955]
+        assert_almost_equal(H.imag, expected_im, decimal=4)
+
+    def test_auto(self):
+        # Test dfreqresp() real part calculation.
+        # 1st order low-pass filter: H(z) = 1 / (z - 0.2),
+        system = TransferFunction(1, [1, -0.2], dt=0.1)
+        w = [0.1, 1, 10, 100]
+        w, H = dfreqresp(system, w=w)
+        jw = np.exp(w * 1j)
+        y = np.polyval(system.num, jw) / np.polyval(system.den, jw)
+
+        # test real
+        expected_re = y.real
+        assert_almost_equal(H.real, expected_re)
+
+        # test imag
+        expected_im = y.imag
+        assert_almost_equal(H.imag, expected_im)
+
+    def test_freq_range(self):
+        # Test that freqresp() finds a reasonable frequency range.
+        # 1st order low-pass filter: H(z) = 1 / (z - 0.2),
+        # Expected range is from 0.01 to 10.
+        system = TransferFunction(1, [1, -0.2], dt=0.1)
+        n = 10
+        expected_w = np.linspace(0, np.pi, 10, endpoint=False)
+        w, H = dfreqresp(system, n=n)
+        assert_almost_equal(w, expected_w)
+
+    def test_pole_one(self):
+        # Test that freqresp() doesn't fail on a system with a pole at 0.
+        # integrator, pole at zero: H(s) = 1 / s
+        system = TransferFunction([1], [1, -1], dt=0.1)
+
+        with suppress_warnings() as sup:
+            sup.filter(RuntimeWarning, message="divide by zero")
+            sup.filter(RuntimeWarning, message="invalid value encountered")
+            w, H = dfreqresp(system, n=2)
+        assert w[0] == 0.   # a fail would give not-a-number
+
+    def test_error(self):
+        # Raise an error for continuous-time systems
+        system = lti([1], [1, 1])
+        assert_raises(AttributeError, dfreqresp, system)
+
+    def test_from_state_space(self):
+        # H(z) = 2 / z^3 - 0.5 * z^2
+
+        system_TF = dlti([2], [1, -0.5, 0, 0])
+
+        A = np.array([[0.5, 0, 0],
+                      [1, 0, 0],
+                      [0, 1, 0]])
+        B = np.array([[1, 0, 0]]).T
+        C = np.array([[0, 0, 2]])
+        D = 0
+
+        system_SS = dlti(A, B, C, D)
+        w = 10.0**np.arange(-3,0,.5)
+        with suppress_warnings() as sup:
+            sup.filter(BadCoefficients)
+            w1, H1 = dfreqresp(system_TF, w=w)
+            w2, H2 = dfreqresp(system_SS, w=w)
+
+        assert_almost_equal(H1, H2)
+
+    def test_from_zpk(self):
+        # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
+        system_ZPK = dlti([],[0.2],0.3)
+        system_TF = dlti(0.3, [1, -0.2])
+        w = [0.1, 1, 10, 100]
+        w1, H1 = dfreqresp(system_ZPK, w=w)
+        w2, H2 = dfreqresp(system_TF, w=w)
+        assert_almost_equal(H1, H2)
+
+
+class Test_bode:
+
+    def test_manual(self):
+        # Test bode() magnitude calculation (manual sanity check).
+        # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
+        dt = 0.1
+        system = TransferFunction(0.3, [1, -0.2], dt=dt)
+        w = [0.1, 0.5, 1, np.pi]
+        w2, mag, phase = dbode(system, w=w)
+
+        # Test mag
+        expected_mag = [-8.5329, -8.8396, -9.6162, -12.0412]
+        assert_almost_equal(mag, expected_mag, decimal=4)
+
+        # Test phase
+        expected_phase = [-7.1575, -35.2814, -67.9809, -180.0000]
+        assert_almost_equal(phase, expected_phase, decimal=4)
+
+        # Test frequency
+        xp_assert_equal(np.array(w) / dt, w2)
+
+    def test_auto(self):
+        # Test bode() magnitude calculation.
+        # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
+        system = TransferFunction(0.3, [1, -0.2], dt=0.1)
+        w = np.array([0.1, 0.5, 1, np.pi])
+        w2, mag, phase = dbode(system, w=w)
+        jw = np.exp(w * 1j)
+        y = np.polyval(system.num, jw) / np.polyval(system.den, jw)
+
+        # Test mag
+        expected_mag = 20.0 * np.log10(abs(y))
+        assert_almost_equal(mag, expected_mag)
+
+        # Test phase
+        expected_phase = np.rad2deg(np.angle(y))
+        assert_almost_equal(phase, expected_phase)
+
+    def test_range(self):
+        # Test that bode() finds a reasonable frequency range.
+        # 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
+        dt = 0.1
+        system = TransferFunction(0.3, [1, -0.2], dt=0.1)
+        n = 10
+        # Expected range is from 0.01 to 10.
+        expected_w = np.linspace(0, np.pi, n, endpoint=False) / dt
+        w, mag, phase = dbode(system, n=n)
+        assert_almost_equal(w, expected_w)
+
+    def test_pole_one(self):
+        # Test that freqresp() doesn't fail on a system with a pole at 0.
+        # integrator, pole at zero: H(s) = 1 / s
+        system = TransferFunction([1], [1, -1], dt=0.1)
+
+        with suppress_warnings() as sup:
+            sup.filter(RuntimeWarning, message="divide by zero")
+            sup.filter(RuntimeWarning, message="invalid value encountered")
+            w, mag, phase = dbode(system, n=2)
+        assert w[0] == 0.  # a fail would give not-a-number
+
+    def test_imaginary(self):
+        # bode() should not fail on a system with pure imaginary poles.
+        # The test passes if bode doesn't raise an exception.
+        system = TransferFunction([1], [1, 0, 100], dt=0.1)
+        dbode(system, n=2)
+
+    def test_error(self):
+        # Raise an error for continuous-time systems
+        system = lti([1], [1, 1])
+        assert_raises(AttributeError, dbode, system)
+
+
+class TestTransferFunctionZConversion:
+    """Test private conversions between 'z' and 'z**-1' polynomials."""
+
+    def test_full(self):
+        # Numerator and denominator same order
+        num = np.asarray([2.0, 3, 4])
+        den = np.asarray([5.0, 6, 7])
+        num2, den2 = TransferFunction._z_to_zinv(num, den)
+        xp_assert_equal(num, num2)
+        xp_assert_equal(den, den2)
+
+        num2, den2 = TransferFunction._zinv_to_z(num, den)
+        xp_assert_equal(num, num2)
+        xp_assert_equal(den, den2)
+
+    def test_numerator(self):
+        # Numerator lower order than denominator
+        num = np.asarray([2.0, 3])
+        den = np.asarray([50, 6, 7])
+        num2, den2 = TransferFunction._z_to_zinv(num, den)
+        xp_assert_equal([0.0, 2, 3], num2)
+        xp_assert_equal(den, den2)
+
+        num2, den2 = TransferFunction._zinv_to_z(num, den)
+        xp_assert_equal([2.0, 3, 0], num2)
+        xp_assert_equal(den, den2)
+
+    def test_denominator(self):
+        # Numerator higher order than denominator
+        num = np.asarray([2., 3, 4])
+        den = np.asarray([5.0, 6])
+        num2, den2 = TransferFunction._z_to_zinv(num, den)
+        xp_assert_equal(num, num2)
+        xp_assert_equal([0.0, 5, 6], den2)
+
+        num2, den2 = TransferFunction._zinv_to_z(num, den)
+        xp_assert_equal(num, num2)
+        xp_assert_equal([5.0, 6, 0], den2)
+

venv/lib/python3.13/site-packages/scipy/signal/tests/test_fir_filter_design.py

@@ -0,0 +1,851 @@
+import math
+import numpy as np
+
+from numpy.testing import assert_warns
+from pytest import raises as assert_raises
+import pytest
+
+import scipy._lib.array_api_extra as xpx
+from scipy._lib._array_api import (
+    xp_assert_close, xp_assert_equal, assert_almost_equal, assert_array_almost_equal,
+    array_namespace, xp_default_dtype
+)
+from scipy.fft import fft, fft2
+from scipy.signal import (kaiser_beta, kaiser_atten, kaiserord,
+    firwin, firwin2, freqz, remez, firls, minimum_phase, convolve2d, firwin_2d
+)
+
+skip_xp_backends = pytest.mark.skip_xp_backends
+xfail_xp_backends = pytest.mark.xfail_xp_backends
+
+
+def test_kaiser_beta():
+    b = kaiser_beta(58.7)
+    assert_almost_equal(b, 0.1102 * 50.0)
+    b = kaiser_beta(22.0)
+    assert_almost_equal(b, 0.5842 + 0.07886)
+    b = kaiser_beta(21.0)
+    assert b == 0.0
+    b = kaiser_beta(10.0)
+    assert b == 0.0
+
+
+def test_kaiser_atten():
+    a = kaiser_atten(1, 1.0)
+    assert a == 7.95
+    a = kaiser_atten(2, 1/np.pi)
+    assert a == 2.285 + 7.95
+
+
+def test_kaiserord():
+    assert_raises(ValueError, kaiserord, 1.0, 1.0)
+    numtaps, beta = kaiserord(2.285 + 7.95 - 0.001, 1/np.pi)
+    assert (numtaps, beta) == (2, 0.0)
+
+
+class TestFirwin:
+
+    def check_response(self, h, expected_response, tol=.05):
+        xp = array_namespace(h)
+        N = h.shape[0]
+        alpha = 0.5 * (N-1)
+        m = xp.arange(0, N) - alpha   # time indices of taps
+        for freq, expected in expected_response:
+            actual = abs(xp.sum(h * xp.exp(-1j * xp.pi * m * freq)))
+            mse = abs(actual - expected)**2
+            assert mse < tol, f'response not as expected, mse={mse:g} > {tol:g}'
+
+    def test_response(self, xp):
+        N = 51
+        f = .5
+
+        # increase length just to try even/odd
+        h = firwin(N, f)  # low-pass from 0 to f
+        self.check_response(h, [(.25,1), (.75,0)])
+
+        h = firwin(N+1, f, window='nuttall')  # specific window
+        self.check_response(h, [(.25,1), (.75,0)])
+
+        h = firwin(N+2, f, pass_zero=False)  # stop from 0 to f --> high-pass
+        self.check_response(h, [(.25,0), (.75,1)])
+
+        f1, f2, f3, f4 = .2, .4, .6, .8
+        h = firwin(N+3, [f1, f2], pass_zero=False)  # band-pass filter
+        self.check_response(h, [(.1,0), (.3,1), (.5,0)])
+
+        h = firwin(N+4, [f1, f2])  # band-stop filter
+        self.check_response(h, [(.1,1), (.3,0), (.5,1)])
+
+        h = firwin(N+5, [f1, f2, f3, f4], pass_zero=False, scale=False)
+        self.check_response(h, [(.1,0), (.3,1), (.5,0), (.7,1), (.9,0)])
+
+        h = firwin(N+6, [f1, f2, f3, f4])  # multiband filter
+        self.check_response(h, [(.1,1), (.3,0), (.5,1), (.7,0), (.9,1)])
+
+        h = firwin(N+7, 0.1, width=.03)  # low-pass
+        self.check_response(h, [(.05,1), (.75,0)])
+
+        h = firwin(N+8, 0.1, pass_zero=False)  # high-pass
+        self.check_response(h, [(.05,0), (.75,1)])
+
+    def mse(self, h, bands):
+        """Compute mean squared error versus ideal response across frequency
+        band.
+          h -- coefficients
+          bands -- list of (left, right) tuples relative to 1==Nyquist of
+            passbands
+        """
+        w, H = freqz(h, worN=1024)
+        f = w/np.pi
+        passIndicator = np.zeros(len(w), bool)
+        for left, right in bands:
+            passIndicator |= (f >= left) & (f < right)
+        Hideal = np.where(passIndicator, 1, 0)
+        mse = np.mean(abs(abs(H)-Hideal)**2)
+        return mse
+
+    def test_scaling(self, xp):
+        """
+        For one lowpass, bandpass, and highpass example filter, this test
+        checks two things:
+          - the mean squared error over the frequency domain of the unscaled
+            filter is smaller than the scaled filter (true for rectangular
+            window)
+          - the response of the scaled filter is exactly unity at the center
+            of the first passband
+        """
+        N = 11
+        cases = [
+            ([.5], True, (0, 1)),
+            ([0.2, .6], False, (.4, 1)),
+            ([.5], False, (1, 1)),
+        ]
+        for cutoff, pass_zero, expected_response in cases:
+            h = firwin(N, cutoff, scale=False, pass_zero=pass_zero, window='ones')
+            hs = firwin(N, cutoff, scale=True, pass_zero=pass_zero, window='ones')
+            if len(cutoff) == 1:
+                if pass_zero:
+                    cutoff = [0] + cutoff
+                else:
+                    cutoff = cutoff + [1]
+            msg = 'least squares violation'
+            assert self.mse(h, [cutoff]) < self.mse(hs, [cutoff]), msg
+            self.check_response(hs, [expected_response], 1e-12)
+
+    def test_fs_validation(self):
+        with pytest.raises(ValueError, match="Sampling.*single scalar"):
+            firwin(51, .5, fs=np.array([10, 20]))
+
+
+class TestFirWinMore:
+    """Different author, different style, different tests..."""
+
+    def test_lowpass(self, xp):
+        width = 0.04
+        ntaps, beta = kaiserord(120, width)
+        cutoff = xp.asarray(0.5)
+        kwargs = dict(cutoff=cutoff, window=('kaiser', beta), scale=False)
+        taps = firwin(ntaps, **kwargs)
+
+        # Check the symmetry of taps.
+        assert_array_almost_equal(taps[:ntaps//2], xp.flip(taps)[:ntaps//2])
+
+        # Check the gain at a few samples where
+        # we know it should be approximately 0 or 1.
+        freq_samples = xp.asarray([0.0, 0.25, 0.5-width/2, 0.5+width/2, 0.75, 1.0])
+        freqs, response = freqz(taps, worN=xp.pi*freq_samples)
+
+        assert_array_almost_equal(
+            xp.abs(response),
+            xp.asarray([1.0, 1.0, 1.0, 0.0, 0.0, 0.0]), decimal=5
+        )
+
+        taps_str = firwin(ntaps, pass_zero='lowpass', **kwargs)
+        xp_assert_close(taps, taps_str)
+
+    def test_highpass(self, xp):
+        width = 0.04
+        ntaps, beta = kaiserord(120, width)
+
+        # Ensure that ntaps is odd.
+        ntaps |= 1
+
+        cutoff = xp.asarray(0.5)
+        kwargs = dict(cutoff=cutoff, window=('kaiser', beta), scale=False)
+        taps = firwin(ntaps, pass_zero=False, **kwargs)
+
+        # Check the symmetry of taps.
+        assert_array_almost_equal(taps[:ntaps//2], xp.flip(taps)[:ntaps//2])
+
+        # Check the gain at a few samples where
+        # we know it should be approximately 0 or 1.
+        freq_samples = xp.asarray([0.0, 0.25, 0.5 - width/2, 0.5 + width/2, 0.75, 1.0])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples)
+
+        assert_array_almost_equal(xp.abs(response),
+                                  xp.asarray([0.0, 0.0, 0.0, 1.0, 1.0, 1.0]), decimal=5)
+
+        taps_str = firwin(ntaps, pass_zero='highpass', **kwargs)
+        xp_assert_close(taps, taps_str)
+
+    def test_bandpass(self, xp):
+        width = 0.04
+        ntaps, beta = kaiserord(120, width)
+        kwargs = dict(
+            cutoff=xp.asarray([0.3, 0.7]), window=('kaiser', beta), scale=False
+        )
+        taps = firwin(ntaps, pass_zero=False, **kwargs)
+
+        # Check the symmetry of taps.
+        assert_array_almost_equal(taps[:ntaps//2], xp.flip(taps)[:ntaps//2])
+
+        # Check the gain at a few samples where
+        # we know it should be approximately 0 or 1.
+        freq_samples = xp.asarray([0.0, 0.2, 0.3 - width/2, 0.3 + width/2, 0.5,
+                                   0.7 - width/2, 0.7 + width/2, 0.8, 1.0])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples)
+
+        assert_array_almost_equal(xp.abs(response),
+                xp.asarray([0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0]), decimal=5)
+
+        taps_str = firwin(ntaps, pass_zero='bandpass', **kwargs)
+        xp_assert_close(taps, taps_str)
+
+    def test_bandstop_multi(self, xp):
+        width = 0.04
+        ntaps, beta = kaiserord(120, width)
+        kwargs = dict(cutoff=xp.asarray([0.2, 0.5, 0.8]), window=('kaiser', beta),
+                      scale=False)
+        taps = firwin(ntaps, **kwargs)
+
+        # Check the symmetry of taps.
+        assert_array_almost_equal(taps[:ntaps//2], xp.flip(taps)[:ntaps//2])
+
+        # Check the gain at a few samples where
+        # we know it should be approximately 0 or 1.
+        freq_samples = xp.asarray([0.0, 0.1, 0.2 - width/2, 0.2 + width/2, 0.35,
+                                   0.5 - width/2, 0.5 + width/2, 0.65,
+                                   0.8 - width/2, 0.8 + width/2, 0.9, 1.0])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples)
+
+        assert_array_almost_equal(
+            xp.abs(response),
+            xp.asarray([1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0]),
+            decimal=5
+        )
+
+        taps_str = firwin(ntaps, pass_zero='bandstop', **kwargs)
+        xp_assert_close(taps, taps_str)
+
+    def test_fs_nyq(self, xp):
+        """Test the fs and nyq keywords."""
+        nyquist = 1000
+        width = 40.0
+        relative_width = width/nyquist
+        ntaps, beta = kaiserord(120, relative_width)
+        taps = firwin(ntaps, cutoff=xp.asarray([300, 700]), window=('kaiser', beta),
+                        pass_zero=False, scale=False, fs=2*nyquist)
+
+        # Check the symmetry of taps.
+        assert_array_almost_equal(taps[:ntaps//2], xp.flip(taps)[:ntaps//2])
+
+        # Check the gain at a few samples where
+        # we know it should be approximately 0 or 1.
+        freq_samples = xp.asarray([0.0, 200, 300 - width/2, 300 + width/2, 500,
+                                   700 - width/2, 700 + width/2, 800, 1000])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples/nyquist)
+
+        assert_array_almost_equal(xp.abs(response),
+                xp.asarray([0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0]), decimal=5)
+
+    def test_array_cutoff(self, xp):
+        taps = firwin(3, xp.asarray([.1, .2]))
+        # smoke test against the value computed by scipy==1.5.2
+        xp_assert_close(
+            taps, xp.asarray([-0.00801395, 1.0160279, -0.00801395]), atol=1e-8
+        )
+
+    def test_bad_cutoff(self):
+        """Test that invalid cutoff argument raises ValueError."""
+        # cutoff values must be greater than 0 and less than 1.
+        assert_raises(ValueError, firwin, 99, -0.5)
+        assert_raises(ValueError, firwin, 99, 1.5)
+        # Don't allow 0 or 1 in cutoff.
+        assert_raises(ValueError, firwin, 99, [0, 0.5])
+        assert_raises(ValueError, firwin, 99, [0.5, 1])
+        # cutoff values must be strictly increasing.
+        assert_raises(ValueError, firwin, 99, [0.1, 0.5, 0.2])
+        assert_raises(ValueError, firwin, 99, [0.1, 0.5, 0.5])
+        # Must have at least one cutoff value.
+        assert_raises(ValueError, firwin, 99, [])
+        # 2D array not allowed.
+        assert_raises(ValueError, firwin, 99, [[0.1, 0.2],[0.3, 0.4]])
+        # cutoff values must be less than nyq.
+        assert_raises(ValueError, firwin, 99, 50.0, fs=80)
+        assert_raises(ValueError, firwin, 99, [10, 20, 30], fs=50)
+
+    def test_even_highpass_raises_value_error(self):
+        """Test that attempt to create a highpass filter with an even number
+        of taps raises a ValueError exception."""
+        assert_raises(ValueError, firwin, 40, 0.5, pass_zero=False)
+        assert_raises(ValueError, firwin, 40, [.25, 0.5])
+
+    def test_bad_pass_zero(self):
+        """Test degenerate pass_zero cases."""
+        with assert_raises(ValueError, match="^Parameter pass_zero='foo' not in "):
+            firwin(41, 0.5, pass_zero='foo')
+        with assert_raises(ValueError, match="^Parameter pass_zero=1.0 not in "):
+            firwin(41, 0.5, pass_zero=1.)
+        for pass_zero in ('lowpass', 'highpass'):
+            with assert_raises(ValueError, match='cutoff must have one'):
+                firwin(41, [0.5, 0.6], pass_zero=pass_zero)
+        for pass_zero in ('bandpass', 'bandstop'):
+            with assert_raises(ValueError, match='must have at least two'):
+                firwin(41, [0.5], pass_zero=pass_zero)
+
+    def test_fs_validation(self):
+        with pytest.raises(ValueError, match="Sampling.*single scalar"):
+            firwin2(51, .5, 1, fs=np.array([10, 20]))
+
+
+@skip_xp_backends(cpu_only=True, reason="firwin2 uses np.interp")
+class TestFirwin2:
+
+    def test_invalid_args(self):
+        # `freq` and `gain` have different lengths.
+        with assert_raises(ValueError, match='must be of same length'):
+            firwin2(50, [0, 0.5, 1], [0.0, 1.0])
+        # `nfreqs` is less than `ntaps`.
+        with assert_raises(ValueError, match='ntaps must be less than nfreqs'):
+            firwin2(50, [0, 0.5, 1], [0.0, 1.0, 1.0], nfreqs=33)
+        # Decreasing value in `freq`
+        with assert_raises(ValueError, match='must be nondecreasing'):
+            firwin2(50, [0, 0.5, 0.4, 1.0], [0, .25, .5, 1.0])
+        # Value in `freq` repeated more than once.
+        with assert_raises(ValueError, match='must not occur more than twice'):
+            firwin2(50, [0, .1, .1, .1, 1.0], [0.0, 0.5, 0.75, 1.0, 1.0])
+        # `freq` does not start at 0.0.
+        with assert_raises(ValueError, match='start with 0'):
+            firwin2(50, [0.5, 1.0], [0.0, 1.0])
+        # `freq` does not end at fs/2.
+        with assert_raises(ValueError, match='end with fs/2'):
+            firwin2(50, [0.0, 0.5], [0.0, 1.0])
+        # Value 0 is repeated in `freq`
+        with assert_raises(ValueError, match='0 must not be repeated'):
+            firwin2(50, [0.0, 0.0, 0.5, 1.0], [1.0, 1.0, 0.0, 0.0])
+        # Value fs/2 is repeated in `freq`
+        with assert_raises(ValueError, match='fs/2 must not be repeated'):
+            firwin2(50, [0.0, 0.5, 1.0, 1.0], [1.0, 1.0, 0.0, 0.0])
+        # Value in `freq` that is too close to a repeated number
+        with assert_raises(ValueError, match='cannot contain numbers '
+                                             'that are too close'):
+            firwin2(50, [0.0, 0.5 - np.finfo(float).eps * 0.5, 0.5, 0.5, 1.0],
+                        [1.0, 1.0, 1.0, 0.0, 0.0])
+
+        # Type II filter, but the gain at nyquist frequency is not zero.
+        with assert_raises(ValueError, match='Type II filter'):
+            firwin2(16, [0.0, 0.5, 1.0], [0.0, 1.0, 1.0])
+
+        # Type III filter, but the gains at nyquist and zero rate are not zero.
+        with assert_raises(ValueError, match='Type III filter'):
+            firwin2(17, [0.0, 0.5, 1.0], [0.0, 1.0, 1.0], antisymmetric=True)
+        with assert_raises(ValueError, match='Type III filter'):
+            firwin2(17, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0], antisymmetric=True)
+        with assert_raises(ValueError, match='Type III filter'):
+            firwin2(17, [0.0, 0.5, 1.0], [1.0, 1.0, 1.0], antisymmetric=True)
+
+        # Type IV filter, but the gain at zero rate is not zero.
+        with assert_raises(ValueError, match='Type IV filter'):
+            firwin2(16, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0], antisymmetric=True)
+
+    def test01(self, xp):
+        width = 0.04
+        beta = 12.0
+        ntaps = 400
+        # Filter is 1 from w=0 to w=0.5, then decreases linearly from 1 to 0 as w
+        # increases from w=0.5 to w=1  (w=1 is the Nyquist frequency).
+        freq = xp.asarray([0.0, 0.5, 1.0])
+        gain = xp.asarray([1.0, 1.0, 0.0])
+        taps = firwin2(ntaps, freq, gain, window=('kaiser', beta))
+        freq_samples = xp.asarray([0.0, 0.25, 0.5 - width/2, 0.5 + width/2,
+                                                        0.75, 1.0 - width/2])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples)
+        freqs, response = xp.asarray(freqs), xp.asarray(response)
+        assert_array_almost_equal(
+            xp.abs(response),
+            xp.asarray([1.0, 1.0, 1.0, 1.0 - width, 0.5, width]), decimal=5
+        )
+
+    @skip_xp_backends("jax.numpy", reason="immutable arrays")
+    def test02(self, xp):
+        width = 0.04
+        beta = 12.0
+        # ntaps must be odd for positive gain at Nyquist.
+        ntaps = 401
+        # An ideal highpass filter.
+        freq = xp.asarray([0.0, 0.5, 0.5, 1.0])
+        gain = xp.asarray([0.0, 0.0, 1.0, 1.0])
+        taps = firwin2(ntaps, freq, gain, window=('kaiser', beta))
+        freq_samples = np.array([0.0, 0.25, 0.5 - width, 0.5 + width, 0.75, 1.0])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples)
+        freqs, response = xp.asarray(freqs), xp.asarray(response)
+        assert_array_almost_equal(
+            xp.abs(response),
+            xp.asarray([0.0, 0.0, 0.0, 1.0, 1.0, 1.0]), decimal=5
+        )
+
+    @skip_xp_backends("jax.numpy", reason="immutable arrays")
+    def test03(self, xp):
+        width = 0.02
+        ntaps, beta = kaiserord(120, width)
+        # ntaps must be odd for positive gain at Nyquist.
+        ntaps = int(ntaps) | 1
+        freq = xp.asarray([0.0, 0.4, 0.4, 0.5, 0.5, 1.0])
+        gain = xp.asarray([1.0, 1.0, 0.0, 0.0, 1.0, 1.0])
+        taps = firwin2(ntaps, freq, gain, window=('kaiser', beta))
+        freq_samples = np.array([0.0, 0.4 - width, 0.4 + width, 0.45,
+                                    0.5 - width, 0.5 + width, 0.75, 1.0])
+        freqs, response = freqz(taps, worN=np.pi*freq_samples)
+        freqs, response = xp.asarray(freqs), xp.asarray(response)
+        assert_array_almost_equal(
+            xp.abs(response),
+            xp.asarray([1.0, 1.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0]), decimal=5
+        )
+
+    @skip_xp_backends("jax.numpy", reason="immutable arrays")
+    def test04(self, xp):
+        """Test firwin2 when window=None."""
+        ntaps = 5
+        # Ideal lowpass: gain is 1 on [0,0.5], and 0 on [0.5, 1.0]
+        freq = xp.asarray([0.0, 0.5, 0.5, 1.0])
+        gain = xp.asarray([1.0, 1.0, 0.0, 0.0])
+
+        taps = firwin2(ntaps, freq, gain, window=None, nfreqs=8193)
+        alpha = 0.5 * (ntaps - 1)
+        m = xp.arange(0, ntaps, dtype=freq.dtype) - alpha
+        h = 0.5 * xpx.sinc(0.5 * m)
+        assert_array_almost_equal(h, taps)
+
+    def test05(self, xp):
+        """Test firwin2 for calculating Type IV filters"""
+        ntaps = 1500
+
+        freq = xp.asarray([0.0, 1.0])
+        gain = xp.asarray([0.0, 1.0])
+        taps = firwin2(ntaps, freq, gain, window=None, antisymmetric=True)
+
+        flip = array_namespace(freq).flip
+        dec = {'decimal': 4.5} if xp_default_dtype(xp) == xp.float32 else {}
+        assert_array_almost_equal(taps[: ntaps // 2], flip(-taps[ntaps // 2:]), **dec)
+
+        freqs, response = freqz(np.asarray(taps), worN=2048)    # XXX convert freqz
+        assert_array_almost_equal(abs(xp.asarray(response)),
+                                  xp.asarray(freqs / np.pi), decimal=4)
+
+    @skip_xp_backends("jax.numpy", reason="immutable arrays")
+    def test06(self, xp):
+        """Test firwin2 for calculating Type III filters"""
+        ntaps = 1501
+
+        freq = xp.asarray([0.0, 0.5, 0.55, 1.0])
+        gain = xp.asarray([0.0, 0.5, 0.0, 0.0])
+        taps = firwin2(ntaps, freq, gain, window=None, antisymmetric=True)
+        assert taps[ntaps // 2] == 0.0
+
+        flip = array_namespace(freq).flip
+        dec = {'decimal': 4.5} if xp_default_dtype(xp) == xp.float32 else {}
+        assert_array_almost_equal(taps[: ntaps // 2],
+                                  flip(-taps[ntaps // 2 + 1:]), **dec
+        )
+
+        freqs, response1 = freqz(np.asarray(taps), worN=2048)  # XXX convert freqz
+        response1 = xp.asarray(response1)
+        response2 = xp.asarray(
+            np.interp(np.asarray(freqs) / np.pi, np.asarray(freq), np.asarray(gain))
+        )
+        assert_array_almost_equal(abs(response1), response2, decimal=3)
+
+    def test_fs_nyq(self, xp):
+        taps1 = firwin2(80, xp.asarray([0.0, 0.5, 1.0]), xp.asarray([1.0, 1.0, 0.0]))
+        taps2 = firwin2(80, xp.asarray([0.0, 30.0, 60.0]), xp.asarray([1.0, 1.0, 0.0]),
+                        fs=120.0)
+        assert_array_almost_equal(taps1, taps2)
+
+    def test_tuple(self):
+        taps1 = firwin2(150, (0.0, 0.5, 0.5, 1.0), (1.0, 1.0, 0.0, 0.0))
+        taps2 = firwin2(150, [0.0, 0.5, 0.5, 1.0], [1.0, 1.0, 0.0, 0.0])
+        assert_array_almost_equal(taps1, taps2)
+
+    @skip_xp_backends("jax.numpy", reason="immutable arrays")
+    def test_input_modyfication(self, xp):
+        freq1 = xp.asarray([0.0, 0.5, 0.5, 1.0])
+        freq2 = xp.asarray(freq1)
+        firwin2(80, freq1, xp.asarray([1.0, 1.0, 0.0, 0.0]))
+        xp_assert_equal(freq1, freq2)
+
+
+@skip_xp_backends(cpu_only=True)
+class TestRemez:
+
+    def test_bad_args(self):
+        assert_raises(ValueError, remez, 11, [0.1, 0.4], [1], type='pooka')
+
+    def test_hilbert(self):
+        N = 11  # number of taps in the filter
+        a = 0.1  # width of the transition band
+
+        # design an unity gain hilbert bandpass filter from w to 0.5-w
+        h = remez(11, [a, 0.5-a], [1], type='hilbert')
+
+        # make sure the filter has correct # of taps
+        assert len(h) == N, "Number of Taps"
+
+        # make sure it is type III (anti-symmetric tap coefficients)
+        assert_array_almost_equal(h[:(N-1)//2], -h[:-(N-1)//2-1:-1])
+
+        # Since the requested response is symmetric, all even coefficients
+        # should be zero (or in this case really small)
+        assert (abs(h[1::2]) < 1e-15).all(), "Even Coefficients Equal Zero"
+
+        # now check the frequency response
+        w, H = freqz(h, 1)
+        f = w/2/np.pi
+        Hmag = abs(H)
+
+        # should have a zero at 0 and pi (in this case close to zero)
+        assert (Hmag[[0, -1]] < 0.02).all(), "Zero at zero and pi"
+
+        # check that the pass band is close to unity
+        idx = np.logical_and(f > a, f < 0.5-a)
+        assert (abs(Hmag[idx] - 1) < 0.015).all(), "Pass Band Close To Unity"
+
+    def test_compare(self, xp):
+        # test comparison to MATLAB
+        k = [0.024590270518440, -0.041314581814658, -0.075943803756711,
+             -0.003530911231040, 0.193140296954975, 0.373400753484939,
+             0.373400753484939, 0.193140296954975, -0.003530911231040,
+             -0.075943803756711, -0.041314581814658, 0.024590270518440]
+        h = remez(12, xp.asarray([0, 0.3, 0.5, 1]), xp.asarray([1, 0]), fs=2.)
+        atol_arg = {'atol': 1e-8} if xp_default_dtype(xp) == xp.float32 else {}
+        xp_assert_close(h, xp.asarray(k, dtype=xp.float64), **atol_arg)
+
+        h = [-0.038976016082299, 0.018704846485491, -0.014644062687875,
+             0.002879152556419, 0.016849978528150, -0.043276706138248,
+             0.073641298245579, -0.103908158578635, 0.129770906801075,
+             -0.147163447297124, 0.153302248456347, -0.147163447297124,
+             0.129770906801075, -0.103908158578635, 0.073641298245579,
+             -0.043276706138248, 0.016849978528150, 0.002879152556419,
+             -0.014644062687875, 0.018704846485491, -0.038976016082299]
+        atol_arg = {'atol': 3e-8} if xp_default_dtype(xp) == xp.float32 else {}
+        xp_assert_close(
+            remez(21, xp.asarray([0, 0.8, 0.9, 1]), xp.asarray([0, 1]), fs=2.),
+            xp.asarray(h, dtype=xp.float64), **atol_arg
+        )
+
+    def test_fs_validation(self):
+        with pytest.raises(ValueError, match="Sampling.*single scalar"):
+            remez(11, .1, 1, fs=np.array([10, 20]))
+
+    def test_gh_23266(self, xp):
+        bands = xp.asarray([0.0, 0.2, 0.3, 0.5])
+        desired = xp.asarray([1.0, 0.0])
+        weight = xp.asarray([1.0, 2.0])
+        remez(21, bands, desired, weight=weight)
+
+
+@skip_xp_backends(cpu_only=True, reason="lstsq")
+class TestFirls:
+
+    def test_bad_args(self):
+        # even numtaps
+        assert_raises(ValueError, firls, 10, [0.1, 0.2], [0, 0])
+        # odd bands
+        assert_raises(ValueError, firls, 11, [0.1, 0.2, 0.4], [0, 0, 0])
+        # len(bands) != len(desired)
+        assert_raises(ValueError, firls, 11, [0.1, 0.2, 0.3, 0.4], [0, 0, 0])
+        # non-monotonic bands
+        assert_raises(ValueError, firls, 11, [0.2, 0.1], [0, 0])
+        assert_raises(ValueError, firls, 11, [0.1, 0.2, 0.3, 0.3], [0] * 4)
+        assert_raises(ValueError, firls, 11, [0.3, 0.4, 0.1, 0.2], [0] * 4)
+        assert_raises(ValueError, firls, 11, [0.1, 0.3, 0.2, 0.4], [0] * 4)
+        # negative desired
+        assert_raises(ValueError, firls, 11, [0.1, 0.2], [-1, 1])
+        # len(weight) != len(pairs)
+        assert_raises(ValueError, firls, 11, [0.1, 0.2], [0, 0], weight=[1, 2])
+        # negative weight
+        assert_raises(ValueError, firls, 11, [0.1, 0.2], [0, 0], weight=[-1])
+
+    @skip_xp_backends("dask.array", reason="dask fancy indexing shape=(nan,)")
+    def test_firls(self, xp):
+        N = 11  # number of taps in the filter
+        a = 0.1  # width of the transition band
+
+        # design a halfband symmetric low-pass filter
+        h = firls(11, xp.asarray([0, a, 0.5 - a, 0.5]), xp.asarray([1, 1, 0, 0]),
+                  fs=1.0)
+
+        # make sure the filter has correct # of taps
+        assert h.shape[0] == N
+
+        # make sure it is symmetric
+        midx = (N-1) // 2
+        flip = array_namespace(h).flip
+        assert_array_almost_equal(h[:midx],  flip(h[midx+1:])) # h[:-midx-1:-1])
+
+        # make sure the center tap is 0.5
+        assert math.isclose(h[midx], 0.5, abs_tol=1e-8) 
+
+        # For halfband symmetric, odd coefficients (except the center)
+        # should be zero (really small)
+        hodd = xp.stack((h[1:midx:2], h[-midx+1::2]))
+        assert_array_almost_equal(hodd, xp.zeros_like(hodd))
+
+        # now check the frequency response
+        w, H = freqz(np.asarray(h), 1)
+        w, H = xp.asarray(w), xp.asarray(H)
+        f = w/2/xp.pi
+        Hmag = xp.abs(H)
+
+        # check that the pass band is close to unity
+        idx = xp.logical_and(f > 0, f < a)
+        assert_array_almost_equal(Hmag[idx], xp.ones_like(Hmag[idx]), decimal=3)
+
+        # check that the stop band is close to zero
+        idx = xp.logical_and(f > 0.5 - a, f < 0.5)
+        assert_array_almost_equal(Hmag[idx], xp.zeros_like(Hmag[idx]), decimal=3)
+
+    def test_compare(self, xp):
+        # compare to OCTAVE output
+        taps = firls(9, xp.asarray([0, 0.5, 0.55, 1]),
+                    xp.asarray([1, 1, 0, 0]), weight=xp.asarray([1, 2]))
+        # >> taps = firls(8, [0 0.5 0.55 1], [1 1 0 0], [1, 2]);
+        known_taps = [-6.26930101730182e-04, -1.03354450635036e-01,
+                      -9.81576747564301e-03, 3.17271686090449e-01,
+                      5.11409425599933e-01, 3.17271686090449e-01,
+                      -9.81576747564301e-03, -1.03354450635036e-01,
+                      -6.26930101730182e-04]
+        atol_arg = {'atol': 5e-8} if xp_default_dtype(xp) == xp.float32 else {}
+        known_taps = xp.asarray(known_taps, dtype=xp.float64)
+        xp_assert_close(taps, known_taps, **atol_arg)
+
+        # compare to MATLAB output
+        taps = firls(11, xp.asarray([0, 0.5, 0.5, 1]),
+                     xp.asarray([1, 1, 0, 0]), weight=xp.asarray([1, 2]))
+        # >> taps = firls(10, [0 0.5 0.5 1], [1 1 0 0], [1, 2]);
+        known_taps = [
+            0.058545300496815, -0.014233383714318, -0.104688258464392,
+            0.012403323025279, 0.317930861136062, 0.488047220029700,
+            0.317930861136062, 0.012403323025279, -0.104688258464392,
+            -0.014233383714318, 0.058545300496815]
+        known_taps = xp.asarray(known_taps, dtype=xp.float64)
+        atol_arg = {'atol': 3e-8} if xp_default_dtype(xp) == xp.float32 else {}
+        xp_assert_close(taps, known_taps, **atol_arg)
+
+        # With linear changes:
+        taps = firls(7, xp.asarray((0, 1, 2, 3, 4, 5)),
+                     xp.asarray([1, 0, 0, 1, 1, 0]), fs=20)
+        # >> taps = firls(6, [0, 0.1, 0.2, 0.3, 0.4, 0.5], [1, 0, 0, 1, 1, 0])
+        known_taps = [
+            1.156090832768218, -4.1385894727395849, 7.5288619164321826,
+            -8.5530572592947856, 7.5288619164321826, -4.1385894727395849,
+            1.156090832768218]
+        known_taps = xp.asarray(known_taps, dtype=xp.float64)
+        xp_assert_close(taps, known_taps)
+
+    def test_rank_deficient(self, xp):
+        # solve() runs but warns (only sometimes, so here we don't use match)
+        x = firls(21, xp.asarray([0, 0.1, 0.9, 1]), xp.asarray([1, 1, 0, 0]))
+        w, h = freqz(np.asarray(x), fs=2.)
+        w, h = map(xp.asarray, (w, h))   # XXX convert freqz
+        absh2 = xp.abs(h[:2])
+        xp_assert_close(absh2, xp.ones_like(absh2), atol=1e-5)
+        absh2 = xp.abs(h[-2:])
+        xp_assert_close(absh2, xp.zeros_like(absh2), atol=1e-6, rtol=1e-7)
+        # switch to pinvh (tolerances could be higher with longer
+        # filters, but using shorter ones is faster computationally and
+        # the idea is the same)
+        x = firls(101, xp.asarray([0, 0.01, 0.99, 1]), xp.asarray([1, 1, 0, 0]))
+        w, h = freqz(np.asarray(x), fs=2.)
+        w, h = map(xp.asarray, (w, h))   # XXX convert freqz
+        mask = xp.asarray(w < 0.01)
+        h = xp.asarray(h)
+        assert xp.sum(xp.astype(mask, xp.int64)) > 3
+        habs = xp.abs(h[mask])
+        xp_assert_close(habs, xp.ones_like(habs), atol=1e-4)
+        mask = xp.asarray(w > 0.99)
+        assert xp.sum(xp.astype(mask, xp.int64)) > 3
+        habs = xp.abs(h[mask])
+        xp_assert_close(habs, xp.zeros_like(habs), atol=1e-4)
+
+    def test_fs_validation(self):
+        with pytest.raises(ValueError, match="Sampling.*single scalar"):
+            firls(11, .1, 1, fs=np.array([10, 20]))
+
+class TestMinimumPhase:
+
+    @pytest.mark.thread_unsafe
+    def test_bad_args(self):
+        # not enough taps
+        assert_raises(ValueError, minimum_phase, [1.])
+        assert_raises(ValueError, minimum_phase, [1., 1.])
+        assert_raises(ValueError, minimum_phase, np.full(10, 1j))
+        assert_raises((ValueError, TypeError), minimum_phase, 'foo')
+        assert_raises(ValueError, minimum_phase, np.ones(10), n_fft=8)
+        assert_raises(ValueError, minimum_phase, np.ones(10), method='foo')
+        assert_warns(RuntimeWarning, minimum_phase, np.arange(3))
+        with pytest.raises(ValueError, match="is only supported when"):
+            minimum_phase(np.ones(3), method='hilbert', half=False)
+
+    def test_homomorphic(self):
+        # check that it can recover frequency responses of arbitrary
+        # linear-phase filters
+
+        # for some cases we can get the actual filter back
+        h = [1, -1]
+        h_new = minimum_phase(np.convolve(h, h[::-1]))
+        xp_assert_close(h_new, np.asarray(h, dtype=np.float64), rtol=0.05)
+
+        # but in general we only guarantee we get the magnitude back
+        rng = np.random.RandomState(0)
+        for n in (2, 3, 10, 11, 15, 16, 17, 20, 21, 100, 101):
+            h = rng.randn(n)
+            h_linear = np.convolve(h, h[::-1])
+            h_new = minimum_phase(h_linear)
+            xp_assert_close(np.abs(fft(h_new)), np.abs(fft(h)), rtol=1e-4)
+            h_new = minimum_phase(h_linear, half=False)
+            assert len(h_linear) == len(h_new)
+            xp_assert_close(np.abs(fft(h_new)), np.abs(fft(h_linear)), rtol=1e-4)
+
+    @skip_xp_backends("dask.array", reason="too slow")
+    @skip_xp_backends("jax.numpy", reason="immutable arrays")
+    def test_hilbert(self, xp):
+        # compare to MATLAB output of reference implementation
+
+        # f=[0 0.3 0.5 1];
+        # a=[1 1 0 0];
+        # h=remez(11,f,a);
+        h = remez(12, [0, 0.3, 0.5, 1], [1, 0], fs=2.)
+        k = [0.349585548646686, 0.373552164395447, 0.326082685363438,
+             0.077152207480935, -0.129943946349364, -0.059355880509749]
+        h = xp.asarray(h)
+        k = xp.asarray(k, dtype=xp.float64)
+        m = minimum_phase(h, 'hilbert')
+        xp_assert_close(m, k, rtol=5e-3)
+
+        # f=[0 0.8 0.9 1];
+        # a=[0 0 1 1];
+        # h=remez(20,f,a);
+        h = remez(21, [0, 0.8, 0.9, 1], [0, 1], fs=2.)
+        k = [0.232486803906329, -0.133551833687071, 0.151871456867244,
+             -0.157957283165866, 0.151739294892963, -0.129293146705090,
+             0.100787844523204, -0.065832656741252, 0.035361328741024,
+             -0.014977068692269, -0.158416139047557]
+        h = xp.asarray(h)
+        k = xp.asarray(k, dtype=xp.float64)
+        m = minimum_phase(h, 'hilbert', n_fft=2**19)
+        xp_assert_close(m, k, rtol=2e-3)
+
+
+class Testfirwin_2d:
+    def test_invalid_args(self):
+        with pytest.raises(ValueError,
+                           match="hsize must be a 2-element tuple or list"):
+            firwin_2d((50,), window=(("kaiser", 5.0), "boxcar"), fc=0.4)
+
+        with pytest.raises(ValueError,
+                           match="window must be a 2-element tuple or list"):
+            firwin_2d((51, 51), window=("hamming",), fc=0.5)
+
+        with pytest.raises(ValueError,
+                           match="window must be a 2-element tuple or list"):
+            firwin_2d((51, 51), window="invalid_window", fc=0.5)
+
+    def test_filter_design(self):
+        hsize = (51, 51)
+        window = (("kaiser", 8.0), ("kaiser", 8.0))
+        fc = 0.4
+        taps_kaiser = firwin_2d(hsize, window, fc=fc)
+        assert taps_kaiser.shape == (51, 51)
+
+        window = ("hamming", "hamming")
+        taps_hamming = firwin_2d(hsize, window, fc=fc)
+        assert taps_hamming.shape == (51, 51)
+
+    def test_impulse_response(self):
+        hsize = (31, 31)
+        window = ("hamming", "hamming")
+        fc = 0.4
+        taps = firwin_2d(hsize, window, fc=fc)
+
+        impulse = np.zeros((63, 63))
+        impulse[31, 31] = 1
+
+        response = convolve2d(impulse, taps, mode='same')
+
+        expected_response = taps
+        xp_assert_close(response[16:47, 16:47], expected_response, rtol=1e-5)
+
+    def test_frequency_response(self):
+        """Compare 1d and 2d frequency response. """
+        hsize = (31, 31)
+        windows = ("hamming", "hamming")
+        fc = 0.4
+        taps_1d = firwin(numtaps=hsize[0], cutoff=fc, window=windows[0])
+        taps_2d = firwin_2d(hsize, windows, fc=fc)
+
+        f_resp_1d = fft(taps_1d)
+        f_resp_2d = fft2(taps_2d)
+
+        xp_assert_close(f_resp_2d[0, :], f_resp_1d,
+                        err_msg='DC Gain at (0, f1) is not unity!')
+        xp_assert_close(f_resp_2d[:, 0], f_resp_1d,
+                        err_msg='DC Gain at (f0, 0) is not unity!')
+        xp_assert_close(f_resp_2d, np.outer(f_resp_1d, f_resp_1d),
+                        atol=np.finfo(f_resp_2d.dtype).resolution,
+                        err_msg='2d frequency response is not product of 1d responses')
+
+    def test_symmetry(self):
+        hsize = (51, 51)
+        window = ("hamming", "hamming")
+        fc = 0.4
+        taps = firwin_2d(hsize, window, fc=fc)
+        xp_assert_close(taps, np.flip(taps), rtol=1e-5)
+
+    def test_circular_symmetry(self):
+        hsize = (51, 51)
+        window = "hamming"
+        taps = firwin_2d(hsize, window, circular=True, fc=0.5)
+        center = hsize[0] // 2
+        for i in range(hsize[0]):
+            for j in range(hsize[1]):
+                xp_assert_close(taps[i, j], 
+                                taps[center - (i - center), center - (j - center)], 
+                                rtol=1e-5)
+
+    def test_edge_case_circular(self):
+        hsize = (3, 3)
+        window = "hamming"
+        taps_small = firwin_2d(hsize, window, circular=True, fc=0.5)
+        assert taps_small.shape == (3, 3)
+
+        hsize = (101, 101)
+        taps_large = firwin_2d(hsize, window, circular=True, fc=0.5)
+        assert taps_large.shape == (101, 101)
+
+    def test_known_result(self):
+        hsize = (5, 5)
+        window = ('kaiser', 8.0)
+        fc = 0.1
+        fs = 2
+
+        row_filter = firwin(hsize[0], cutoff=fc, window=window, fs=fs)
+        col_filter = firwin(hsize[1], cutoff=fc, window=window, fs=fs)
+        known_result = np.outer(row_filter, col_filter)
+
+        taps = firwin_2d(hsize, (window, window), fc=fc)
+        assert taps.shape == known_result.shape, (
+            f"Shape mismatch: {taps.shape} vs {known_result.shape}"
+        )
+        assert np.allclose(taps, known_result, rtol=1e-1), (
+            f"Filter shape mismatch: {taps} vs {known_result}"
+        )