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@@ -164,3 +164,5 @@ env-llmeval/lib/python3.10/site-packages/nvidia/nccl/lib/libnccl.so.2 filter=lfs
 env-llmeval/lib/python3.10/site-packages/scipy/special/cython_special.cpython-310-x86_64-linux-gnu.so filter=lfs diff=lfs merge=lfs -text
 env-llmeval/lib/python3.10/site-packages/scipy/stats/_unuran/unuran_wrapper.cpython-310-x86_64-linux-gnu.so filter=lfs diff=lfs merge=lfs -text
 env-llmeval/lib/python3.10/site-packages/scipy/special/_ufuncs.cpython-310-x86_64-linux-gnu.so filter=lfs diff=lfs merge=lfs -text
+env-llmeval/lib/python3.10/site-packages/scipy/spatial/_ckdtree.cpython-310-x86_64-linux-gnu.so filter=lfs diff=lfs merge=lfs -text
+env-llmeval/lib/python3.10/site-packages/scipy/spatial/_qhull.cpython-310-x86_64-linux-gnu.so filter=lfs diff=lfs merge=lfs -text

env-llmeval/lib/python3.10/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)

env-llmeval/lib/python3.10/site-packages/scipy/signal/_fir_filter_design.py

@@ -0,0 +1,1301 @@
+"""Functions for FIR filter design."""
+
+from math import ceil, log
+import operator
+import warnings
+
+import numpy as np
+from numpy.fft import irfft, fft, ifft
+from scipy.special import sinc
+from scipy.linalg import (toeplitz, hankel, solve, LinAlgError, LinAlgWarning,
+                          lstsq)
+from scipy._lib.deprecation import _NoValue, _deprecate_positional_args
+from scipy.signal._arraytools import _validate_fs
+
+from . import _sigtools
+
+__all__ = ['kaiser_beta', 'kaiser_atten', 'kaiserord',
+           'firwin', 'firwin2', 'remez', 'firls', 'minimum_phase']
+
+
+def _get_fs(fs, nyq):
+    """
+    Utility for replacing the argument 'nyq' (with default 1) with 'fs'.
+    """
+    if nyq is _NoValue and fs is None:
+        fs = 2
+    elif nyq is not _NoValue:
+        if fs is not None:
+            raise ValueError("Values cannot be given for both 'nyq' and 'fs'.")
+        msg = ("Keyword argument 'nyq' is deprecated in favour of 'fs' and "
+               "will be removed in SciPy 1.14.0.")
+        warnings.warn(msg, DeprecationWarning, stacklevel=3)
+        if nyq is None:
+            fs = 2
+        else:
+            fs = 2*nyq
+    return fs
+
+
+# Some notes on function parameters:
+#
+# `cutoff` and `width` are given as numbers between 0 and 1.  These are
+# relative frequencies, expressed as a fraction of the Nyquist frequency.
+# For example, if the Nyquist frequency is 2 KHz, then width=0.15 is a width
+# of 300 Hz.
+#
+# The `order` of a FIR filter is one less than the number of taps.
+# This is a potential source of confusion, so in the following code,
+# we will always use the number of taps as the parameterization of
+# the 'size' of the filter. The "number of taps" means the number
+# of coefficients, which is the same as the length of the impulse
+# response of the filter.
+
+
+def kaiser_beta(a):
+    """Compute the Kaiser parameter `beta`, given the attenuation `a`.
+
+    Parameters
+    ----------
+    a : float
+        The desired attenuation in the stopband and maximum ripple in
+        the passband, in dB.  This should be a *positive* number.
+
+    Returns
+    -------
+    beta : float
+        The `beta` parameter to be used in the formula for a Kaiser window.
+
+    References
+    ----------
+    Oppenheim, Schafer, "Discrete-Time Signal Processing", p.475-476.
+
+    Examples
+    --------
+    Suppose we want to design a lowpass filter, with 65 dB attenuation
+    in the stop band.  The Kaiser window parameter to be used in the
+    window method is computed by ``kaiser_beta(65)``:
+
+    >>> from scipy.signal import kaiser_beta
+    >>> kaiser_beta(65)
+    6.20426
+
+    """
+    if a > 50:
+        beta = 0.1102 * (a - 8.7)
+    elif a > 21:
+        beta = 0.5842 * (a - 21) ** 0.4 + 0.07886 * (a - 21)
+    else:
+        beta = 0.0
+    return beta
+
+
+def kaiser_atten(numtaps, width):
+    """Compute the attenuation of a Kaiser FIR filter.
+
+    Given the number of taps `N` and the transition width `width`, compute the
+    attenuation `a` in dB, given by Kaiser's formula:
+
+        a = 2.285 * (N - 1) * pi * width + 7.95
+
+    Parameters
+    ----------
+    numtaps : int
+        The number of taps in the FIR filter.
+    width : float
+        The desired width of the transition region between passband and
+        stopband (or, in general, at any discontinuity) for the filter,
+        expressed as a fraction of the Nyquist frequency.
+
+    Returns
+    -------
+    a : float
+        The attenuation of the ripple, in dB.
+
+    See Also
+    --------
+    kaiserord, kaiser_beta
+
+    Examples
+    --------
+    Suppose we want to design a FIR filter using the Kaiser window method
+    that will have 211 taps and a transition width of 9 Hz for a signal that
+    is sampled at 480 Hz. Expressed as a fraction of the Nyquist frequency,
+    the width is 9/(0.5*480) = 0.0375. The approximate attenuation (in dB)
+    is computed as follows:
+
+    >>> from scipy.signal import kaiser_atten
+    >>> kaiser_atten(211, 0.0375)
+    64.48099630593983
+
+    """
+    a = 2.285 * (numtaps - 1) * np.pi * width + 7.95
+    return a
+
+
+def kaiserord(ripple, width):
+    """
+    Determine the filter window parameters for the Kaiser window method.
+
+    The parameters returned by this function are generally used to create
+    a finite impulse response filter using the window method, with either
+    `firwin` or `firwin2`.
+
+    Parameters
+    ----------
+    ripple : float
+        Upper bound for the deviation (in dB) of the magnitude of the
+        filter's frequency response from that of the desired filter (not
+        including frequencies in any transition intervals). That is, if w
+        is the frequency expressed as a fraction of the Nyquist frequency,
+        A(w) is the actual frequency response of the filter and D(w) is the
+        desired frequency response, the design requirement is that::
+
+            abs(A(w) - D(w))) < 10**(-ripple/20)
+
+        for 0 <= w <= 1 and w not in a transition interval.
+    width : float
+        Width of transition region, normalized so that 1 corresponds to pi
+        radians / sample. That is, the frequency is expressed as a fraction
+        of the Nyquist frequency.
+
+    Returns
+    -------
+    numtaps : int
+        The length of the Kaiser window.
+    beta : float
+        The beta parameter for the Kaiser window.
+
+    See Also
+    --------
+    kaiser_beta, kaiser_atten
+
+    Notes
+    -----
+    There are several ways to obtain the Kaiser window:
+
+    - ``signal.windows.kaiser(numtaps, beta, sym=True)``
+    - ``signal.get_window(beta, numtaps)``
+    - ``signal.get_window(('kaiser', beta), numtaps)``
+
+    The empirical equations discovered by Kaiser are used.
+
+    References
+    ----------
+    Oppenheim, Schafer, "Discrete-Time Signal Processing", pp.475-476.
+
+    Examples
+    --------
+    We will use the Kaiser window method to design a lowpass FIR filter
+    for a signal that is sampled at 1000 Hz.
+
+    We want at least 65 dB rejection in the stop band, and in the pass
+    band the gain should vary no more than 0.5%.
+
+    We want a cutoff frequency of 175 Hz, with a transition between the
+    pass band and the stop band of 24 Hz. That is, in the band [0, 163],
+    the gain varies no more than 0.5%, and in the band [187, 500], the
+    signal is attenuated by at least 65 dB.
+
+    >>> import numpy as np
+    >>> from scipy.signal import kaiserord, firwin, freqz
+    >>> import matplotlib.pyplot as plt
+    >>> fs = 1000.0
+    >>> cutoff = 175
+    >>> width = 24
+
+    The Kaiser method accepts just a single parameter to control the pass
+    band ripple and the stop band rejection, so we use the more restrictive
+    of the two. In this case, the pass band ripple is 0.005, or 46.02 dB,
+    so we will use 65 dB as the design parameter.
+
+    Use `kaiserord` to determine the length of the filter and the
+    parameter for the Kaiser window.
+
+    >>> numtaps, beta = kaiserord(65, width/(0.5*fs))
+    >>> numtaps
+    167
+    >>> beta
+    6.20426
+
+    Use `firwin` to create the FIR filter.
+
+    >>> taps = firwin(numtaps, cutoff, window=('kaiser', beta),
+    ...               scale=False, fs=fs)
+
+    Compute the frequency response of the filter.  ``w`` is the array of
+    frequencies, and ``h`` is the corresponding complex array of frequency
+    responses.
+
+    >>> w, h = freqz(taps, worN=8000)
+    >>> w *= 0.5*fs/np.pi  # Convert w to Hz.
+
+    Compute the deviation of the magnitude of the filter's response from
+    that of the ideal lowpass filter. Values in the transition region are
+    set to ``nan``, so they won't appear in the plot.
+
+    >>> ideal = w < cutoff  # The "ideal" frequency response.
+    >>> deviation = np.abs(np.abs(h) - ideal)
+    >>> deviation[(w > cutoff - 0.5*width) & (w < cutoff + 0.5*width)] = np.nan
+
+    Plot the deviation. A close look at the left end of the stop band shows
+    that the requirement for 65 dB attenuation is violated in the first lobe
+    by about 0.125 dB. This is not unusual for the Kaiser window method.
+
+    >>> plt.plot(w, 20*np.log10(np.abs(deviation)))
+    >>> plt.xlim(0, 0.5*fs)
+    >>> plt.ylim(-90, -60)
+    >>> plt.grid(alpha=0.25)
+    >>> plt.axhline(-65, color='r', ls='--', alpha=0.3)
+    >>> plt.xlabel('Frequency (Hz)')
+    >>> plt.ylabel('Deviation from ideal (dB)')
+    >>> plt.title('Lowpass Filter Frequency Response')
+    >>> plt.show()
+
+    """
+    A = abs(ripple)  # in case somebody is confused as to what's meant
+    if A < 8:
+        # Formula for N is not valid in this range.
+        raise ValueError("Requested maximum ripple attenuation %f is too "
+                         "small for the Kaiser formula." % A)
+    beta = kaiser_beta(A)
+
+    # Kaiser's formula (as given in Oppenheim and Schafer) is for the filter
+    # order, so we have to add 1 to get the number of taps.
+    numtaps = (A - 7.95) / 2.285 / (np.pi * width) + 1
+
+    return int(ceil(numtaps)), beta
+
+
+@_deprecate_positional_args(version="1.14")
+def firwin(numtaps, cutoff, *, width=None, window='hamming', pass_zero=True,
+           scale=True, nyq=_NoValue, fs=None):
+    """
+    FIR filter design using the window method.
+
+    This function computes the coefficients of a finite impulse response
+    filter. The filter will have linear phase; it will be Type I if
+    `numtaps` is odd and Type II if `numtaps` is even.
+
+    Type II filters always have zero response at the Nyquist frequency, so a
+    ValueError exception is raised if firwin is called with `numtaps` even and
+    having a passband whose right end is at the Nyquist frequency.
+
+    Parameters
+    ----------
+    numtaps : int
+        Length of the filter (number of coefficients, i.e. the filter
+        order + 1).  `numtaps` must be odd if a passband includes the
+        Nyquist frequency.
+    cutoff : float or 1-D array_like
+        Cutoff frequency of filter (expressed in the same units as `fs`)
+        OR an array of cutoff frequencies (that is, band edges). In the
+        latter case, the frequencies in `cutoff` should be positive and
+        monotonically increasing between 0 and `fs/2`. The values 0 and
+        `fs/2` must not be included in `cutoff`.
+    width : float or None, optional
+        If `width` is not None, then assume it is the approximate width
+        of the transition region (expressed in the same units as `fs`)
+        for use in Kaiser FIR filter design. In this case, the `window`
+        argument is ignored.
+    window : string or tuple of string and parameter values, optional
+        Desired window to use. See `scipy.signal.get_window` for a list
+        of windows and required parameters.
+    pass_zero : {True, False, 'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional
+        If True, the gain at the frequency 0 (i.e., the "DC gain") is 1.
+        If False, the DC gain is 0. Can also be a string argument for the
+        desired filter type (equivalent to ``btype`` in IIR design functions).
+
+        .. versionadded:: 1.3.0
+           Support for string arguments.
+    scale : bool, optional
+        Set to True to scale the coefficients so that the frequency
+        response is exactly unity at a certain frequency.
+        That frequency is either:
+
+        - 0 (DC) if the first passband starts at 0 (i.e. pass_zero
+          is True)
+        - `fs/2` (the Nyquist frequency) if the first passband ends at
+          `fs/2` (i.e the filter is a single band highpass filter);
+          center of first passband otherwise
+
+    nyq : float, optional, deprecated
+        This is the Nyquist frequency. Each frequency in `cutoff` must be
+        between 0 and `nyq`. Default is 1.
+
+        .. deprecated:: 1.0.0
+           `firwin` keyword argument `nyq` is deprecated in favour of `fs` and
+           will be removed in SciPy 1.14.0.
+    fs : float, optional
+        The sampling frequency of the signal. Each frequency in `cutoff`
+        must be between 0 and ``fs/2``.  Default is 2.
+
+    Returns
+    -------
+    h : (numtaps,) ndarray
+        Coefficients of length `numtaps` FIR filter.
+
+    Raises
+    ------
+    ValueError
+        If any value in `cutoff` is less than or equal to 0 or greater
+        than or equal to ``fs/2``, if the values in `cutoff` are not strictly
+        monotonically increasing, or if `numtaps` is even but a passband
+        includes the Nyquist frequency.
+
+    See Also
+    --------
+    firwin2
+    firls
+    minimum_phase
+    remez
+
+    Examples
+    --------
+    Low-pass from 0 to f:
+
+    >>> from scipy import signal
+    >>> numtaps = 3
+    >>> f = 0.1
+    >>> signal.firwin(numtaps, f)
+    array([ 0.06799017,  0.86401967,  0.06799017])
+
+    Use a specific window function:
+
+    >>> signal.firwin(numtaps, f, window='nuttall')
+    array([  3.56607041e-04,   9.99286786e-01,   3.56607041e-04])
+
+    High-pass ('stop' from 0 to f):
+
+    >>> signal.firwin(numtaps, f, pass_zero=False)
+    array([-0.00859313,  0.98281375, -0.00859313])
+
+    Band-pass:
+
+    >>> f1, f2 = 0.1, 0.2
+    >>> signal.firwin(numtaps, [f1, f2], pass_zero=False)
+    array([ 0.06301614,  0.88770441,  0.06301614])
+
+    Band-stop:
+
+    >>> signal.firwin(numtaps, [f1, f2])
+    array([-0.00801395,  1.0160279 , -0.00801395])
+
+    Multi-band (passbands are [0, f1], [f2, f3] and [f4, 1]):
+
+    >>> f3, f4 = 0.3, 0.4
+    >>> signal.firwin(numtaps, [f1, f2, f3, f4])
+    array([-0.01376344,  1.02752689, -0.01376344])
+
+    Multi-band (passbands are [f1, f2] and [f3,f4]):
+
+    >>> signal.firwin(numtaps, [f1, f2, f3, f4], pass_zero=False)
+    array([ 0.04890915,  0.91284326,  0.04890915])
+
+    """
+    # The major enhancements to this function added in November 2010 were
+    # developed by Tom Krauss (see ticket #902).
+    fs = _validate_fs(fs, allow_none=True)
+
+    nyq = 0.5 * _get_fs(fs, nyq)
+
+    cutoff = np.atleast_1d(cutoff) / float(nyq)
+
+    # Check for invalid input.
+    if cutoff.ndim > 1:
+        raise ValueError("The cutoff argument must be at most "
+                         "one-dimensional.")
+    if cutoff.size == 0:
+        raise ValueError("At least one cutoff frequency must be given.")
+    if cutoff.min() <= 0 or cutoff.max() >= 1:
+        raise ValueError("Invalid cutoff frequency: frequencies must be "
+                         "greater than 0 and less than fs/2.")
+    if np.any(np.diff(cutoff) <= 0):
+        raise ValueError("Invalid cutoff frequencies: the frequencies "
+                         "must be strictly increasing.")
+
+    if width is not None:
+        # A width was given.  Find the beta parameter of the Kaiser window
+        # and set `window`.  This overrides the value of `window` passed in.
+        atten = kaiser_atten(numtaps, float(width) / nyq)
+        beta = kaiser_beta(atten)
+        window = ('kaiser', beta)
+
+    if isinstance(pass_zero, str):
+        if pass_zero in ('bandstop', 'lowpass'):
+            if pass_zero == 'lowpass':
+                if cutoff.size != 1:
+                    raise ValueError('cutoff must have one element if '
+                                     f'pass_zero=="lowpass", got {cutoff.shape}')
+            elif cutoff.size <= 1:
+                raise ValueError('cutoff must have at least two elements if '
+                                 f'pass_zero=="bandstop", got {cutoff.shape}')
+            pass_zero = True
+        elif pass_zero in ('bandpass', 'highpass'):
+            if pass_zero == 'highpass':
+                if cutoff.size != 1:
+                    raise ValueError('cutoff must have one element if '
+                                     f'pass_zero=="highpass", got {cutoff.shape}')
+            elif cutoff.size <= 1:
+                raise ValueError('cutoff must have at least two elements if '
+                                 f'pass_zero=="bandpass", got {cutoff.shape}')
+            pass_zero = False
+        else:
+            raise ValueError('pass_zero must be True, False, "bandpass", '
+                             '"lowpass", "highpass", or "bandstop", got '
+                             f'{pass_zero}')
+    pass_zero = bool(operator.index(pass_zero))  # ensure bool-like
+
+    pass_nyquist = bool(cutoff.size & 1) ^ pass_zero
+    if pass_nyquist and numtaps % 2 == 0:
+        raise ValueError("A filter with an even number of coefficients must "
+                         "have zero response at the Nyquist frequency.")
+
+    # Insert 0 and/or 1 at the ends of cutoff so that the length of cutoff
+    # is even, and each pair in cutoff corresponds to passband.
+    cutoff = np.hstack(([0.0] * pass_zero, cutoff, [1.0] * pass_nyquist))
+
+    # `bands` is a 2-D array; each row gives the left and right edges of
+    # a passband.
+    bands = cutoff.reshape(-1, 2)
+
+    # Build up the coefficients.
+    alpha = 0.5 * (numtaps - 1)
+    m = np.arange(0, numtaps) - alpha
+    h = 0
+    for left, right in bands:
+        h += right * sinc(right * m)
+        h -= left * sinc(left * m)
+
+    # Get and apply the window function.
+    from .windows import get_window
+    win = get_window(window, numtaps, fftbins=False)
+    h *= win
+
+    # Now handle scaling if desired.
+    if scale:
+        # Get the first passband.
+        left, right = bands[0]
+        if left == 0:
+            scale_frequency = 0.0
+        elif right == 1:
+            scale_frequency = 1.0
+        else:
+            scale_frequency = 0.5 * (left + right)
+        c = np.cos(np.pi * m * scale_frequency)
+        s = np.sum(h * c)
+        h /= s
+
+    return h
+
+
+# Original version of firwin2 from scipy ticket #457, submitted by "tash".
+#
+# Rewritten by Warren Weckesser, 2010.
+@_deprecate_positional_args(version="1.14")
+def firwin2(numtaps, freq, gain, *, nfreqs=None, window='hamming', nyq=_NoValue,
+            antisymmetric=False, fs=None):
+    """
+    FIR filter design using the window method.
+
+    From the given frequencies `freq` and corresponding gains `gain`,
+    this function constructs an FIR filter with linear phase and
+    (approximately) the given frequency response.
+
+    Parameters
+    ----------
+    numtaps : int
+        The number of taps in the FIR filter.  `numtaps` must be less than
+        `nfreqs`.
+    freq : array_like, 1-D
+        The frequency sampling points. Typically 0.0 to 1.0 with 1.0 being
+        Nyquist.  The Nyquist frequency is half `fs`.
+        The values in `freq` must be nondecreasing. A value can be repeated
+        once to implement a discontinuity. The first value in `freq` must
+        be 0, and the last value must be ``fs/2``. Values 0 and ``fs/2`` must
+        not be repeated.
+    gain : array_like
+        The filter gains at the frequency sampling points. Certain
+        constraints to gain values, depending on the filter type, are applied,
+        see Notes for details.
+    nfreqs : int, optional
+        The size of the interpolation mesh used to construct the filter.
+        For most efficient behavior, this should be a power of 2 plus 1
+        (e.g, 129, 257, etc). The default is one more than the smallest
+        power of 2 that is not less than `numtaps`. `nfreqs` must be greater
+        than `numtaps`.
+    window : string or (string, float) or float, or None, optional
+        Window function to use. Default is "hamming". See
+        `scipy.signal.get_window` for the complete list of possible values.
+        If None, no window function is applied.
+    nyq : float, optional, deprecated
+        This is the Nyquist frequency. Each frequency in `freq` must be
+        between 0 and `nyq`. Default is 1.
+
+        .. deprecated:: 1.0.0
+           `firwin2` keyword argument `nyq` is deprecated in favour of `fs` and
+           will be removed in SciPy 1.14.0.
+    antisymmetric : bool, optional
+        Whether resulting impulse response is symmetric/antisymmetric.
+        See Notes for more details.
+    fs : float, optional
+        The sampling frequency of the signal. Each frequency in `cutoff`
+        must be between 0 and ``fs/2``. Default is 2.
+
+    Returns
+    -------
+    taps : ndarray
+        The filter coefficients of the FIR filter, as a 1-D array of length
+        `numtaps`.
+
+    See Also
+    --------
+    firls
+    firwin
+    minimum_phase
+    remez
+
+    Notes
+    -----
+    From the given set of frequencies and gains, the desired response is
+    constructed in the frequency domain. The inverse FFT is applied to the
+    desired response to create the associated convolution kernel, and the
+    first `numtaps` coefficients of this kernel, scaled by `window`, are
+    returned.
+
+    The FIR filter will have linear phase. The type of filter is determined by
+    the value of 'numtaps` and `antisymmetric` flag.
+    There are four possible combinations:
+
+       - odd  `numtaps`, `antisymmetric` is False, type I filter is produced
+       - even `numtaps`, `antisymmetric` is False, type II filter is produced
+       - odd  `numtaps`, `antisymmetric` is True, type III filter is produced
+       - even `numtaps`, `antisymmetric` is True, type IV filter is produced
+
+    Magnitude response of all but type I filters are subjects to following
+    constraints:
+
+       - type II  -- zero at the Nyquist frequency
+       - type III -- zero at zero and Nyquist frequencies
+       - type IV  -- zero at zero frequency
+
+    .. versionadded:: 0.9.0
+
+    References
+    ----------
+    .. [1] Oppenheim, A. V. and Schafer, R. W., "Discrete-Time Signal
+       Processing", Prentice-Hall, Englewood Cliffs, New Jersey (1989).
+       (See, for example, Section 7.4.)
+
+    .. [2] Smith, Steven W., "The Scientist and Engineer's Guide to Digital
+       Signal Processing", Ch. 17. http://www.dspguide.com/ch17/1.htm
+
+    Examples
+    --------
+    A lowpass FIR filter with a response that is 1 on [0.0, 0.5], and
+    that decreases linearly on [0.5, 1.0] from 1 to 0:
+
+    >>> from scipy import signal
+    >>> taps = signal.firwin2(150, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0])
+    >>> print(taps[72:78])
+    [-0.02286961 -0.06362756  0.57310236  0.57310236 -0.06362756 -0.02286961]
+
+    """
+    fs = _validate_fs(fs, allow_none=True)
+    nyq = 0.5 * _get_fs(fs, nyq)
+
+    if len(freq) != len(gain):
+        raise ValueError('freq and gain must be of same length.')
+
+    if nfreqs is not None and numtaps >= nfreqs:
+        raise ValueError(('ntaps must be less than nfreqs, but firwin2 was '
+                          'called with ntaps=%d and nfreqs=%s') %
+                         (numtaps, nfreqs))
+
+    if freq[0] != 0 or freq[-1] != nyq:
+        raise ValueError('freq must start with 0 and end with fs/2.')
+    d = np.diff(freq)
+    if (d < 0).any():
+        raise ValueError('The values in freq must be nondecreasing.')
+    d2 = d[:-1] + d[1:]
+    if (d2 == 0).any():
+        raise ValueError('A value in freq must not occur more than twice.')
+    if freq[1] == 0:
+        raise ValueError('Value 0 must not be repeated in freq')
+    if freq[-2] == nyq:
+        raise ValueError('Value fs/2 must not be repeated in freq')
+
+    if antisymmetric:
+        if numtaps % 2 == 0:
+            ftype = 4
+        else:
+            ftype = 3
+    else:
+        if numtaps % 2 == 0:
+            ftype = 2
+        else:
+            ftype = 1
+
+    if ftype == 2 and gain[-1] != 0.0:
+        raise ValueError("A Type II filter must have zero gain at the "
+                         "Nyquist frequency.")
+    elif ftype == 3 and (gain[0] != 0.0 or gain[-1] != 0.0):
+        raise ValueError("A Type III filter must have zero gain at zero "
+                         "and Nyquist frequencies.")
+    elif ftype == 4 and gain[0] != 0.0:
+        raise ValueError("A Type IV filter must have zero gain at zero "
+                         "frequency.")
+
+    if nfreqs is None:
+        nfreqs = 1 + 2 ** int(ceil(log(numtaps, 2)))
+
+    if (d == 0).any():
+        # Tweak any repeated values in freq so that interp works.
+        freq = np.array(freq, copy=True)
+        eps = np.finfo(float).eps * nyq
+        for k in range(len(freq) - 1):
+            if freq[k] == freq[k + 1]:
+                freq[k] = freq[k] - eps
+                freq[k + 1] = freq[k + 1] + eps
+        # Check if freq is strictly increasing after tweak
+        d = np.diff(freq)
+        if (d <= 0).any():
+            raise ValueError("freq cannot contain numbers that are too close "
+                             "(within eps * (fs/2): "
+                             f"{eps}) to a repeated value")
+
+    # Linearly interpolate the desired response on a uniform mesh `x`.
+    x = np.linspace(0.0, nyq, nfreqs)
+    fx = np.interp(x, freq, gain)
+
+    # Adjust the phases of the coefficients so that the first `ntaps` of the
+    # inverse FFT are the desired filter coefficients.
+    shift = np.exp(-(numtaps - 1) / 2. * 1.j * np.pi * x / nyq)
+    if ftype > 2:
+        shift *= 1j
+
+    fx2 = fx * shift
+
+    # Use irfft to compute the inverse FFT.
+    out_full = irfft(fx2)
+
+    if window is not None:
+        # Create the window to apply to the filter coefficients.
+        from .windows import get_window
+        wind = get_window(window, numtaps, fftbins=False)
+    else:
+        wind = 1
+
+    # Keep only the first `numtaps` coefficients in `out`, and multiply by
+    # the window.
+    out = out_full[:numtaps] * wind
+
+    if ftype == 3:
+        out[out.size // 2] = 0.0
+
+    return out
+
+
+@_deprecate_positional_args(version="1.14")
+def remez(numtaps, bands, desired, *, weight=None, Hz=_NoValue, type='bandpass',
+          maxiter=25, grid_density=16, fs=None):
+    """
+    Calculate the minimax optimal filter using the Remez exchange algorithm.
+
+    Calculate the filter-coefficients for the finite impulse response
+    (FIR) filter whose transfer function minimizes the maximum error
+    between the desired gain and the realized gain in the specified
+    frequency bands using the Remez exchange algorithm.
+
+    Parameters
+    ----------
+    numtaps : int
+        The desired number of taps in the filter. The number of taps is
+        the number of terms in the filter, or the filter order plus one.
+    bands : array_like
+        A monotonic sequence containing the band edges.
+        All elements must be non-negative and less than half the sampling
+        frequency as given by `fs`.
+    desired : array_like
+        A sequence half the size of bands containing the desired gain
+        in each of the specified bands.
+    weight : array_like, optional
+        A relative weighting to give to each band region. The length of
+        `weight` has to be half the length of `bands`.
+    Hz : scalar, optional, deprecated
+        The sampling frequency in Hz. Default is 1.
+
+        .. deprecated:: 1.0.0
+           `remez` keyword argument `Hz` is deprecated in favour of `fs` and
+           will be removed in SciPy 1.14.0.
+    type : {'bandpass', 'differentiator', 'hilbert'}, optional
+        The type of filter:
+
+          * 'bandpass' : flat response in bands. This is the default.
+
+          * 'differentiator' : frequency proportional response in bands.
+
+          * 'hilbert' : filter with odd symmetry, that is, type III
+                        (for even order) or type IV (for odd order)
+                        linear phase filters.
+
+    maxiter : int, optional
+        Maximum number of iterations of the algorithm. Default is 25.
+    grid_density : int, optional
+        Grid density. The dense grid used in `remez` is of size
+        ``(numtaps + 1) * grid_density``. Default is 16.
+    fs : float, optional
+        The sampling frequency of the signal.  Default is 1.
+
+    Returns
+    -------
+    out : ndarray
+        A rank-1 array containing the coefficients of the optimal
+        (in a minimax sense) filter.
+
+    See Also
+    --------
+    firls
+    firwin
+    firwin2
+    minimum_phase
+
+    References
+    ----------
+    .. [1] J. H. McClellan and T. W. Parks, "A unified approach to the
+           design of optimum FIR linear phase digital filters",
+           IEEE Trans. Circuit Theory, vol. CT-20, pp. 697-701, 1973.
+    .. [2] J. H. McClellan, T. W. Parks and L. R. Rabiner, "A Computer
+           Program for Designing Optimum FIR Linear Phase Digital
+           Filters", IEEE Trans. Audio Electroacoust., vol. AU-21,
+           pp. 506-525, 1973.
+
+    Examples
+    --------
+    In these examples, `remez` is used to design low-pass, high-pass,
+    band-pass and band-stop filters.  The parameters that define each filter
+    are the filter order, the band boundaries, the transition widths of the
+    boundaries, the desired gains in each band, and the sampling frequency.
+
+    We'll use a sample frequency of 22050 Hz in all the examples.  In each
+    example, the desired gain in each band is either 0 (for a stop band)
+    or 1 (for a pass band).
+
+    `freqz` is used to compute the frequency response of each filter, and
+    the utility function ``plot_response`` defined below is used to plot
+    the response.
+
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+
+    >>> fs = 22050   # Sample rate, Hz
+
+    >>> def plot_response(w, h, title):
+    ...     "Utility function to plot response functions"
+    ...     fig = plt.figure()
+    ...     ax = fig.add_subplot(111)
+    ...     ax.plot(w, 20*np.log10(np.abs(h)))
+    ...     ax.set_ylim(-40, 5)
+    ...     ax.grid(True)
+    ...     ax.set_xlabel('Frequency (Hz)')
+    ...     ax.set_ylabel('Gain (dB)')
+    ...     ax.set_title(title)
+
+    The first example is a low-pass filter, with cutoff frequency 8 kHz.
+    The filter length is 325, and the transition width from pass to stop
+    is 100 Hz.
+
+    >>> cutoff = 8000.0    # Desired cutoff frequency, Hz
+    >>> trans_width = 100  # Width of transition from pass to stop, Hz
+    >>> numtaps = 325      # Size of the FIR filter.
+    >>> taps = signal.remez(numtaps, [0, cutoff, cutoff + trans_width, 0.5*fs],
+    ...                     [1, 0], fs=fs)
+    >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
+    >>> plot_response(w, h, "Low-pass Filter")
+    >>> plt.show()
+
+    This example shows a high-pass filter:
+
+    >>> cutoff = 2000.0    # Desired cutoff frequency, Hz
+    >>> trans_width = 250  # Width of transition from pass to stop, Hz
+    >>> numtaps = 125      # Size of the FIR filter.
+    >>> taps = signal.remez(numtaps, [0, cutoff - trans_width, cutoff, 0.5*fs],
+    ...                     [0, 1], fs=fs)
+    >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
+    >>> plot_response(w, h, "High-pass Filter")
+    >>> plt.show()
+
+    This example shows a band-pass filter with a pass-band from 2 kHz to
+    5 kHz.  The transition width is 260 Hz and the length of the filter
+    is 63, which is smaller than in the other examples:
+
+    >>> band = [2000, 5000]  # Desired pass band, Hz
+    >>> trans_width = 260    # Width of transition from pass to stop, Hz
+    >>> numtaps = 63         # Size of the FIR filter.
+    >>> edges = [0, band[0] - trans_width, band[0], band[1],
+    ...          band[1] + trans_width, 0.5*fs]
+    >>> taps = signal.remez(numtaps, edges, [0, 1, 0], fs=fs)
+    >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
+    >>> plot_response(w, h, "Band-pass Filter")
+    >>> plt.show()
+
+    The low order leads to higher ripple and less steep transitions.
+
+    The next example shows a band-stop filter.
+
+    >>> band = [6000, 8000]  # Desired stop band, Hz
+    >>> trans_width = 200    # Width of transition from pass to stop, Hz
+    >>> numtaps = 175        # Size of the FIR filter.
+    >>> edges = [0, band[0] - trans_width, band[0], band[1],
+    ...          band[1] + trans_width, 0.5*fs]
+    >>> taps = signal.remez(numtaps, edges, [1, 0, 1], fs=fs)
+    >>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
+    >>> plot_response(w, h, "Band-stop Filter")
+    >>> plt.show()
+
+    """
+    fs = _validate_fs(fs, allow_none=True)
+    if Hz is _NoValue and fs is None:
+        fs = 1.0
+    elif Hz is not _NoValue:
+        if fs is not None:
+            raise ValueError("Values cannot be given for both 'Hz' and 'fs'.")
+        msg = ("'remez' keyword argument 'Hz' is deprecated in favour of 'fs'"
+               " and will be removed in SciPy 1.14.0.")
+        warnings.warn(msg, DeprecationWarning, stacklevel=2)
+        fs = Hz
+
+    # Convert type
+    try:
+        tnum = {'bandpass': 1, 'differentiator': 2, 'hilbert': 3}[type]
+    except KeyError as e:
+        raise ValueError("Type must be 'bandpass', 'differentiator', "
+                         "or 'hilbert'") from e
+
+    # Convert weight
+    if weight is None:
+        weight = [1] * len(desired)
+
+    bands = np.asarray(bands).copy()
+    return _sigtools._remez(numtaps, bands, desired, weight, tnum, fs,
+                            maxiter, grid_density)
+
+
+@_deprecate_positional_args(version="1.14")
+def firls(numtaps, bands, desired, *, weight=None, nyq=_NoValue, fs=None):
+    """
+    FIR filter design using least-squares error minimization.
+
+    Calculate the filter coefficients for the linear-phase finite
+    impulse response (FIR) filter which has the best approximation
+    to the desired frequency response described by `bands` and
+    `desired` in the least squares sense (i.e., the integral of the
+    weighted mean-squared error within the specified bands is
+    minimized).
+
+    Parameters
+    ----------
+    numtaps : int
+        The number of taps in the FIR filter. `numtaps` must be odd.
+    bands : array_like
+        A monotonic nondecreasing sequence containing the band edges in
+        Hz. All elements must be non-negative and less than or equal to
+        the Nyquist frequency given by `nyq`. The bands are specified as
+        frequency pairs, thus, if using a 1D array, its length must be
+        even, e.g., `np.array([0, 1, 2, 3, 4, 5])`. Alternatively, the
+        bands can be specified as an nx2 sized 2D array, where n is the
+        number of bands, e.g, `np.array([[0, 1], [2, 3], [4, 5]])`.
+    desired : array_like
+        A sequence the same size as `bands` containing the desired gain
+        at the start and end point of each band.
+    weight : array_like, optional
+        A relative weighting to give to each band region when solving
+        the least squares problem. `weight` has to be half the size of
+        `bands`.
+    nyq : float, optional, deprecated
+        This is the Nyquist frequency. Each frequency in `bands` must be
+        between 0 and `nyq` (inclusive). Default is 1.
+
+        .. deprecated:: 1.0.0
+           `firls` keyword argument `nyq` is deprecated in favour of `fs` and
+           will be removed in SciPy 1.14.0.
+    fs : float, optional
+        The sampling frequency of the signal. Each frequency in `bands`
+        must be between 0 and ``fs/2`` (inclusive). Default is 2.
+
+    Returns
+    -------
+    coeffs : ndarray
+        Coefficients of the optimal (in a least squares sense) FIR filter.
+
+    See Also
+    --------
+    firwin
+    firwin2
+    minimum_phase
+    remez
+
+    Notes
+    -----
+    This implementation follows the algorithm given in [1]_.
+    As noted there, least squares design has multiple advantages:
+
+        1. Optimal in a least-squares sense.
+        2. Simple, non-iterative method.
+        3. The general solution can obtained by solving a linear
+           system of equations.
+        4. Allows the use of a frequency dependent weighting function.
+
+    This function constructs a Type I linear phase FIR filter, which
+    contains an odd number of `coeffs` satisfying for :math:`n < numtaps`:
+
+    .. math:: coeffs(n) = coeffs(numtaps - 1 - n)
+
+    The odd number of coefficients and filter symmetry avoid boundary
+    conditions that could otherwise occur at the Nyquist and 0 frequencies
+    (e.g., for Type II, III, or IV variants).
+
+    .. versionadded:: 0.18
+
+    References
+    ----------
+    .. [1] Ivan Selesnick, Linear-Phase Fir Filter Design By Least Squares.
+           OpenStax CNX. Aug 9, 2005.
+           http://cnx.org/contents/eb1ecb35-03a9-4610-ba87-41cd771c95f2@7
+
+    Examples
+    --------
+    We want to construct a band-pass filter. Note that the behavior in the
+    frequency ranges between our stop bands and pass bands is unspecified,
+    and thus may overshoot depending on the parameters of our filter:
+
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+    >>> fig, axs = plt.subplots(2)
+    >>> fs = 10.0  # Hz
+    >>> desired = (0, 0, 1, 1, 0, 0)
+    >>> for bi, bands in enumerate(((0, 1, 2, 3, 4, 5), (0, 1, 2, 4, 4.5, 5))):
+    ...     fir_firls = signal.firls(73, bands, desired, fs=fs)
+    ...     fir_remez = signal.remez(73, bands, desired[::2], fs=fs)
+    ...     fir_firwin2 = signal.firwin2(73, bands, desired, fs=fs)
+    ...     hs = list()
+    ...     ax = axs[bi]
+    ...     for fir in (fir_firls, fir_remez, fir_firwin2):
+    ...         freq, response = signal.freqz(fir)
+    ...         hs.append(ax.semilogy(0.5*fs*freq/np.pi, np.abs(response))[0])
+    ...     for band, gains in zip(zip(bands[::2], bands[1::2]),
+    ...                            zip(desired[::2], desired[1::2])):
+    ...         ax.semilogy(band, np.maximum(gains, 1e-7), 'k--', linewidth=2)
+    ...     if bi == 0:
+    ...         ax.legend(hs, ('firls', 'remez', 'firwin2'),
+    ...                   loc='lower center', frameon=False)
+    ...     else:
+    ...         ax.set_xlabel('Frequency (Hz)')
+    ...     ax.grid(True)
+    ...     ax.set(title='Band-pass %d-%d Hz' % bands[2:4], ylabel='Magnitude')
+    ...
+    >>> fig.tight_layout()
+    >>> plt.show()
+
+    """
+    fs = _validate_fs(fs, allow_none=True)
+    nyq = 0.5 * _get_fs(fs, nyq)
+
+    numtaps = int(numtaps)
+    if numtaps % 2 == 0 or numtaps < 1:
+        raise ValueError("numtaps must be odd and >= 1")
+    M = (numtaps-1) // 2
+
+    # normalize bands 0->1 and make it 2 columns
+    nyq = float(nyq)
+    if nyq <= 0:
+        raise ValueError('nyq must be positive, got %s <= 0.' % nyq)
+    bands = np.asarray(bands).flatten() / nyq
+    if len(bands) % 2 != 0:
+        raise ValueError("bands must contain frequency pairs.")
+    if (bands < 0).any() or (bands > 1).any():
+        raise ValueError("bands must be between 0 and 1 relative to Nyquist")
+    bands.shape = (-1, 2)
+
+    # check remaining params
+    desired = np.asarray(desired).flatten()
+    if bands.size != desired.size:
+        raise ValueError("desired must have one entry per frequency, got {} "
+                         "gains for {} frequencies.".format(desired.size, bands.size))
+    desired.shape = (-1, 2)
+    if (np.diff(bands) <= 0).any() or (np.diff(bands[:, 0]) < 0).any():
+        raise ValueError("bands must be monotonically nondecreasing and have "
+                         "width > 0.")
+    if (bands[:-1, 1] > bands[1:, 0]).any():
+        raise ValueError("bands must not overlap.")
+    if (desired < 0).any():
+        raise ValueError("desired must be non-negative.")
+    if weight is None:
+        weight = np.ones(len(desired))
+    weight = np.asarray(weight).flatten()
+    if len(weight) != len(desired):
+        raise ValueError("weight must be the same size as the number of "
+                         f"band pairs ({len(bands)}).")
+    if (weight < 0).any():
+        raise ValueError("weight must be non-negative.")
+
+    # Set up the linear matrix equation to be solved, Qa = b
+
+    # We can express Q(k,n) = 0.5 Q1(k,n) + 0.5 Q2(k,n)
+    # where Q1(k,n)=q(k-n) and Q2(k,n)=q(k+n), i.e. a Toeplitz plus Hankel.
+
+    # We omit the factor of 0.5 above, instead adding it during coefficient
+    # calculation.
+
+    # We also omit the 1/π from both Q and b equations, as they cancel
+    # during solving.
+
+    # We have that:
+    #     q(n) = 1/π ∫W(ω)cos(nω)dω (over 0->π)
+    # Using our normalization ω=πf and with a constant weight W over each
+    # interval f1->f2 we get:
+    #     q(n) = W∫cos(πnf)df (0->1) = Wf sin(πnf)/πnf
+    # integrated over each f1->f2 pair (i.e., value at f2 - value at f1).
+    n = np.arange(numtaps)[:, np.newaxis, np.newaxis]
+    q = np.dot(np.diff(np.sinc(bands * n) * bands, axis=2)[:, :, 0], weight)
+
+    # Now we assemble our sum of Toeplitz and Hankel
+    Q1 = toeplitz(q[:M+1])
+    Q2 = hankel(q[:M+1], q[M:])
+    Q = Q1 + Q2
+
+    # Now for b(n) we have that:
+    #     b(n) = 1/π ∫ W(ω)D(ω)cos(nω)dω (over 0->π)
+    # Using our normalization ω=πf and with a constant weight W over each
+    # interval and a linear term for D(ω) we get (over each f1->f2 interval):
+    #     b(n) = W ∫ (mf+c)cos(πnf)df
+    #          = f(mf+c)sin(πnf)/πnf + mf**2 cos(nπf)/(πnf)**2
+    # integrated over each f1->f2 pair (i.e., value at f2 - value at f1).
+    n = n[:M + 1]  # only need this many coefficients here
+    # Choose m and c such that we are at the start and end weights
+    m = (np.diff(desired, axis=1) / np.diff(bands, axis=1))
+    c = desired[:, [0]] - bands[:, [0]] * m
+    b = bands * (m*bands + c) * np.sinc(bands * n)
+    # Use L'Hospital's rule here for cos(nπf)/(πnf)**2 @ n=0
+    b[0] -= m * bands * bands / 2.
+    b[1:] += m * np.cos(n[1:] * np.pi * bands) / (np.pi * n[1:]) ** 2
+    b = np.dot(np.diff(b, axis=2)[:, :, 0], weight)
+
+    # Now we can solve the equation
+    try:  # try the fast way
+        with warnings.catch_warnings(record=True) as w:
+            warnings.simplefilter('always')
+            a = solve(Q, b, assume_a="pos", check_finite=False)
+        for ww in w:
+            if (ww.category == LinAlgWarning and
+                    str(ww.message).startswith('Ill-conditioned matrix')):
+                raise LinAlgError(str(ww.message))
+    except LinAlgError:  # in case Q is rank deficient
+        # This is faster than pinvh, even though we don't explicitly use
+        # the symmetry here. gelsy was faster than gelsd and gelss in
+        # some non-exhaustive tests.
+        a = lstsq(Q, b, lapack_driver='gelsy')[0]
+
+    # make coefficients symmetric (linear phase)
+    coeffs = np.hstack((a[:0:-1], 2 * a[0], a[1:]))
+    return coeffs
+
+
+def _dhtm(mag):
+    """Compute the modified 1-D discrete Hilbert transform
+
+    Parameters
+    ----------
+    mag : ndarray
+        The magnitude spectrum. Should be 1-D with an even length, and
+        preferably a fast length for FFT/IFFT.
+    """
+    # Adapted based on code by Niranjan Damera-Venkata,
+    # Brian L. Evans and Shawn R. McCaslin (see refs for `minimum_phase`)
+    sig = np.zeros(len(mag))
+    # Leave Nyquist and DC at 0, knowing np.abs(fftfreq(N)[midpt]) == 0.5
+    midpt = len(mag) // 2
+    sig[1:midpt] = 1
+    sig[midpt+1:] = -1
+    # eventually if we want to support complex filters, we will need a
+    # np.abs() on the mag inside the log, and should remove the .real
+    recon = ifft(mag * np.exp(fft(sig * ifft(np.log(mag))))).real
+    return recon
+
+
+def minimum_phase(h, method='homomorphic', n_fft=None):
+    """Convert a linear-phase FIR filter to minimum phase
+
+    Parameters
+    ----------
+    h : array
+        Linear-phase FIR filter coefficients.
+    method : {'hilbert', 'homomorphic'}
+        The method to use:
+
+            'homomorphic' (default)
+                This method [4]_ [5]_ works best with filters with an
+                odd number of taps, and the resulting minimum phase filter
+                will have a magnitude response that approximates the square
+                root of the original filter's magnitude response.
+
+            'hilbert'
+                This method [1]_ is designed to be used with equiripple
+                filters (e.g., from `remez`) with unity or zero gain
+                regions.
+
+    n_fft : int
+        The number of points to use for the FFT. Should be at least a
+        few times larger than the signal length (see Notes).
+
+    Returns
+    -------
+    h_minimum : array
+        The minimum-phase version of the filter, with length
+        ``(length(h) + 1) // 2``.
+
+    See Also
+    --------
+    firwin
+    firwin2
+    remez
+
+    Notes
+    -----
+    Both the Hilbert [1]_ or homomorphic [4]_ [5]_ methods require selection
+    of an FFT length to estimate the complex cepstrum of the filter.
+
+    In the case of the Hilbert method, the deviation from the ideal
+    spectrum ``epsilon`` is related to the number of stopband zeros
+    ``n_stop`` and FFT length ``n_fft`` as::
+
+        epsilon = 2. * n_stop / n_fft
+
+    For example, with 100 stopband zeros and a FFT length of 2048,
+    ``epsilon = 0.0976``. If we conservatively assume that the number of
+    stopband zeros is one less than the filter length, we can take the FFT
+    length to be the next power of 2 that satisfies ``epsilon=0.01`` as::
+
+        n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01)))
+
+    This gives reasonable results for both the Hilbert and homomorphic
+    methods, and gives the value used when ``n_fft=None``.
+
+    Alternative implementations exist for creating minimum-phase filters,
+    including zero inversion [2]_ and spectral factorization [3]_ [4]_.
+    For more information, see:
+
+        http://dspguru.com/dsp/howtos/how-to-design-minimum-phase-fir-filters
+
+    References
+    ----------
+    .. [1] N. Damera-Venkata and B. L. Evans, "Optimal design of real and
+           complex minimum phase digital FIR filters," Acoustics, Speech,
+           and Signal Processing, 1999. Proceedings., 1999 IEEE International
+           Conference on, Phoenix, AZ, 1999, pp. 1145-1148 vol.3.
+           :doi:`10.1109/ICASSP.1999.756179`
+    .. [2] X. Chen and T. W. Parks, "Design of optimal minimum phase FIR
+           filters by direct factorization," Signal Processing,
+           vol. 10, no. 4, pp. 369-383, Jun. 1986.
+    .. [3] T. Saramaki, "Finite Impulse Response Filter Design," in
+           Handbook for Digital Signal Processing, chapter 4,
+           New York: Wiley-Interscience, 1993.
+    .. [4] J. S. Lim, Advanced Topics in Signal Processing.
+           Englewood Cliffs, N.J.: Prentice Hall, 1988.
+    .. [5] A. V. Oppenheim, R. W. Schafer, and J. R. Buck,
+           "Discrete-Time Signal Processing," 2nd edition.
+           Upper Saddle River, N.J.: Prentice Hall, 1999.
+
+    Examples
+    --------
+    Create an optimal linear-phase filter, then convert it to minimum phase:
+
+    >>> import numpy as np
+    >>> from scipy.signal import remez, minimum_phase, freqz, group_delay
+    >>> import matplotlib.pyplot as plt
+    >>> freq = [0, 0.2, 0.3, 1.0]
+    >>> desired = [1, 0]
+    >>> h_linear = remez(151, freq, desired, fs=2.)
+
+    Convert it to minimum phase:
+
+    >>> h_min_hom = minimum_phase(h_linear, method='homomorphic')
+    >>> h_min_hil = minimum_phase(h_linear, method='hilbert')
+
+    Compare the three filters:
+
+    >>> fig, axs = plt.subplots(4, figsize=(4, 8))
+    >>> for h, style, color in zip((h_linear, h_min_hom, h_min_hil),
+    ...                            ('-', '-', '--'), ('k', 'r', 'c')):
+    ...     w, H = freqz(h)
+    ...     w, gd = group_delay((h, 1))
+    ...     w /= np.pi
+    ...     axs[0].plot(h, color=color, linestyle=style)
+    ...     axs[1].plot(w, np.abs(H), color=color, linestyle=style)
+    ...     axs[2].plot(w, 20 * np.log10(np.abs(H)), color=color, linestyle=style)
+    ...     axs[3].plot(w, gd, color=color, linestyle=style)
+    >>> for ax in axs:
+    ...     ax.grid(True, color='0.5')
+    ...     ax.fill_between(freq[1:3], *ax.get_ylim(), color='#ffeeaa', zorder=1)
+    >>> axs[0].set(xlim=[0, len(h_linear) - 1], ylabel='Amplitude', xlabel='Samples')
+    >>> axs[1].legend(['Linear', 'Min-Hom', 'Min-Hil'], title='Phase')
+    >>> for ax, ylim in zip(axs[1:], ([0, 1.1], [-150, 10], [-60, 60])):
+    ...     ax.set(xlim=[0, 1], ylim=ylim, xlabel='Frequency')
+    >>> axs[1].set(ylabel='Magnitude')
+    >>> axs[2].set(ylabel='Magnitude (dB)')
+    >>> axs[3].set(ylabel='Group delay')
+    >>> plt.tight_layout()
+
+    """
+    h = np.asarray(h)
+    if np.iscomplexobj(h):
+        raise ValueError('Complex filters not supported')
+    if h.ndim != 1 or h.size <= 2:
+        raise ValueError('h must be 1-D and at least 2 samples long')
+    n_half = len(h) // 2
+    if not np.allclose(h[-n_half:][::-1], h[:n_half]):
+        warnings.warn('h does not appear to by symmetric, conversion may fail',
+                      RuntimeWarning, stacklevel=2)
+    if not isinstance(method, str) or method not in \
+            ('homomorphic', 'hilbert',):
+        raise ValueError(f'method must be "homomorphic" or "hilbert", got {method!r}')
+    if n_fft is None:
+        n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01)))
+    n_fft = int(n_fft)
+    if n_fft < len(h):
+        raise ValueError('n_fft must be at least len(h)==%s' % len(h))
+    if method == 'hilbert':
+        w = np.arange(n_fft) * (2 * np.pi / n_fft * n_half)
+        H = np.real(fft(h, n_fft) * np.exp(1j * w))
+        dp = max(H) - 1
+        ds = 0 - min(H)
+        S = 4. / (np.sqrt(1+dp+ds) + np.sqrt(1-dp+ds)) ** 2
+        H += ds
+        H *= S
+        H = np.sqrt(H, out=H)
+        H += 1e-10  # ensure that the log does not explode
+        h_minimum = _dhtm(H)
+    else:  # method == 'homomorphic'
+        # zero-pad; calculate the DFT
+        h_temp = np.abs(fft(h, n_fft))
+        # take 0.25*log(|H|**2) = 0.5*log(|H|)
+        h_temp += 1e-7 * h_temp[h_temp > 0].min()  # don't let log blow up
+        np.log(h_temp, out=h_temp)
+        h_temp *= 0.5
+        # IDFT
+        h_temp = ifft(h_temp).real
+        # multiply pointwise by the homomorphic filter
+        # lmin[n] = 2u[n] - d[n]
+        win = np.zeros(n_fft)
+        win[0] = 1
+        stop = (len(h) + 1) // 2
+        win[1:stop] = 2
+        if len(h) % 2:
+            win[stop] = 1
+        h_temp *= win
+        h_temp = ifft(np.exp(fft(h_temp)))
+        h_minimum = h_temp.real
+    n_out = n_half + len(h) % 2
+    return h_minimum[:n_out]

env-llmeval/lib/python3.10/site-packages/scipy/signal/_peak_finding.py

@@ -0,0 +1,1312 @@
+"""
+Functions for identifying peaks in signals.
+"""
+import math
+import numpy as np
+
+from scipy.signal._wavelets import _cwt, _ricker
+from scipy.stats import scoreatpercentile
+
+from ._peak_finding_utils import (
+    _local_maxima_1d,
+    _select_by_peak_distance,
+    _peak_prominences,
+    _peak_widths
+)
+
+
+__all__ = ['argrelmin', 'argrelmax', 'argrelextrema', 'peak_prominences',
+           'peak_widths', 'find_peaks', 'find_peaks_cwt']
+
+
+def _boolrelextrema(data, comparator, axis=0, order=1, mode='clip'):
+    """
+    Calculate the relative extrema of `data`.
+
+    Relative extrema are calculated by finding locations where
+    ``comparator(data[n], data[n+1:n+order+1])`` is True.
+
+    Parameters
+    ----------
+    data : ndarray
+        Array in which to find the relative extrema.
+    comparator : callable
+        Function to use to compare two data points.
+        Should take two arrays as arguments.
+    axis : int, optional
+        Axis over which to select from `data`. Default is 0.
+    order : int, optional
+        How many points on each side to use for the comparison
+        to consider ``comparator(n,n+x)`` to be True.
+    mode : str, optional
+        How the edges of the vector are treated. 'wrap' (wrap around) or
+        'clip' (treat overflow as the same as the last (or first) element).
+        Default 'clip'. See numpy.take.
+
+    Returns
+    -------
+    extrema : ndarray
+        Boolean array of the same shape as `data` that is True at an extrema,
+        False otherwise.
+
+    See also
+    --------
+    argrelmax, argrelmin
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._peak_finding import _boolrelextrema
+    >>> testdata = np.array([1,2,3,2,1])
+    >>> _boolrelextrema(testdata, np.greater, axis=0)
+    array([False, False,  True, False, False], dtype=bool)
+
+    """
+    if (int(order) != order) or (order < 1):
+        raise ValueError('Order must be an int >= 1')
+
+    datalen = data.shape[axis]
+    locs = np.arange(0, datalen)
+
+    results = np.ones(data.shape, dtype=bool)
+    main = data.take(locs, axis=axis, mode=mode)
+    for shift in range(1, order + 1):
+        plus = data.take(locs + shift, axis=axis, mode=mode)
+        minus = data.take(locs - shift, axis=axis, mode=mode)
+        results &= comparator(main, plus)
+        results &= comparator(main, minus)
+        if ~results.any():
+            return results
+    return results
+
+
+def argrelmin(data, axis=0, order=1, mode='clip'):
+    """
+    Calculate the relative minima of `data`.
+
+    Parameters
+    ----------
+    data : ndarray
+        Array in which to find the relative minima.
+    axis : int, optional
+        Axis over which to select from `data`. Default is 0.
+    order : int, optional
+        How many points on each side to use for the comparison
+        to consider ``comparator(n, n+x)`` to be True.
+    mode : str, optional
+        How the edges of the vector are treated.
+        Available options are 'wrap' (wrap around) or 'clip' (treat overflow
+        as the same as the last (or first) element).
+        Default 'clip'. See numpy.take.
+
+    Returns
+    -------
+    extrema : tuple of ndarrays
+        Indices of the minima in arrays of integers. ``extrema[k]`` is
+        the array of indices of axis `k` of `data`. Note that the
+        return value is a tuple even when `data` is 1-D.
+
+    See Also
+    --------
+    argrelextrema, argrelmax, find_peaks
+
+    Notes
+    -----
+    This function uses `argrelextrema` with np.less as comparator. Therefore, it
+    requires a strict inequality on both sides of a value to consider it a
+    minimum. This means flat minima (more than one sample wide) are not detected.
+    In case of 1-D `data` `find_peaks` can be used to detect all
+    local minima, including flat ones, by calling it with negated `data`.
+
+    .. versionadded:: 0.11.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import argrelmin
+    >>> x = np.array([2, 1, 2, 3, 2, 0, 1, 0])
+    >>> argrelmin(x)
+    (array([1, 5]),)
+    >>> y = np.array([[1, 2, 1, 2],
+    ...               [2, 2, 0, 0],
+    ...               [5, 3, 4, 4]])
+    ...
+    >>> argrelmin(y, axis=1)
+    (array([0, 2]), array([2, 1]))
+
+    """
+    return argrelextrema(data, np.less, axis, order, mode)
+
+
+def argrelmax(data, axis=0, order=1, mode='clip'):
+    """
+    Calculate the relative maxima of `data`.
+
+    Parameters
+    ----------
+    data : ndarray
+        Array in which to find the relative maxima.
+    axis : int, optional
+        Axis over which to select from `data`. Default is 0.
+    order : int, optional
+        How many points on each side to use for the comparison
+        to consider ``comparator(n, n+x)`` to be True.
+    mode : str, optional
+        How the edges of the vector are treated.
+        Available options are 'wrap' (wrap around) or 'clip' (treat overflow
+        as the same as the last (or first) element).
+        Default 'clip'. See `numpy.take`.
+
+    Returns
+    -------
+    extrema : tuple of ndarrays
+        Indices of the maxima in arrays of integers. ``extrema[k]`` is
+        the array of indices of axis `k` of `data`. Note that the
+        return value is a tuple even when `data` is 1-D.
+
+    See Also
+    --------
+    argrelextrema, argrelmin, find_peaks
+
+    Notes
+    -----
+    This function uses `argrelextrema` with np.greater as comparator. Therefore,
+    it  requires a strict inequality on both sides of a value to consider it a
+    maximum. This means flat maxima (more than one sample wide) are not detected.
+    In case of 1-D `data` `find_peaks` can be used to detect all
+    local maxima, including flat ones.
+
+    .. versionadded:: 0.11.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import argrelmax
+    >>> x = np.array([2, 1, 2, 3, 2, 0, 1, 0])
+    >>> argrelmax(x)
+    (array([3, 6]),)
+    >>> y = np.array([[1, 2, 1, 2],
+    ...               [2, 2, 0, 0],
+    ...               [5, 3, 4, 4]])
+    ...
+    >>> argrelmax(y, axis=1)
+    (array([0]), array([1]))
+    """
+    return argrelextrema(data, np.greater, axis, order, mode)
+
+
+def argrelextrema(data, comparator, axis=0, order=1, mode='clip'):
+    """
+    Calculate the relative extrema of `data`.
+
+    Parameters
+    ----------
+    data : ndarray
+        Array in which to find the relative extrema.
+    comparator : callable
+        Function to use to compare two data points.
+        Should take two arrays as arguments.
+    axis : int, optional
+        Axis over which to select from `data`. Default is 0.
+    order : int, optional
+        How many points on each side to use for the comparison
+        to consider ``comparator(n, n+x)`` to be True.
+    mode : str, optional
+        How the edges of the vector are treated. 'wrap' (wrap around) or
+        'clip' (treat overflow as the same as the last (or first) element).
+        Default is 'clip'. See `numpy.take`.
+
+    Returns
+    -------
+    extrema : tuple of ndarrays
+        Indices of the maxima in arrays of integers. ``extrema[k]`` is
+        the array of indices of axis `k` of `data`. Note that the
+        return value is a tuple even when `data` is 1-D.
+
+    See Also
+    --------
+    argrelmin, argrelmax
+
+    Notes
+    -----
+
+    .. versionadded:: 0.11.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import argrelextrema
+    >>> x = np.array([2, 1, 2, 3, 2, 0, 1, 0])
+    >>> argrelextrema(x, np.greater)
+    (array([3, 6]),)
+    >>> y = np.array([[1, 2, 1, 2],
+    ...               [2, 2, 0, 0],
+    ...               [5, 3, 4, 4]])
+    ...
+    >>> argrelextrema(y, np.less, axis=1)
+    (array([0, 2]), array([2, 1]))
+
+    """
+    results = _boolrelextrema(data, comparator,
+                              axis, order, mode)
+    return np.nonzero(results)
+
+
+def _arg_x_as_expected(value):
+    """Ensure argument `x` is a 1-D C-contiguous array of dtype('float64').
+
+    Used in `find_peaks`, `peak_prominences` and `peak_widths` to make `x`
+    compatible with the signature of the wrapped Cython functions.
+
+    Returns
+    -------
+    value : ndarray
+        A 1-D C-contiguous array with dtype('float64').
+    """
+    value = np.asarray(value, order='C', dtype=np.float64)
+    if value.ndim != 1:
+        raise ValueError('`x` must be a 1-D array')
+    return value
+
+
+def _arg_peaks_as_expected(value):
+    """Ensure argument `peaks` is a 1-D C-contiguous array of dtype('intp').
+
+    Used in `peak_prominences` and `peak_widths` to make `peaks` compatible
+    with the signature of the wrapped Cython functions.
+
+    Returns
+    -------
+    value : ndarray
+        A 1-D C-contiguous array with dtype('intp').
+    """
+    value = np.asarray(value)
+    if value.size == 0:
+        # Empty arrays default to np.float64 but are valid input
+        value = np.array([], dtype=np.intp)
+    try:
+        # Safely convert to C-contiguous array of type np.intp
+        value = value.astype(np.intp, order='C', casting='safe',
+                             subok=False, copy=False)
+    except TypeError as e:
+        raise TypeError("cannot safely cast `peaks` to dtype('intp')") from e
+    if value.ndim != 1:
+        raise ValueError('`peaks` must be a 1-D array')
+    return value
+
+
+def _arg_wlen_as_expected(value):
+    """Ensure argument `wlen` is of type `np.intp` and larger than 1.
+
+    Used in `peak_prominences` and `peak_widths`.
+
+    Returns
+    -------
+    value : np.intp
+        The original `value` rounded up to an integer or -1 if `value` was
+        None.
+    """
+    if value is None:
+        # _peak_prominences expects an intp; -1 signals that no value was
+        # supplied by the user
+        value = -1
+    elif 1 < value:
+        # Round up to a positive integer
+        if isinstance(value, float):
+            value = math.ceil(value)
+        value = np.intp(value)
+    else:
+        raise ValueError(f'`wlen` must be larger than 1, was {value}')
+    return value
+
+
+def peak_prominences(x, peaks, wlen=None):
+    """
+    Calculate the prominence of each peak in a signal.
+
+    The prominence of a peak measures how much a peak stands out from the
+    surrounding baseline of the signal and is defined as the vertical distance
+    between the peak and its lowest contour line.
+
+    Parameters
+    ----------
+    x : sequence
+        A signal with peaks.
+    peaks : sequence
+        Indices of peaks in `x`.
+    wlen : int, optional
+        A window length in samples that optionally limits the evaluated area for
+        each peak to a subset of `x`. The peak is always placed in the middle of
+        the window therefore the given length is rounded up to the next odd
+        integer. This parameter can speed up the calculation (see Notes).
+
+    Returns
+    -------
+    prominences : ndarray
+        The calculated prominences for each peak in `peaks`.
+    left_bases, right_bases : ndarray
+        The peaks' bases as indices in `x` to the left and right of each peak.
+        The higher base of each pair is a peak's lowest contour line.
+
+    Raises
+    ------
+    ValueError
+        If a value in `peaks` is an invalid index for `x`.
+
+    Warns
+    -----
+    PeakPropertyWarning
+        For indices in `peaks` that don't point to valid local maxima in `x`,
+        the returned prominence will be 0 and this warning is raised. This
+        also happens if `wlen` is smaller than the plateau size of a peak.
+
+    Warnings
+    --------
+    This function may return unexpected results for data containing NaNs. To
+    avoid this, NaNs should either be removed or replaced.
+
+    See Also
+    --------
+    find_peaks
+        Find peaks inside a signal based on peak properties.
+    peak_widths
+        Calculate the width of peaks.
+
+    Notes
+    -----
+    Strategy to compute a peak's prominence:
+
+    1. Extend a horizontal line from the current peak to the left and right
+       until the line either reaches the window border (see `wlen`) or
+       intersects the signal again at the slope of a higher peak. An
+       intersection with a peak of the same height is ignored.
+    2. On each side find the minimal signal value within the interval defined
+       above. These points are the peak's bases.
+    3. The higher one of the two bases marks the peak's lowest contour line. The
+       prominence can then be calculated as the vertical difference between the
+       peaks height itself and its lowest contour line.
+
+    Searching for the peak's bases can be slow for large `x` with periodic
+    behavior because large chunks or even the full signal need to be evaluated
+    for the first algorithmic step. This evaluation area can be limited with the
+    parameter `wlen` which restricts the algorithm to a window around the
+    current peak and can shorten the calculation time if the window length is
+    short in relation to `x`.
+    However, this may stop the algorithm from finding the true global contour
+    line if the peak's true bases are outside this window. Instead, a higher
+    contour line is found within the restricted window leading to a smaller
+    calculated prominence. In practice, this is only relevant for the highest set
+    of peaks in `x`. This behavior may even be used intentionally to calculate
+    "local" prominences.
+
+    .. versionadded:: 1.1.0
+
+    References
+    ----------
+    .. [1] Wikipedia Article for Topographic Prominence:
+       https://en.wikipedia.org/wiki/Topographic_prominence
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import find_peaks, peak_prominences
+    >>> import matplotlib.pyplot as plt
+
+    Create a test signal with two overlaid harmonics
+
+    >>> x = np.linspace(0, 6 * np.pi, 1000)
+    >>> x = np.sin(x) + 0.6 * np.sin(2.6 * x)
+
+    Find all peaks and calculate prominences
+
+    >>> peaks, _ = find_peaks(x)
+    >>> prominences = peak_prominences(x, peaks)[0]
+    >>> prominences
+    array([1.24159486, 0.47840168, 0.28470524, 3.10716793, 0.284603  ,
+           0.47822491, 2.48340261, 0.47822491])
+
+    Calculate the height of each peak's contour line and plot the results
+
+    >>> contour_heights = x[peaks] - prominences
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.vlines(x=peaks, ymin=contour_heights, ymax=x[peaks])
+    >>> plt.show()
+
+    Let's evaluate a second example that demonstrates several edge cases for
+    one peak at index 5.
+
+    >>> x = np.array([0, 1, 0, 3, 1, 3, 0, 4, 0])
+    >>> peaks = np.array([5])
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.show()
+    >>> peak_prominences(x, peaks)  # -> (prominences, left_bases, right_bases)
+    (array([3.]), array([2]), array([6]))
+
+    Note how the peak at index 3 of the same height is not considered as a
+    border while searching for the left base. Instead, two minima at 0 and 2
+    are found in which case the one closer to the evaluated peak is always
+    chosen. On the right side, however, the base must be placed at 6 because the
+    higher peak represents the right border to the evaluated area.
+
+    >>> peak_prominences(x, peaks, wlen=3.1)
+    (array([2.]), array([4]), array([6]))
+
+    Here, we restricted the algorithm to a window from 3 to 7 (the length is 5
+    samples because `wlen` was rounded up to the next odd integer). Thus, the
+    only two candidates in the evaluated area are the two neighboring samples
+    and a smaller prominence is calculated.
+    """
+    x = _arg_x_as_expected(x)
+    peaks = _arg_peaks_as_expected(peaks)
+    wlen = _arg_wlen_as_expected(wlen)
+    return _peak_prominences(x, peaks, wlen)
+
+
+def peak_widths(x, peaks, rel_height=0.5, prominence_data=None, wlen=None):
+    """
+    Calculate the width of each peak in a signal.
+
+    This function calculates the width of a peak in samples at a relative
+    distance to the peak's height and prominence.
+
+    Parameters
+    ----------
+    x : sequence
+        A signal with peaks.
+    peaks : sequence
+        Indices of peaks in `x`.
+    rel_height : float, optional
+        Chooses the relative height at which the peak width is measured as a
+        percentage of its prominence. 1.0 calculates the width of the peak at
+        its lowest contour line while 0.5 evaluates at half the prominence
+        height. Must be at least 0. See notes for further explanation.
+    prominence_data : tuple, optional
+        A tuple of three arrays matching the output of `peak_prominences` when
+        called with the same arguments `x` and `peaks`. This data are calculated
+        internally if not provided.
+    wlen : int, optional
+        A window length in samples passed to `peak_prominences` as an optional
+        argument for internal calculation of `prominence_data`. This argument
+        is ignored if `prominence_data` is given.
+
+    Returns
+    -------
+    widths : ndarray
+        The widths for each peak in samples.
+    width_heights : ndarray
+        The height of the contour lines at which the `widths` where evaluated.
+    left_ips, right_ips : ndarray
+        Interpolated positions of left and right intersection points of a
+        horizontal line at the respective evaluation height.
+
+    Raises
+    ------
+    ValueError
+        If `prominence_data` is supplied but doesn't satisfy the condition
+        ``0 <= left_base <= peak <= right_base < x.shape[0]`` for each peak,
+        has the wrong dtype, is not C-contiguous or does not have the same
+        shape.
+
+    Warns
+    -----
+    PeakPropertyWarning
+        Raised if any calculated width is 0. This may stem from the supplied
+        `prominence_data` or if `rel_height` is set to 0.
+
+    Warnings
+    --------
+    This function may return unexpected results for data containing NaNs. To
+    avoid this, NaNs should either be removed or replaced.
+
+    See Also
+    --------
+    find_peaks
+        Find peaks inside a signal based on peak properties.
+    peak_prominences
+        Calculate the prominence of peaks.
+
+    Notes
+    -----
+    The basic algorithm to calculate a peak's width is as follows:
+
+    * Calculate the evaluation height :math:`h_{eval}` with the formula
+      :math:`h_{eval} = h_{Peak} - P \\cdot R`, where :math:`h_{Peak}` is the
+      height of the peak itself, :math:`P` is the peak's prominence and
+      :math:`R` a positive ratio specified with the argument `rel_height`.
+    * Draw a horizontal line at the evaluation height to both sides, starting at
+      the peak's current vertical position until the lines either intersect a
+      slope, the signal border or cross the vertical position of the peak's
+      base (see `peak_prominences` for an definition). For the first case,
+      intersection with the signal, the true intersection point is estimated
+      with linear interpolation.
+    * Calculate the width as the horizontal distance between the chosen
+      endpoints on both sides. As a consequence of this the maximal possible
+      width for each peak is the horizontal distance between its bases.
+
+    As shown above to calculate a peak's width its prominence and bases must be
+    known. You can supply these yourself with the argument `prominence_data`.
+    Otherwise, they are internally calculated (see `peak_prominences`).
+
+    .. versionadded:: 1.1.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal import chirp, find_peaks, peak_widths
+    >>> import matplotlib.pyplot as plt
+
+    Create a test signal with two overlaid harmonics
+
+    >>> x = np.linspace(0, 6 * np.pi, 1000)
+    >>> x = np.sin(x) + 0.6 * np.sin(2.6 * x)
+
+    Find all peaks and calculate their widths at the relative height of 0.5
+    (contour line at half the prominence height) and 1 (at the lowest contour
+    line at full prominence height).
+
+    >>> peaks, _ = find_peaks(x)
+    >>> results_half = peak_widths(x, peaks, rel_height=0.5)
+    >>> results_half[0]  # widths
+    array([ 64.25172825,  41.29465463,  35.46943289, 104.71586081,
+            35.46729324,  41.30429622, 181.93835853,  45.37078546])
+    >>> results_full = peak_widths(x, peaks, rel_height=1)
+    >>> results_full[0]  # widths
+    array([181.9396084 ,  72.99284945,  61.28657872, 373.84622694,
+        61.78404617,  72.48822812, 253.09161876,  79.36860878])
+
+    Plot signal, peaks and contour lines at which the widths where calculated
+
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.hlines(*results_half[1:], color="C2")
+    >>> plt.hlines(*results_full[1:], color="C3")
+    >>> plt.show()
+    """
+    x = _arg_x_as_expected(x)
+    peaks = _arg_peaks_as_expected(peaks)
+    if prominence_data is None:
+        # Calculate prominence if not supplied and use wlen if supplied.
+        wlen = _arg_wlen_as_expected(wlen)
+        prominence_data = _peak_prominences(x, peaks, wlen)
+    return _peak_widths(x, peaks, rel_height, *prominence_data)
+
+
+def _unpack_condition_args(interval, x, peaks):
+    """
+    Parse condition arguments for `find_peaks`.
+
+    Parameters
+    ----------
+    interval : number or ndarray or sequence
+        Either a number or ndarray or a 2-element sequence of the former. The
+        first value is always interpreted as `imin` and the second, if supplied,
+        as `imax`.
+    x : ndarray
+        The signal with `peaks`.
+    peaks : ndarray
+        An array with indices used to reduce `imin` and / or `imax` if those are
+        arrays.
+
+    Returns
+    -------
+    imin, imax : number or ndarray or None
+        Minimal and maximal value in `argument`.
+
+    Raises
+    ------
+    ValueError :
+        If interval border is given as array and its size does not match the size
+        of `x`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.1.0
+    """
+    try:
+        imin, imax = interval
+    except (TypeError, ValueError):
+        imin, imax = (interval, None)
+
+    # Reduce arrays if arrays
+    if isinstance(imin, np.ndarray):
+        if imin.size != x.size:
+            raise ValueError('array size of lower interval border must match x')
+        imin = imin[peaks]
+    if isinstance(imax, np.ndarray):
+        if imax.size != x.size:
+            raise ValueError('array size of upper interval border must match x')
+        imax = imax[peaks]
+
+    return imin, imax
+
+
+def _select_by_property(peak_properties, pmin, pmax):
+    """
+    Evaluate where the generic property of peaks confirms to an interval.
+
+    Parameters
+    ----------
+    peak_properties : ndarray
+        An array with properties for each peak.
+    pmin : None or number or ndarray
+        Lower interval boundary for `peak_properties`. ``None`` is interpreted as
+        an open border.
+    pmax : None or number or ndarray
+        Upper interval boundary for `peak_properties`. ``None`` is interpreted as
+        an open border.
+
+    Returns
+    -------
+    keep : bool
+        A boolean mask evaluating to true where `peak_properties` confirms to the
+        interval.
+
+    See Also
+    --------
+    find_peaks
+
+    Notes
+    -----
+
+    .. versionadded:: 1.1.0
+    """
+    keep = np.ones(peak_properties.size, dtype=bool)
+    if pmin is not None:
+        keep &= (pmin <= peak_properties)
+    if pmax is not None:
+        keep &= (peak_properties <= pmax)
+    return keep
+
+
+def _select_by_peak_threshold(x, peaks, tmin, tmax):
+    """
+    Evaluate which peaks fulfill the threshold condition.
+
+    Parameters
+    ----------
+    x : ndarray
+        A 1-D array which is indexable by `peaks`.
+    peaks : ndarray
+        Indices of peaks in `x`.
+    tmin, tmax : scalar or ndarray or None
+         Minimal and / or maximal required thresholds. If supplied as ndarrays
+         their size must match `peaks`. ``None`` is interpreted as an open
+         border.
+
+    Returns
+    -------
+    keep : bool
+        A boolean mask evaluating to true where `peaks` fulfill the threshold
+        condition.
+    left_thresholds, right_thresholds : ndarray
+        Array matching `peak` containing the thresholds of each peak on
+        both sides.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.1.0
+    """
+    # Stack thresholds on both sides to make min / max operations easier:
+    # tmin is compared with the smaller, and tmax with the greater threshold to
+    # each peak's side
+    stacked_thresholds = np.vstack([x[peaks] - x[peaks - 1],
+                                    x[peaks] - x[peaks + 1]])
+    keep = np.ones(peaks.size, dtype=bool)
+    if tmin is not None:
+        min_thresholds = np.min(stacked_thresholds, axis=0)
+        keep &= (tmin <= min_thresholds)
+    if tmax is not None:
+        max_thresholds = np.max(stacked_thresholds, axis=0)
+        keep &= (max_thresholds <= tmax)
+
+    return keep, stacked_thresholds[0], stacked_thresholds[1]
+
+
+def find_peaks(x, height=None, threshold=None, distance=None,
+               prominence=None, width=None, wlen=None, rel_height=0.5,
+               plateau_size=None):
+    """
+    Find peaks inside a signal based on peak properties.
+
+    This function takes a 1-D array and finds all local maxima by
+    simple comparison of neighboring values. Optionally, a subset of these
+    peaks can be selected by specifying conditions for a peak's properties.
+
+    Parameters
+    ----------
+    x : sequence
+        A signal with peaks.
+    height : number or ndarray or sequence, optional
+        Required height of peaks. Either a number, ``None``, an array matching
+        `x` or a 2-element sequence of the former. The first element is
+        always interpreted as the  minimal and the second, if supplied, as the
+        maximal required height.
+    threshold : number or ndarray or sequence, optional
+        Required threshold of peaks, the vertical distance to its neighboring
+        samples. Either a number, ``None``, an array matching `x` or a
+        2-element sequence of the former. The first element is always
+        interpreted as the  minimal and the second, if supplied, as the maximal
+        required threshold.
+    distance : number, optional
+        Required minimal horizontal distance (>= 1) in samples between
+        neighbouring peaks. Smaller peaks are removed first until the condition
+        is fulfilled for all remaining peaks.
+    prominence : number or ndarray or sequence, optional
+        Required prominence of peaks. Either a number, ``None``, an array
+        matching `x` or a 2-element sequence of the former. The first
+        element is always interpreted as the  minimal and the second, if
+        supplied, as the maximal required prominence.
+    width : number or ndarray or sequence, optional
+        Required width of peaks in samples. Either a number, ``None``, an array
+        matching `x` or a 2-element sequence of the former. The first
+        element is always interpreted as the  minimal and the second, if
+        supplied, as the maximal required width.
+    wlen : int, optional
+        Used for calculation of the peaks prominences, thus it is only used if
+        one of the arguments `prominence` or `width` is given. See argument
+        `wlen` in `peak_prominences` for a full description of its effects.
+    rel_height : float, optional
+        Used for calculation of the peaks width, thus it is only used if `width`
+        is given. See argument  `rel_height` in `peak_widths` for a full
+        description of its effects.
+    plateau_size : number or ndarray or sequence, optional
+        Required size of the flat top of peaks in samples. Either a number,
+        ``None``, an array matching `x` or a 2-element sequence of the former.
+        The first element is always interpreted as the minimal and the second,
+        if supplied as the maximal required plateau size.
+
+        .. versionadded:: 1.2.0
+
+    Returns
+    -------
+    peaks : ndarray
+        Indices of peaks in `x` that satisfy all given conditions.
+    properties : dict
+        A dictionary containing properties of the returned peaks which were
+        calculated as intermediate results during evaluation of the specified
+        conditions:
+
+        * 'peak_heights'
+              If `height` is given, the height of each peak in `x`.
+        * 'left_thresholds', 'right_thresholds'
+              If `threshold` is given, these keys contain a peaks vertical
+              distance to its neighbouring samples.
+        * 'prominences', 'right_bases', 'left_bases'
+              If `prominence` is given, these keys are accessible. See
+              `peak_prominences` for a description of their content.
+        * 'width_heights', 'left_ips', 'right_ips'
+              If `width` is given, these keys are accessible. See `peak_widths`
+              for a description of their content.
+        * 'plateau_sizes', left_edges', 'right_edges'
+              If `plateau_size` is given, these keys are accessible and contain
+              the indices of a peak's edges (edges are still part of the
+              plateau) and the calculated plateau sizes.
+
+              .. versionadded:: 1.2.0
+
+        To calculate and return properties without excluding peaks, provide the
+        open interval ``(None, None)`` as a value to the appropriate argument
+        (excluding `distance`).
+
+    Warns
+    -----
+    PeakPropertyWarning
+        Raised if a peak's properties have unexpected values (see
+        `peak_prominences` and `peak_widths`).
+
+    Warnings
+    --------
+    This function may return unexpected results for data containing NaNs. To
+    avoid this, NaNs should either be removed or replaced.
+
+    See Also
+    --------
+    find_peaks_cwt
+        Find peaks using the wavelet transformation.
+    peak_prominences
+        Directly calculate the prominence of peaks.
+    peak_widths
+        Directly calculate the width of peaks.
+
+    Notes
+    -----
+    In the context of this function, a peak or local maximum is defined as any
+    sample whose two direct neighbours have a smaller amplitude. For flat peaks
+    (more than one sample of equal amplitude wide) the index of the middle
+    sample is returned (rounded down in case the number of samples is even).
+    For noisy signals the peak locations can be off because the noise might
+    change the position of local maxima. In those cases consider smoothing the
+    signal before searching for peaks or use other peak finding and fitting
+    methods (like `find_peaks_cwt`).
+
+    Some additional comments on specifying conditions:
+
+    * Almost all conditions (excluding `distance`) can be given as half-open or
+      closed intervals, e.g., ``1`` or ``(1, None)`` defines the half-open
+      interval :math:`[1, \\infty]` while ``(None, 1)`` defines the interval
+      :math:`[-\\infty, 1]`. The open interval ``(None, None)`` can be specified
+      as well, which returns the matching properties without exclusion of peaks.
+    * The border is always included in the interval used to select valid peaks.
+    * For several conditions the interval borders can be specified with
+      arrays matching `x` in shape which enables dynamic constrains based on
+      the sample position.
+    * The conditions are evaluated in the following order: `plateau_size`,
+      `height`, `threshold`, `distance`, `prominence`, `width`. In most cases
+      this order is the fastest one because faster operations are applied first
+      to reduce the number of peaks that need to be evaluated later.
+    * While indices in `peaks` are guaranteed to be at least `distance` samples
+      apart, edges of flat peaks may be closer than the allowed `distance`.
+    * Use `wlen` to reduce the time it takes to evaluate the conditions for
+      `prominence` or `width` if `x` is large or has many local maxima
+      (see `peak_prominences`).
+
+    .. versionadded:: 1.1.0
+
+    Examples
+    --------
+    To demonstrate this function's usage we use a signal `x` supplied with
+    SciPy (see `scipy.datasets.electrocardiogram`). Let's find all peaks (local
+    maxima) in `x` whose amplitude lies above 0.
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.datasets import electrocardiogram
+    >>> from scipy.signal import find_peaks
+    >>> x = electrocardiogram()[2000:4000]
+    >>> peaks, _ = find_peaks(x, height=0)
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.plot(np.zeros_like(x), "--", color="gray")
+    >>> plt.show()
+
+    We can select peaks below 0 with ``height=(None, 0)`` or use arrays matching
+    `x` in size to reflect a changing condition for different parts of the
+    signal.
+
+    >>> border = np.sin(np.linspace(0, 3 * np.pi, x.size))
+    >>> peaks, _ = find_peaks(x, height=(-border, border))
+    >>> plt.plot(x)
+    >>> plt.plot(-border, "--", color="gray")
+    >>> plt.plot(border, ":", color="gray")
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.show()
+
+    Another useful condition for periodic signals can be given with the
+    `distance` argument. In this case, we can easily select the positions of
+    QRS complexes within the electrocardiogram (ECG) by demanding a distance of
+    at least 150 samples.
+
+    >>> peaks, _ = find_peaks(x, distance=150)
+    >>> np.diff(peaks)
+    array([186, 180, 177, 171, 177, 169, 167, 164, 158, 162, 172])
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.show()
+
+    Especially for noisy signals peaks can be easily grouped by their
+    prominence (see `peak_prominences`). E.g., we can select all peaks except
+    for the mentioned QRS complexes by limiting the allowed prominence to 0.6.
+
+    >>> peaks, properties = find_peaks(x, prominence=(None, 0.6))
+    >>> properties["prominences"].max()
+    0.5049999999999999
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.show()
+
+    And, finally, let's examine a different section of the ECG which contains
+    beat forms of different shape. To select only the atypical heart beats, we
+    combine two conditions: a minimal prominence of 1 and width of at least 20
+    samples.
+
+    >>> x = electrocardiogram()[17000:18000]
+    >>> peaks, properties = find_peaks(x, prominence=1, width=20)
+    >>> properties["prominences"], properties["widths"]
+    (array([1.495, 2.3  ]), array([36.93773946, 39.32723577]))
+    >>> plt.plot(x)
+    >>> plt.plot(peaks, x[peaks], "x")
+    >>> plt.vlines(x=peaks, ymin=x[peaks] - properties["prominences"],
+    ...            ymax = x[peaks], color = "C1")
+    >>> plt.hlines(y=properties["width_heights"], xmin=properties["left_ips"],
+    ...            xmax=properties["right_ips"], color = "C1")
+    >>> plt.show()
+    """
+    # _argmaxima1d expects array of dtype 'float64'
+    x = _arg_x_as_expected(x)
+    if distance is not None and distance < 1:
+        raise ValueError('`distance` must be greater or equal to 1')
+
+    peaks, left_edges, right_edges = _local_maxima_1d(x)
+    properties = {}
+
+    if plateau_size is not None:
+        # Evaluate plateau size
+        plateau_sizes = right_edges - left_edges + 1
+        pmin, pmax = _unpack_condition_args(plateau_size, x, peaks)
+        keep = _select_by_property(plateau_sizes, pmin, pmax)
+        peaks = peaks[keep]
+        properties["plateau_sizes"] = plateau_sizes
+        properties["left_edges"] = left_edges
+        properties["right_edges"] = right_edges
+        properties = {key: array[keep] for key, array in properties.items()}
+
+    if height is not None:
+        # Evaluate height condition
+        peak_heights = x[peaks]
+        hmin, hmax = _unpack_condition_args(height, x, peaks)
+        keep = _select_by_property(peak_heights, hmin, hmax)
+        peaks = peaks[keep]
+        properties["peak_heights"] = peak_heights
+        properties = {key: array[keep] for key, array in properties.items()}
+
+    if threshold is not None:
+        # Evaluate threshold condition
+        tmin, tmax = _unpack_condition_args(threshold, x, peaks)
+        keep, left_thresholds, right_thresholds = _select_by_peak_threshold(
+            x, peaks, tmin, tmax)
+        peaks = peaks[keep]
+        properties["left_thresholds"] = left_thresholds
+        properties["right_thresholds"] = right_thresholds
+        properties = {key: array[keep] for key, array in properties.items()}
+
+    if distance is not None:
+        # Evaluate distance condition
+        keep = _select_by_peak_distance(peaks, x[peaks], distance)
+        peaks = peaks[keep]
+        properties = {key: array[keep] for key, array in properties.items()}
+
+    if prominence is not None or width is not None:
+        # Calculate prominence (required for both conditions)
+        wlen = _arg_wlen_as_expected(wlen)
+        properties.update(zip(
+            ['prominences', 'left_bases', 'right_bases'],
+            _peak_prominences(x, peaks, wlen=wlen)
+        ))
+
+    if prominence is not None:
+        # Evaluate prominence condition
+        pmin, pmax = _unpack_condition_args(prominence, x, peaks)
+        keep = _select_by_property(properties['prominences'], pmin, pmax)
+        peaks = peaks[keep]
+        properties = {key: array[keep] for key, array in properties.items()}
+
+    if width is not None:
+        # Calculate widths
+        properties.update(zip(
+            ['widths', 'width_heights', 'left_ips', 'right_ips'],
+            _peak_widths(x, peaks, rel_height, properties['prominences'],
+                         properties['left_bases'], properties['right_bases'])
+        ))
+        # Evaluate width condition
+        wmin, wmax = _unpack_condition_args(width, x, peaks)
+        keep = _select_by_property(properties['widths'], wmin, wmax)
+        peaks = peaks[keep]
+        properties = {key: array[keep] for key, array in properties.items()}
+
+    return peaks, properties
+
+
+def _identify_ridge_lines(matr, max_distances, gap_thresh):
+    """
+    Identify ridges in the 2-D matrix.
+
+    Expect that the width of the wavelet feature increases with increasing row
+    number.
+
+    Parameters
+    ----------
+    matr : 2-D ndarray
+        Matrix in which to identify ridge lines.
+    max_distances : 1-D sequence
+        At each row, a ridge line is only connected
+        if the relative max at row[n] is within
+        `max_distances`[n] from the relative max at row[n+1].
+    gap_thresh : int
+        If a relative maximum is not found within `max_distances`,
+        there will be a gap. A ridge line is discontinued if
+        there are more than `gap_thresh` points without connecting
+        a new relative maximum.
+
+    Returns
+    -------
+    ridge_lines : tuple
+        Tuple of 2 1-D sequences. `ridge_lines`[ii][0] are the rows of the
+        ii-th ridge-line, `ridge_lines`[ii][1] are the columns. Empty if none
+        found.  Each ridge-line will be sorted by row (increasing), but the
+        order of the ridge lines is not specified.
+
+    References
+    ----------
+    .. [1] Bioinformatics (2006) 22 (17): 2059-2065.
+       :doi:`10.1093/bioinformatics/btl355`
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.signal._peak_finding import _identify_ridge_lines
+    >>> rng = np.random.default_rng()
+    >>> data = rng.random((5,5))
+    >>> max_dist = 3
+    >>> max_distances = np.full(20, max_dist)
+    >>> ridge_lines = _identify_ridge_lines(data, max_distances, 1)
+
+    Notes
+    -----
+    This function is intended to be used in conjunction with `cwt`
+    as part of `find_peaks_cwt`.
+
+    """
+    if len(max_distances) < matr.shape[0]:
+        raise ValueError('Max_distances must have at least as many rows '
+                         'as matr')
+
+    all_max_cols = _boolrelextrema(matr, np.greater, axis=1, order=1)
+    # Highest row for which there are any relative maxima
+    has_relmax = np.nonzero(all_max_cols.any(axis=1))[0]
+    if len(has_relmax) == 0:
+        return []
+    start_row = has_relmax[-1]
+    # Each ridge line is a 3-tuple:
+    # rows, cols,Gap number
+    ridge_lines = [[[start_row],
+                   [col],
+                   0] for col in np.nonzero(all_max_cols[start_row])[0]]
+    final_lines = []
+    rows = np.arange(start_row - 1, -1, -1)
+    cols = np.arange(0, matr.shape[1])
+    for row in rows:
+        this_max_cols = cols[all_max_cols[row]]
+
+        # Increment gap number of each line,
+        # set it to zero later if appropriate
+        for line in ridge_lines:
+            line[2] += 1
+
+        # XXX These should always be all_max_cols[row]
+        # But the order might be different. Might be an efficiency gain
+        # to make sure the order is the same and avoid this iteration
+        prev_ridge_cols = np.array([line[1][-1] for line in ridge_lines])
+        # Look through every relative maximum found at current row
+        # Attempt to connect them with existing ridge lines.
+        for ind, col in enumerate(this_max_cols):
+            # If there is a previous ridge line within
+            # the max_distance to connect to, do so.
+            # Otherwise start a new one.
+            line = None
+            if len(prev_ridge_cols) > 0:
+                diffs = np.abs(col - prev_ridge_cols)
+                closest = np.argmin(diffs)
+                if diffs[closest] <= max_distances[row]:
+                    line = ridge_lines[closest]
+            if line is not None:
+                # Found a point close enough, extend current ridge line
+                line[1].append(col)
+                line[0].append(row)
+                line[2] = 0
+            else:
+                new_line = [[row],
+                            [col],
+                            0]
+                ridge_lines.append(new_line)
+
+        # Remove the ridge lines with gap_number too high
+        # XXX Modifying a list while iterating over it.
+        # Should be safe, since we iterate backwards, but
+        # still tacky.
+        for ind in range(len(ridge_lines) - 1, -1, -1):
+            line = ridge_lines[ind]
+            if line[2] > gap_thresh:
+                final_lines.append(line)
+                del ridge_lines[ind]
+
+    out_lines = []
+    for line in (final_lines + ridge_lines):
+        sortargs = np.array(np.argsort(line[0]))
+        rows, cols = np.zeros_like(sortargs), np.zeros_like(sortargs)
+        rows[sortargs] = line[0]
+        cols[sortargs] = line[1]
+        out_lines.append([rows, cols])
+
+    return out_lines
+
+
+def _filter_ridge_lines(cwt, ridge_lines, window_size=None, min_length=None,
+                        min_snr=1, noise_perc=10):
+    """
+    Filter ridge lines according to prescribed criteria. Intended
+    to be used for finding relative maxima.
+
+    Parameters
+    ----------
+    cwt : 2-D ndarray
+        Continuous wavelet transform from which the `ridge_lines` were defined.
+    ridge_lines : 1-D sequence
+        Each element should contain 2 sequences, the rows and columns
+        of the ridge line (respectively).
+    window_size : int, optional
+        Size of window to use to calculate noise floor.
+        Default is ``cwt.shape[1] / 20``.
+    min_length : int, optional
+        Minimum length a ridge line needs to be acceptable.
+        Default is ``cwt.shape[0] / 4``, ie 1/4-th the number of widths.
+    min_snr : float, optional
+        Minimum SNR ratio. Default 1. The signal is the value of
+        the cwt matrix at the shortest length scale (``cwt[0, loc]``), the
+        noise is the `noise_perc`\\ th percentile of datapoints contained within a
+        window of `window_size` around ``cwt[0, loc]``.
+    noise_perc : float, optional
+        When calculating the noise floor, percentile of data points
+        examined below which to consider noise. Calculated using
+        scipy.stats.scoreatpercentile.
+
+    References
+    ----------
+    .. [1] Bioinformatics (2006) 22 (17): 2059-2065.
+       :doi:`10.1093/bioinformatics/btl355`
+
+    """
+    num_points = cwt.shape[1]
+    if min_length is None:
+        min_length = np.ceil(cwt.shape[0] / 4)
+    if window_size is None:
+        window_size = np.ceil(num_points / 20)
+
+    window_size = int(window_size)
+    hf_window, odd = divmod(window_size, 2)
+
+    # Filter based on SNR
+    row_one = cwt[0, :]
+    noises = np.empty_like(row_one)
+    for ind, val in enumerate(row_one):
+        window_start = max(ind - hf_window, 0)
+        window_end = min(ind + hf_window + odd, num_points)
+        noises[ind] = scoreatpercentile(row_one[window_start:window_end],
+                                        per=noise_perc)
+
+    def filt_func(line):
+        if len(line[0]) < min_length:
+            return False
+        snr = abs(cwt[line[0][0], line[1][0]] / noises[line[1][0]])
+        if snr < min_snr:
+            return False
+        return True
+
+    return list(filter(filt_func, ridge_lines))
+
+
+def find_peaks_cwt(vector, widths, wavelet=None, max_distances=None,
+                   gap_thresh=None, min_length=None,
+                   min_snr=1, noise_perc=10, window_size=None):
+    """
+    Find peaks in a 1-D array with wavelet transformation.
+
+    The general approach is to smooth `vector` by convolving it with
+    `wavelet(width)` for each width in `widths`. Relative maxima which
+    appear at enough length scales, and with sufficiently high SNR, are
+    accepted.
+
+    Parameters
+    ----------
+    vector : ndarray
+        1-D array in which to find the peaks.
+    widths : float or sequence
+        Single width or 1-D array-like of widths to use for calculating
+        the CWT matrix. In general,
+        this range should cover the expected width of peaks of interest.
+    wavelet : callable, optional
+        Should take two parameters and return a 1-D array to convolve
+        with `vector`. The first parameter determines the number of points
+        of the returned wavelet array, the second parameter is the scale
+        (`width`) of the wavelet. Should be normalized and symmetric.
+        Default is the ricker wavelet.
+    max_distances : ndarray, optional
+        At each row, a ridge line is only connected if the relative max at
+        row[n] is within ``max_distances[n]`` from the relative max at
+        ``row[n+1]``.  Default value is ``widths/4``.
+    gap_thresh : float, optional
+        If a relative maximum is not found within `max_distances`,
+        there will be a gap. A ridge line is discontinued if there are more
+        than `gap_thresh` points without connecting a new relative maximum.
+        Default is the first value of the widths array i.e. widths[0].
+    min_length : int, optional
+        Minimum length a ridge line needs to be acceptable.
+        Default is ``cwt.shape[0] / 4``, ie 1/4-th the number of widths.
+    min_snr : float, optional
+        Minimum SNR ratio. Default 1. The signal is the maximum CWT coefficient
+        on the largest ridge line. The noise is `noise_perc` th percentile of
+        datapoints contained within the same ridge line.
+    noise_perc : float, optional
+        When calculating the noise floor, percentile of data points
+        examined below which to consider noise. Calculated using
+        `stats.scoreatpercentile`.  Default is 10.
+    window_size : int, optional
+        Size of window to use to calculate noise floor.
+        Default is ``cwt.shape[1] / 20``.
+
+    Returns
+    -------
+    peaks_indices : ndarray
+        Indices of the locations in the `vector` where peaks were found.
+        The list is sorted.
+
+    See Also
+    --------
+    cwt
+        Continuous wavelet transform.
+    find_peaks
+        Find peaks inside a signal based on peak properties.
+
+    Notes
+    -----
+    This approach was designed for finding sharp peaks among noisy data,
+    however with proper parameter selection it should function well for
+    different peak shapes.
+
+    The algorithm is as follows:
+     1. Perform a continuous wavelet transform on `vector`, for the supplied
+        `widths`. This is a convolution of `vector` with `wavelet(width)` for
+        each width in `widths`. See `cwt`.
+     2. Identify "ridge lines" in the cwt matrix. These are relative maxima
+        at each row, connected across adjacent rows. See identify_ridge_lines
+     3. Filter the ridge_lines using filter_ridge_lines.
+
+    .. versionadded:: 0.11.0
+
+    References
+    ----------
+    .. [1] Bioinformatics (2006) 22 (17): 2059-2065.
+       :doi:`10.1093/bioinformatics/btl355`
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> xs = np.arange(0, np.pi, 0.05)
+    >>> data = np.sin(xs)
+    >>> peakind = signal.find_peaks_cwt(data, np.arange(1,10))
+    >>> peakind, xs[peakind], data[peakind]
+    ([32], array([ 1.6]), array([ 0.9995736]))
+
+    """
+    widths = np.atleast_1d(np.asarray(widths))
+
+    if gap_thresh is None:
+        gap_thresh = np.ceil(widths[0])
+    if max_distances is None:
+        max_distances = widths / 4.0
+    if wavelet is None:
+        wavelet = _ricker
+
+    cwt_dat = _cwt(vector, wavelet, widths)
+    ridge_lines = _identify_ridge_lines(cwt_dat, max_distances, gap_thresh)
+    filtered = _filter_ridge_lines(cwt_dat, ridge_lines, min_length=min_length,
+                                   window_size=window_size, min_snr=min_snr,
+                                   noise_perc=noise_perc)
+    max_locs = np.asarray([x[1][0] for x in filtered])
+    max_locs.sort()
+
+    return max_locs

env-llmeval/lib/python3.10/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
+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

env-llmeval/lib/python3.10/site-packages/scipy/signal/_upfirdn.py

@@ -0,0 +1,216 @@
+# 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 ._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.]])
+    """
+    x = np.asarray(x)
+    ufd = _UpFIRDn(h, x.dtype, up, down)
+    # This is equivalent to (but faster than) using np.apply_along_axis
+    return ufd.apply_filter(x, axis, mode, cval)

env-llmeval/lib/python3.10/site-packages/scipy/signal/_wavelets.py

@@ -0,0 +1,556 @@
+import warnings
+
+import numpy as np
+from scipy.linalg import eig
+from scipy.special import comb
+from scipy.signal import convolve
+
+__all__ = ['daub', 'qmf', 'cascade', 'morlet', 'ricker', 'morlet2', 'cwt']
+
+
+_msg="""scipy.signal.%s is deprecated in SciPy 1.12 and will be removed
+in SciPy 1.15. We recommend using PyWavelets instead.
+"""
+
+
+def daub(p):
+    """
+    The coefficients for the FIR low-pass filter producing Daubechies wavelets.
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.daub is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    p>=1 gives the order of the zero at f=1/2.
+    There are 2p filter coefficients.
+
+    Parameters
+    ----------
+    p : int
+        Order of the zero at f=1/2, can have values from 1 to 34.
+
+    Returns
+    -------
+    daub : ndarray
+        Return
+
+    """
+    warnings.warn(_msg % 'daub', DeprecationWarning, stacklevel=2)
+
+    sqrt = np.sqrt
+    if p < 1:
+        raise ValueError("p must be at least 1.")
+    if p == 1:
+        c = 1 / sqrt(2)
+        return np.array([c, c])
+    elif p == 2:
+        f = sqrt(2) / 8
+        c = sqrt(3)
+        return f * np.array([1 + c, 3 + c, 3 - c, 1 - c])
+    elif p == 3:
+        tmp = 12 * sqrt(10)
+        z1 = 1.5 + sqrt(15 + tmp) / 6 - 1j * (sqrt(15) + sqrt(tmp - 15)) / 6
+        z1c = np.conj(z1)
+        f = sqrt(2) / 8
+        d0 = np.real((1 - z1) * (1 - z1c))
+        a0 = np.real(z1 * z1c)
+        a1 = 2 * np.real(z1)
+        return f / d0 * np.array([a0, 3 * a0 - a1, 3 * a0 - 3 * a1 + 1,
+                                  a0 - 3 * a1 + 3, 3 - a1, 1])
+    elif p < 35:
+        # construct polynomial and factor it
+        if p < 35:
+            P = [comb(p - 1 + k, k, exact=1) for k in range(p)][::-1]
+            yj = np.roots(P)
+        else:  # try different polynomial --- needs work
+            P = [comb(p - 1 + k, k, exact=1) / 4.0**k
+                 for k in range(p)][::-1]
+            yj = np.roots(P) / 4
+        # for each root, compute two z roots, select the one with |z|>1
+        # Build up final polynomial
+        c = np.poly1d([1, 1])**p
+        q = np.poly1d([1])
+        for k in range(p - 1):
+            yval = yj[k]
+            part = 2 * sqrt(yval * (yval - 1))
+            const = 1 - 2 * yval
+            z1 = const + part
+            if (abs(z1)) < 1:
+                z1 = const - part
+            q = q * [1, -z1]
+
+        q = c * np.real(q)
+        # Normalize result
+        q = q / np.sum(q) * sqrt(2)
+        return q.c[::-1]
+    else:
+        raise ValueError("Polynomial factorization does not work "
+                         "well for p too large.")
+
+
+def qmf(hk):
+    """
+    Return high-pass qmf filter from low-pass
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.qmf is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    Parameters
+    ----------
+    hk : array_like
+        Coefficients of high-pass filter.
+
+    Returns
+    -------
+    array_like
+        High-pass filter coefficients.
+
+    """
+    warnings.warn(_msg % 'qmf', DeprecationWarning, stacklevel=2)
+
+    N = len(hk) - 1
+    asgn = [{0: 1, 1: -1}[k % 2] for k in range(N + 1)]
+    return hk[::-1] * np.array(asgn)
+
+
+def cascade(hk, J=7):
+    """
+    Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients.
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.cascade is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    Parameters
+    ----------
+    hk : array_like
+        Coefficients of low-pass filter.
+    J : int, optional
+        Values will be computed at grid points ``K/2**J``. Default is 7.
+
+    Returns
+    -------
+    x : ndarray
+        The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where
+        ``len(hk) = len(gk) = N+1``.
+    phi : ndarray
+        The scaling function ``phi(x)`` at `x`:
+        ``phi(x) = sum(hk * phi(2x-k))``, where k is from 0 to N.
+    psi : ndarray, optional
+        The wavelet function ``psi(x)`` at `x`:
+        ``phi(x) = sum(gk * phi(2x-k))``, where k is from 0 to N.
+        `psi` is only returned if `gk` is not None.
+
+    Notes
+    -----
+    The algorithm uses the vector cascade algorithm described by Strang and
+    Nguyen in "Wavelets and Filter Banks".  It builds a dictionary of values
+    and slices for quick reuse.  Then inserts vectors into final vector at the
+    end.
+
+    """
+    warnings.warn(_msg % 'cascade', DeprecationWarning, stacklevel=2)
+
+    N = len(hk) - 1
+
+    if (J > 30 - np.log2(N + 1)):
+        raise ValueError("Too many levels.")
+    if (J < 1):
+        raise ValueError("Too few levels.")
+
+    # construct matrices needed
+    nn, kk = np.ogrid[:N, :N]
+    s2 = np.sqrt(2)
+    # append a zero so that take works
+    thk = np.r_[hk, 0]
+    gk = qmf(hk)
+    tgk = np.r_[gk, 0]
+
+    indx1 = np.clip(2 * nn - kk, -1, N + 1)
+    indx2 = np.clip(2 * nn - kk + 1, -1, N + 1)
+    m = np.empty((2, 2, N, N), 'd')
+    m[0, 0] = np.take(thk, indx1, 0)
+    m[0, 1] = np.take(thk, indx2, 0)
+    m[1, 0] = np.take(tgk, indx1, 0)
+    m[1, 1] = np.take(tgk, indx2, 0)
+    m *= s2
+
+    # construct the grid of points
+    x = np.arange(0, N * (1 << J), dtype=float) / (1 << J)
+    phi = 0 * x
+
+    psi = 0 * x
+
+    # find phi0, and phi1
+    lam, v = eig(m[0, 0])
+    ind = np.argmin(np.absolute(lam - 1))
+    # a dictionary with a binary representation of the
+    #   evaluation points x < 1 -- i.e. position is 0.xxxx
+    v = np.real(v[:, ind])
+    # need scaling function to integrate to 1 so find
+    #  eigenvector normalized to sum(v,axis=0)=1
+    sm = np.sum(v)
+    if sm < 0:  # need scaling function to integrate to 1
+        v = -v
+        sm = -sm
+    bitdic = {'0': v / sm}
+    bitdic['1'] = np.dot(m[0, 1], bitdic['0'])
+    step = 1 << J
+    phi[::step] = bitdic['0']
+    phi[(1 << (J - 1))::step] = bitdic['1']
+    psi[::step] = np.dot(m[1, 0], bitdic['0'])
+    psi[(1 << (J - 1))::step] = np.dot(m[1, 1], bitdic['0'])
+    # descend down the levels inserting more and more values
+    #  into bitdic -- store the values in the correct location once we
+    #  have computed them -- stored in the dictionary
+    #  for quicker use later.
+    prevkeys = ['1']
+    for level in range(2, J + 1):
+        newkeys = ['%d%s' % (xx, yy) for xx in [0, 1] for yy in prevkeys]
+        fac = 1 << (J - level)
+        for key in newkeys:
+            # convert key to number
+            num = 0
+            for pos in range(level):
+                if key[pos] == '1':
+                    num += (1 << (level - 1 - pos))
+            pastphi = bitdic[key[1:]]
+            ii = int(key[0])
+            temp = np.dot(m[0, ii], pastphi)
+            bitdic[key] = temp
+            phi[num * fac::step] = temp
+            psi[num * fac::step] = np.dot(m[1, ii], pastphi)
+        prevkeys = newkeys
+
+    return x, phi, psi
+
+
+def morlet(M, w=5.0, s=1.0, complete=True):
+    """
+    Complex Morlet wavelet.
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.morlet is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    Parameters
+    ----------
+    M : int
+        Length of the wavelet.
+    w : float, optional
+        Omega0. Default is 5
+    s : float, optional
+        Scaling factor, windowed from ``-s*2*pi`` to ``+s*2*pi``. Default is 1.
+    complete : bool, optional
+        Whether to use the complete or the standard version.
+
+    Returns
+    -------
+    morlet : (M,) ndarray
+
+    See Also
+    --------
+    morlet2 : Implementation of Morlet wavelet, compatible with `cwt`.
+    scipy.signal.gausspulse
+
+    Notes
+    -----
+    The standard version::
+
+        pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2))
+
+    This commonly used wavelet is often referred to simply as the
+    Morlet wavelet.  Note that this simplified version can cause
+    admissibility problems at low values of `w`.
+
+    The complete version::
+
+        pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2))
+
+    This version has a correction
+    term to improve admissibility. For `w` greater than 5, the
+    correction term is negligible.
+
+    Note that the energy of the return wavelet is not normalised
+    according to `s`.
+
+    The fundamental frequency of this wavelet in Hz is given
+    by ``f = 2*s*w*r / M`` where `r` is the sampling rate.
+
+    Note: This function was created before `cwt` and is not compatible
+    with it.
+
+    Examples
+    --------
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+
+    >>> M = 100
+    >>> s = 4.0
+    >>> w = 2.0
+    >>> wavelet = signal.morlet(M, s, w)
+    >>> plt.plot(wavelet.real, label="real")
+    >>> plt.plot(wavelet.imag, label="imag")
+    >>> plt.legend()
+    >>> plt.show()
+
+    """
+    warnings.warn(_msg % 'morlet', DeprecationWarning, stacklevel=2)
+
+    x = np.linspace(-s * 2 * np.pi, s * 2 * np.pi, M)
+    output = np.exp(1j * w * x)
+
+    if complete:
+        output -= np.exp(-0.5 * (w**2))
+
+    output *= np.exp(-0.5 * (x**2)) * np.pi**(-0.25)
+
+    return output
+
+
+def ricker(points, a):
+    """
+    Return a Ricker wavelet, also known as the "Mexican hat wavelet".
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.ricker is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    It models the function:
+
+        ``A * (1 - (x/a)**2) * exp(-0.5*(x/a)**2)``,
+
+    where ``A = 2/(sqrt(3*a)*(pi**0.25))``.
+
+    Parameters
+    ----------
+    points : int
+        Number of points in `vector`.
+        Will be centered around 0.
+    a : scalar
+        Width parameter of the wavelet.
+
+    Returns
+    -------
+    vector : (N,) ndarray
+        Array of length `points` in shape of ricker curve.
+
+    Examples
+    --------
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+
+    >>> points = 100
+    >>> a = 4.0
+    >>> vec2 = signal.ricker(points, a)
+    >>> print(len(vec2))
+    100
+    >>> plt.plot(vec2)
+    >>> plt.show()
+
+    """
+    warnings.warn(_msg % 'ricker', DeprecationWarning, stacklevel=2)
+    return _ricker(points, a)
+
+
+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 morlet2(M, s, w=5):
+    """
+    Complex Morlet wavelet, designed to work with `cwt`.
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.morlet2 is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    Returns the complete version of morlet wavelet, normalised
+    according to `s`::
+
+        exp(1j*w*x/s) * exp(-0.5*(x/s)**2) * pi**(-0.25) * sqrt(1/s)
+
+    Parameters
+    ----------
+    M : int
+        Length of the wavelet.
+    s : float
+        Width parameter of the wavelet.
+    w : float, optional
+        Omega0. Default is 5
+
+    Returns
+    -------
+    morlet : (M,) ndarray
+
+    See Also
+    --------
+    morlet : Implementation of Morlet wavelet, incompatible with `cwt`
+
+    Notes
+    -----
+
+    .. versionadded:: 1.4.0
+
+    This function was designed to work with `cwt`. Because `morlet2`
+    returns an array of complex numbers, the `dtype` argument of `cwt`
+    should be set to `complex128` for best results.
+
+    Note the difference in implementation with `morlet`.
+    The fundamental frequency of this wavelet in Hz is given by::
+
+        f = w*fs / (2*s*np.pi)
+
+    where ``fs`` is the sampling rate and `s` is the wavelet width parameter.
+    Similarly we can get the wavelet width parameter at ``f``::
+
+        s = w*fs / (2*f*np.pi)
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+
+    >>> M = 100
+    >>> s = 4.0
+    >>> w = 2.0
+    >>> wavelet = signal.morlet2(M, s, w)
+    >>> plt.plot(abs(wavelet))
+    >>> plt.show()
+
+    This example shows basic use of `morlet2` with `cwt` in time-frequency
+    analysis:
+
+    >>> t, dt = np.linspace(0, 1, 200, retstep=True)
+    >>> fs = 1/dt
+    >>> w = 6.
+    >>> sig = np.cos(2*np.pi*(50 + 10*t)*t) + np.sin(40*np.pi*t)
+    >>> freq = np.linspace(1, fs/2, 100)
+    >>> widths = w*fs / (2*freq*np.pi)
+    >>> cwtm = signal.cwt(sig, signal.morlet2, widths, w=w)
+    >>> plt.pcolormesh(t, freq, np.abs(cwtm), cmap='viridis', shading='gouraud')
+    >>> plt.show()
+
+    """
+    warnings.warn(_msg % 'morlet2', DeprecationWarning, stacklevel=2)
+
+    x = np.arange(0, M) - (M - 1.0) / 2
+    x = x / s
+    wavelet = np.exp(1j * w * x) * np.exp(-0.5 * x**2) * np.pi**(-0.25)
+    output = np.sqrt(1/s) * wavelet
+    return output
+
+
+def cwt(data, wavelet, widths, dtype=None, **kwargs):
+    """
+    Continuous wavelet transform.
+
+    .. deprecated:: 1.12.0
+
+        scipy.signal.cwt is deprecated in SciPy 1.12 and will be removed
+        in SciPy 1.15. We recommend using PyWavelets instead.
+
+    Performs a continuous wavelet transform on `data`,
+    using the `wavelet` function. A CWT performs a convolution
+    with `data` using the `wavelet` function, which is characterized
+    by a width parameter and length parameter. The `wavelet` function
+    is allowed to be complex.
+
+    Parameters
+    ----------
+    data : (N,) ndarray
+        data on which to perform the transform.
+    wavelet : function
+        Wavelet function, which should take 2 arguments.
+        The first argument is the number of points that the returned vector
+        will have (len(wavelet(length,width)) == length).
+        The second is a width parameter, defining the size of the wavelet
+        (e.g. standard deviation of a gaussian). See `ricker`, which
+        satisfies these requirements.
+    widths : (M,) sequence
+        Widths to use for transform.
+    dtype : data-type, optional
+        The desired data type of output. Defaults to ``float64`` if the
+        output of `wavelet` is real and ``complex128`` if it is complex.
+
+        .. versionadded:: 1.4.0
+
+    kwargs
+        Keyword arguments passed to wavelet function.
+
+        .. versionadded:: 1.4.0
+
+    Returns
+    -------
+    cwt: (M, N) ndarray
+        Will have shape of (len(widths), len(data)).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.4.0
+
+    For non-symmetric, complex-valued wavelets, the input signal is convolved
+    with the time-reversed complex-conjugate of the wavelet data [1].
+
+    ::
+
+        length = min(10 * width[ii], len(data))
+        cwt[ii,:] = signal.convolve(data, np.conj(wavelet(length, width[ii],
+                                        **kwargs))[::-1], mode='same')
+
+    References
+    ----------
+    .. [1] S. Mallat, "A Wavelet Tour of Signal Processing (3rd Edition)",
+        Academic Press, 2009.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import signal
+    >>> import matplotlib.pyplot as plt
+    >>> t = np.linspace(-1, 1, 200, endpoint=False)
+    >>> sig  = np.cos(2 * np.pi * 7 * t) + signal.gausspulse(t - 0.4, fc=2)
+    >>> widths = np.arange(1, 31)
+    >>> cwtmatr = signal.cwt(sig, signal.ricker, widths)
+
+    .. note:: For cwt matrix plotting it is advisable to flip the y-axis
+
+    >>> cwtmatr_yflip = np.flipud(cwtmatr)
+    >>> plt.imshow(cwtmatr_yflip, extent=[-1, 1, 1, 31], cmap='PRGn', aspect='auto',
+    ...            vmax=abs(cwtmatr).max(), vmin=-abs(cwtmatr).max())
+    >>> plt.show()
+    """
+    warnings.warn(_msg % 'cwt', DeprecationWarning, stacklevel=2)
+    return _cwt(data, wavelet, widths, dtype, **kwargs)
+
+
+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

env-llmeval/lib/python3.10/site-packages/scipy/signal/bsplines.py

@@ -0,0 +1,23 @@
+# 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',
+    'zeros_like', 'array', 'arctan2',
+    'tan', 'arange', 'floor', 'exp', 'greater', 'add',
+    'cspline2d', 'sepfir2d'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="signal", module="bsplines",
+                                   private_modules=["_bsplines"], all=__all__,
+                                   attribute=name)

env-llmeval/lib/python3.10/site-packages/scipy/signal/filter_design.py

@@ -0,0 +1,34 @@
+# 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', 'iirnotch', 'iirpeak', 'bilinear_zpk',
+    'lp2lp_zpk', 'lp2hp_zpk', 'lp2bp_zpk', 'lp2bs_zpk',
+    'gammatone', 'iircomb',
+    'atleast_1d', 'poly', 'polyval', 'roots', 'resize', 'absolute',
+    'tan', 'log10', 'arcsinh', 'exp', 'arccosh',
+    'ceil', 'conjugate', 'append', 'prod', 'full', 'array', 'mintypecode',
+    'npp_polyval', 'polyvalfromroots', 'optimize', 'sp_fft', 'comb',
+    'float_factorial', 'abs', 'maxflat', 'yulewalk',
+    'EPSILON', 'filter_dict', 'band_dict', 'bessel_norms'
+]
+
+
+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)

env-llmeval/lib/python3.10/site-packages/scipy/signal/fir_filter_design.py

@@ -0,0 +1,22 @@
+# 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',
+    'ceil', 'log', 'irfft', 'fft', 'ifft', 'sinc', 'toeplitz',
+    'hankel', 'solve', 'LinAlgError', 'LinAlgWarning', 'lstsq'
+]
+
+
+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)

env-llmeval/lib/python3.10/site-packages/scipy/signal/signaltools.py

@@ -0,0 +1,29 @@
+# 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',
+    'cmplx_sort', 'unique_roots', 'invres', 'invresz', 'residue',
+    'residuez', 'resample', 'resample_poly', 'detrend',
+    'lfilter_zi', 'sosfilt_zi', 'sosfiltfilt', 'choose_conv_method',
+    'filtfilt', 'decimate', 'vectorstrength',
+    'timeit', 'cKDTree', 'dlti', 'upfirdn', 'linalg',
+    'sp_fft', 'lambertw', 'get_window', 'axis_slice', 'axis_reverse',
+    'odd_ext', 'even_ext', 'const_ext', '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)