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venv/lib/python3.12/site-packages/scipy-1.18.0.dist-info/INSTALLER

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+pip

venv/lib/python3.12/site-packages/scipy-1.18.0.dist-info/LICENSE.txt

@@ -0,0 +1,937 @@
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+    https://www.gnu.org/licenses/old-licenses/lgpl-2.1.html

venv/lib/python3.12/site-packages/scipy-1.18.0.dist-info/METADATA

@@ -0,0 +1,1091 @@
+Metadata-Version: 2.1
+Name: scipy
+Version: 1.18.0
+Summary: Fundamental algorithms for scientific computing in Python
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+         tangible personal property which is normally used for personal, family,
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+         
+           "Installation Information" for a User Product means any methods,
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+           If you convey an object code work under this section in, or with, or
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+         modified object code on the User Product (for example, the work has
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+           The requirement to provide Installation Information does not include a
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+         unpacking, reading or copying.
+         
+           7. Additional Terms.
+         
+           "Additional permissions" are terms that supplement the terms of this
+         License by making exceptions from one or more of its conditions.
+         Additional permissions that are applicable to the entire Program shall
+         be treated as though they were included in this License, to the extent
+         that they are valid under applicable law.  If additional permissions
+         apply only to part of the Program, that part may be used separately
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+         
+           When you convey a copy of a covered work, you may at your option
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+           Notwithstanding any other provision of this License, for material you
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+         that material) supplement the terms of this License with terms:
+         
+             a) Disclaiming warranty or limiting liability differently from the
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+         
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+           However, if you cease all violation of this License, then your
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+           Termination of your rights under this section does not terminate the
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+           9. Acceptance Not Required for Having Copies.
+         
+           You are not required to accept this License in order to receive or
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+           10. Automatic Licensing of Downstream Recipients.
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+         sale, or importing the Program or any portion of it.
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+           11. Patents.
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+           A "contributor" is a copyright holder who authorizes use under this
+         License of the Program or a work on which the Program is based.  The
+         work thus licensed is called the contributor's "contributor version".
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+           A contributor's "essential patent claims" are all patent claims
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+         consequence of further modification of the contributor version.  For
+         purposes of this definition, "control" includes the right to grant
+         patent sublicenses in a manner consistent with the requirements of
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+           In the following three paragraphs, a "patent license" is any express
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+           If you convey a covered work, knowingly relying on a patent license,
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+         available, or (2) arrange to deprive yourself of the benefit of the
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+         consistent with the requirements of this License, to extend the patent
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+         covered work in a country, or your recipient's use of the covered work
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+         
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+         arrangement, you convey, or propagate by procuring conveyance of, a
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+         
+           A patent license is "discriminatory" if it does not include within
+         the scope of its coverage, prohibits the exercise of, or is
+         conditioned on the non-exercise of one or more of the rights that are
+         specifically granted under this License.  You may not convey a covered
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+         in the business of distributing software, under which you make payment
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+         for and in connection with specific products or compilations that
+         contain the covered work, unless you entered into that arrangement,
+         or that patent license was granted, prior to 28 March 2007.
+         
+           Nothing in this License shall be construed as excluding or limiting
+         any implied license or other defenses to infringement that may
+         otherwise be available to you under applicable patent law.
+         
+           12. No Surrender of Others' Freedom.
+         
+           If conditions are imposed on you (whether by court order, agreement or
+         otherwise) that contradict the conditions of this License, they do not
+         excuse you from the conditions of this License.  If you cannot convey a
+         covered work so as to satisfy simultaneously your obligations under this
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+         not convey it at all.  For example, if you agree to terms that obligate you
+         to collect a royalty for further conveying from those to whom you convey
+         the Program, the only way you could satisfy both those terms and this
+         License would be to refrain entirely from conveying the Program.
+         
+           13. Use with the GNU Affero General Public License.
+         
+           Notwithstanding any other provision of this License, you have
+         permission to link or combine any covered work with a work licensed
+         under version 3 of the GNU Affero General Public License into a single
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+         but the special requirements of the GNU Affero General Public License,
+         section 13, concerning interaction through a network will apply to the
+         combination as such.
+         
+           14. Revised Versions of this License.
+         
+           The Free Software Foundation may publish revised and/or new versions of
+         the GNU General Public License from time to time.  Such new versions will
+         be similar in spirit to the present version, but may differ in detail to
+         address new problems or concerns.
+         
+           Each version is given a distinguishing version number.  If the
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+         Public License "or any later version" applies to it, you have the
+         option of following the terms and conditions either of that numbered
+         version or of any later version published by the Free Software
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+         
+           If the Program specifies that a proxy can decide which future
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+         permissions.  However, no additional obligations are imposed on any
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+         later version.
+         
+           15. Disclaimer of Warranty.
+         
+           THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
+         APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
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+           16. Limitation of Liability.
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+         
+           17. Interpretation of Sections 15 and 16.
+         
+           If the disclaimer of warranty and limitation of liability provided
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+         copy of the Program in return for a fee.
+         
+                              END OF TERMS AND CONDITIONS
+         
+                     How to Apply These Terms to Your New Programs
+         
+           If you develop a new program, and you want it to be of the greatest
+         possible use to the public, the best way to achieve this is to make it
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+         
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+         the "copyright" line and a pointer to where the full notice is found.
+         
+             <one line to give the program's name and a brief idea of what it does.>
+             Copyright (C) <year>  <name of author>
+         
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+             it under the terms of the GNU General Public License as published by
+             the Free Software Foundation, either version 3 of the License, or
+             (at your option) any later version.
+         
+             This program is distributed in the hope that it will be useful,
+             but WITHOUT ANY WARRANTY; without even the implied warranty of
+             MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
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+         
+             You should have received a copy of the GNU General Public License
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+         
+         Also add information on how to contact you by electronic and paper mail.
+         
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+         
+             <program>  Copyright (C) <year>  <name of author>
+             This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+             This is free software, and you are welcome to redistribute it
+             under certain conditions; type `show c' for details.
+         
+         The hypothetical commands `show w' and `show c' should show the appropriate
+         parts of the General Public License.  Of course, your program's commands
+         might be different; for a GUI interface, you would use an "about box".
+         
+           You should also get your employer (if you work as a programmer) or school,
+         if any, to sign a "copyright disclaimer" for the program, if necessary.
+         For more information on this, and how to apply and follow the GNU GPL, see
+         <http://www.gnu.org/licenses/>.
+         
+           The GNU General Public License does not permit incorporating your program
+         into proprietary programs.  If your program is a subroutine library, you
+         may consider it more useful to permit linking proprietary applications with
+         the library.  If this is what you want to do, use the GNU Lesser General
+         Public License instead of this License.  But first, please read
+         <http://www.gnu.org/philosophy/why-not-lgpl.html>.
+         
+         
+         Name: libquadmath
+         Files: scipy.libs/libquadmath*.so
+         Description: dynamically linked to files compiled with gcc
+         Availability: https://gcc.gnu.org/git/?p=gcc.git;a=tree;f=libquadmath
+         License: LGPL-2.1-or-later
+         
+             GCC Quad-Precision Math Library
+             Copyright (C) 2010-2019 Free Software Foundation, Inc.
+             Written by Francois-Xavier Coudert  <fxcoudert@gcc.gnu.org>
+         
+             This file is part of the libquadmath library.
+             Libquadmath is free software; you can redistribute it and/or
+             modify it under the terms of the GNU Library General Public
+             License as published by the Free Software Foundation; either
+             version 2.1 of the License, or (at your option) any later version.
+         
+             Libquadmath is distributed in the hope that it will be useful,
+             but WITHOUT ANY WARRANTY; without even the implied warranty of
+             MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+             Lesser General Public License for more details.
+             https://www.gnu.org/licenses/old-licenses/lgpl-2.1.html
+         
+Classifier: Development Status :: 5 - Production/Stable
+Classifier: Intended Audience :: Science/Research
+Classifier: Intended Audience :: Developers
+Classifier: License :: OSI Approved :: BSD License
+Classifier: Programming Language :: C
+Classifier: Programming Language :: Python
+Classifier: Programming Language :: Python :: 3
+Classifier: Programming Language :: Python :: 3.12
+Classifier: Programming Language :: Python :: 3.13
+Classifier: Programming Language :: Python :: 3.14
+Classifier: Topic :: Software Development :: Libraries
+Classifier: Topic :: Scientific/Engineering
+Classifier: Operating System :: Microsoft :: Windows
+Classifier: Operating System :: POSIX :: Linux
+Classifier: Operating System :: POSIX
+Classifier: Operating System :: Unix
+Classifier: Operating System :: MacOS
+Project-URL: homepage, https://scipy.org/
+Project-URL: documentation, https://docs.scipy.org/doc/scipy/
+Project-URL: source, https://github.com/scipy/scipy
+Project-URL: download, https://github.com/scipy/scipy/releases
+Project-URL: tracker, https://github.com/scipy/scipy/issues
+Requires-Python: >=3.12
+Requires-Dist: numpy<2.8,>=2.0.0
+Provides-Extra: test
+Requires-Dist: pytest>=8.0.0; extra == "test"
+Requires-Dist: pytest-cov; extra == "test"
+Requires-Dist: pytest-timeout; extra == "test"
+Requires-Dist: pytest-xdist; extra == "test"
+Requires-Dist: asv; extra == "test"
+Requires-Dist: mpmath; extra == "test"
+Requires-Dist: gmpy2; extra == "test"
+Requires-Dist: threadpoolctl; extra == "test"
+Requires-Dist: scikit-umfpack; extra == "test"
+Requires-Dist: pooch; extra == "test"
+Requires-Dist: hypothesis>=6.30; extra == "test"
+Requires-Dist: array-api-strict>=2.3.1; extra == "test"
+Requires-Dist: Cython; extra == "test"
+Requires-Dist: meson; extra == "test"
+Requires-Dist: ninja; sys_platform != "emscripten" and extra == "test"
+Requires-Dist: scipy-doctest>=2.0.0; extra == "test"
+Provides-Extra: doc
+Requires-Dist: sphinx<8.2.0,>=5.0.0; extra == "doc"
+Requires-Dist: intersphinx_registry; extra == "doc"
+Requires-Dist: pydata-sphinx-theme>=0.15.2; extra == "doc"
+Requires-Dist: sphinx-copybutton; extra == "doc"
+Requires-Dist: sphinx-design>=0.4.0; extra == "doc"
+Requires-Dist: matplotlib>=3.5; extra == "doc"
+Requires-Dist: numpydoc; extra == "doc"
+Requires-Dist: jupytext; extra == "doc"
+Requires-Dist: myst-nb>=1.2.0; extra == "doc"
+Requires-Dist: pooch; extra == "doc"
+Requires-Dist: jupyterlite-sphinx>=0.19.1; extra == "doc"
+Requires-Dist: jupyterlite-pyodide-kernel; extra == "doc"
+Requires-Dist: linkify-it-py; extra == "doc"
+Requires-Dist: tabulate; extra == "doc"
+Provides-Extra: dev
+Requires-Dist: click<8.3.0; extra == "dev"
+Requires-Dist: spin; extra == "dev"
+Requires-Dist: mypy==1.19.1; extra == "dev"
+Requires-Dist: pyrefly==0.63.0; extra == "dev"
+Requires-Dist: typing_extensions; extra == "dev"
+Requires-Dist: types-psutil; extra == "dev"
+Requires-Dist: pycodestyle; extra == "dev"
+Requires-Dist: ruff>=0.12.0; extra == "dev"
+Requires-Dist: cython-lint>=0.12.2; extra == "dev"
+Description-Content-Type: text/x-rst
+
+.. image:: https://raw.githubusercontent.com/scipy/scipy/main/doc/source/_static/logo.svg
+  :target: https://scipy.org
+  :width: 110
+  :height: 110
+  :align: left
+
+.. image:: https://img.shields.io/badge/powered%20by-NumFOCUS-orange.svg?style=flat&colorA=E1523D&colorB=007D8A
+  :target: https://numfocus.org
+
+.. image:: https://img.shields.io/pypi/dm/scipy.svg?label=Pypi%20downloads
+  :target: https://pypi.org/project/scipy/
+
+.. image:: https://img.shields.io/conda/dn/conda-forge/scipy.svg?label=Conda%20downloads
+  :target: https://anaconda.org/conda-forge/scipy
+
+.. image:: https://img.shields.io/badge/stackoverflow-Ask%20questions-blue.svg
+  :target: https://stackoverflow.com/questions/tagged/scipy
+
+.. image:: https://img.shields.io/badge/DOI-10.1038%2Fs41592--019--0686--2-blue.svg
+  :target: https://www.nature.com/articles/s41592-019-0686-2
+
+.. image:: https://insights.linuxfoundation.org/api/badge/health-score?project=scipy
+  :target: https://insights.linuxfoundation.org/project/scipy
+
+SciPy (pronounced "Sigh Pie") is an open-source software for mathematics,
+science, and engineering. It includes modules for statistics, optimization,
+integration, linear algebra, Fourier transforms, signal and image processing,
+ODE solvers, and more.
+
+- **Website:** https://scipy.org
+- **Documentation:** https://docs.scipy.org/doc/scipy/
+- **Development version of the documentation:** https://scipy.github.io/devdocs
+- **SciPy development forum:** https://discuss.scientific-python.org/c/contributor/scipy
+- **Stack Overflow:** https://stackoverflow.com/questions/tagged/scipy
+- **Source code:** https://github.com/scipy/scipy
+- **Contributing:** https://scipy.github.io/devdocs/dev/index.html
+- **Bug reports:** https://github.com/scipy/scipy/issues
+- **Code of Conduct:** https://docs.scipy.org/doc/scipy/dev/conduct/code_of_conduct.html
+- **Report a security vulnerability:** via Tidelift, as explained in
+  `our docs on Security <https://scipy.github.io/devdocs/tutorial/security.html>`__
+- **Citing in your work:** https://www.scipy.org/citing-scipy/
+
+SciPy is built to work with
+NumPy arrays, and provides many user-friendly and efficient numerical routines,
+such as routines for numerical integration and optimization. Together, they
+run on all popular operating systems, are quick to install, and are free of
+charge. NumPy and SciPy are easy to use, but powerful enough to be depended
+upon by some of the world's leading scientists and engineers. If you need to
+manipulate numbers on a computer and display or publish the results, give
+SciPy a try!
+
+For the installation instructions, see `our install
+guide <https://scipy.org/install/>`__.
+
+
+Call for Contributions
+----------------------
+
+We appreciate and welcome contributions. Small improvements or fixes are always appreciated; issues labeled as "good
+first issue" may be a good starting point. Have a look at `our contributing
+guide <https://scipy.github.io/devdocs/dev/index.html>`__.
+
+Writing code isn’t the only way to contribute to SciPy. You can also:
+
+- review pull requests
+- triage issues
+- develop tutorials, presentations, and other educational materials
+- maintain and improve `our website <https://github.com/scipy/scipy.org>`__
+- develop graphic design for our brand assets and promotional materials
+- help with outreach and onboard new contributors
+- write grant proposals and help with other fundraising efforts
+
+If you’re unsure where to start or how your skills fit in, reach out! You can
+ask on the `forum <https://discuss.scientific-python.org/c/contributor/scipy>`__
+or here, on GitHub, by leaving a comment on a relevant issue that is already
+open.
+
+If you are new to contributing to open source, `this
+guide <https://opensource.guide/how-to-contribute/>`__ helps explain why, what,
+and how to get involved.

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venv/lib/python3.12/site-packages/scipy/signal/_arraytools.py

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

venv/lib/python3.12/site-packages/scipy/signal/_czt.py

@@ -0,0 +1,580 @@
+# 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.
+
+        Returns
+        -------
+        ndarray
+            The points in the Z plane at which `CZT` samples the z-transform.
+        """
+        return czt_points(self.m, self.w, self.a)
+
+
+class ZoomFFT(CZT):
+    """
+    Create a callable zoom FFT transform function.
+
+    This is a specialization of the chirp z-transform (`CZT`) for a set of
+    equally-spaced frequencies around the unit circle, used to calculate a
+    section of the FFT more efficiently than calculating the entire FFT and
+    truncating.
+
+    Parameters
+    ----------
+    n : int
+        The size of the signal.
+    fn : array_like
+        A length-2 sequence [`f1`, `f2`] giving the frequency range, or a
+        scalar, for which the range [0, `fn`] is assumed.
+    m : int, optional
+        The number of points to evaluate.  Default is `n`.
+    fs : float, optional
+        The sampling frequency.  If ``fs=10`` represented 10 kHz, for example,
+        then `f1` and `f2` would also be given in kHz.
+        The default sampling frequency is 2, so `f1` and `f2` should be
+        in the range [0, 1] to keep the transform below the Nyquist
+        frequency.
+    endpoint : bool, optional
+        If True, `f2` is the last sample. Otherwise, it is not included.
+        Default is False.
+
+    Returns
+    -------
+    f : ZoomFFT
+        Callable object ``f(x, axis=-1)`` for computing the zoom FFT on `x`.
+
+    See Also
+    --------
+    zoom_fft : Convenience function for calculating a zoom FFT.
+
+    Notes
+    -----
+    The defaults are chosen such that ``f(x, 2)`` is equivalent to
+    ``fft.fft(x)`` and, if ``m > len(x)``, that ``f(x, 2, m)`` is equivalent to
+    ``fft.fft(x, m)``.
+
+    Sampling frequency is 1/dt, the time step between samples in the
+    signal `x`.  The unit circle corresponds to frequencies from 0 up
+    to the sampling frequency.  The default sampling frequency of 2
+    means that `f1`, `f2` values up to the Nyquist frequency are in the
+    range [0, 1). For `f1`, `f2` values expressed in radians, a sampling
+    frequency of 2*pi should be used.
+
+    Remember that a zoom FFT can only interpolate the points of the existing
+    FFT.  It cannot help to resolve two separate nearby frequencies.
+    Frequency resolution can only be increased by increasing acquisition
+    time.
+
+    These functions are implemented using Bluestein's algorithm (as is
+    `scipy.fft`). [2]_
+
+    References
+    ----------
+    .. [1] Steve Alan Shilling, "A study of the chirp z-transform and its
+           applications", pg 29 (1970)
+           https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf
+    .. [2] Leo I. Bluestein, "A linear filtering approach to the computation
+           of the discrete Fourier transform," Northeast Electronics Research
+           and Engineering Meeting Record 10, 218-219 (1968).
+
+    Examples
+    --------
+    To plot the transform results use something like the following:
+
+    >>> import numpy as np
+    >>> from scipy.signal import ZoomFFT
+    >>> t = np.linspace(0, 1, 1021)
+    >>> x = np.cos(2*np.pi*15*t) + np.sin(2*np.pi*17*t)
+    >>> f1, f2 = 5, 27
+    >>> transform = ZoomFFT(len(x), [f1, f2], len(x), fs=1021)
+    >>> X = transform(x)
+    >>> f = np.linspace(f1, f2, len(x))
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(f, 20*np.log10(np.abs(X)))
+    >>> plt.show()
+    """
+
+    def __init__(self, n, fn, m=None, *, fs=2, endpoint=False):
+        m = _validate_sizes(n, m)
+
+        k = arange(max(m, n), dtype=np.min_scalar_type(-max(m, n)**2))
+
+        if np.size(fn) == 2:
+            f1, f2 = fn
+        elif np.size(fn) == 1:
+            f1, f2 = 0.0, fn
+        else:
+            raise ValueError('fn must be a scalar or 2-length sequence')
+
+        self.f1, self.f2, self.fs = f1, f2, fs
+
+        if endpoint:
+            scale = ((f2 - f1) * m) / (fs * (m - 1))
+        else:
+            scale = (f2 - f1) / fs
+        a = cmath.exp(2j * pi * f1/fs)
+        wk2 = np.exp(-(1j * pi * scale * k**2) / m)
+
+        self.w = cmath.exp(-2j*pi/m * scale)
+        self.a = a
+        self.m, self.n = m, n
+
+        ak = np.exp(-2j * pi * f1/fs * k[:n])
+        self._Awk2 = ak * wk2[:n]
+
+        nfft = next_fast_len(n + m - 1)
+        self._nfft = nfft
+        self._Fwk2 = fft(1/np.hstack((wk2[n-1:0:-1], wk2[:m])), nfft)
+        self._wk2 = wk2[:m]
+        self._yidx = slice(n-1, n+m-1)
+
+
+def czt(x, m=None, w=None, a=1+0j, *, axis=-1):
+    """
+    Compute the frequency response around a spiral in the Z plane.
+
+    Parameters
+    ----------
+    x : array
+        The signal to transform.
+    m : int, optional
+        The number of output points desired.  Default is the length of the
+        input data.
+    w : complex, optional
+        The ratio between points in each step.  This must be precise or the
+        accumulated error will degrade the tail of the output sequence.
+        Defaults to equally spaced points around the entire unit circle.
+    a : complex, optional
+        The starting point in the complex plane.  Default is 1+0j.
+    axis : int, optional
+        Axis over which to compute the FFT. If not given, the last axis is
+        used.
+
+    Returns
+    -------
+    out : ndarray
+        An array of the same dimensions as `x`, but with the length of the
+        transformed axis set to `m`.
+
+    See Also
+    --------
+    CZT : Class that creates a callable chirp z-transform function.
+    zoom_fft : Convenience function for partial FFT calculations.
+
+    Notes
+    -----
+    The defaults are chosen such that ``signal.czt(x)`` is equivalent to
+    ``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.czt(x, m)`` is
+    equivalent to ``fft.fft(x, m)``.
+
+    If the transform needs to be repeated, use `CZT` to construct a
+    specialized transform function which can be reused without
+    recomputing constants.
+
+    An example application is in system identification, repeatedly evaluating
+    small slices of the z-transform of a system, around where a pole is
+    expected to exist, to refine the estimate of the pole's true location. [1]_
+
+    References
+    ----------
+    .. [1] Steve Alan Shilling, "A study of the chirp z-transform and its
+           applications", pg 20 (1970)
+           https://krex.k-state.edu/dspace/bitstream/handle/2097/7844/LD2668R41972S43.pdf
+
+    Examples
+    --------
+    Generate a sinusoid:
+
+    >>> import numpy as np
+    >>> f1, f2, fs = 8, 10, 200  # Hz
+    >>> t = np.linspace(0, 1, fs, endpoint=False)
+    >>> x = np.sin(2*np.pi*t*f2)
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(t, x)
+    >>> plt.axis([0, 1, -1.1, 1.1])
+    >>> plt.show()
+
+    Its discrete Fourier transform has all of its energy in a single frequency
+    bin:
+
+    >>> from scipy.fft import rfft, rfftfreq
+    >>> from scipy.signal import czt, czt_points
+    >>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x)))
+    >>> plt.margins(0, 0.1)
+    >>> plt.show()
+
+    However, if the sinusoid is logarithmically-decaying:
+
+    >>> x = np.exp(-t*f1) * np.sin(2*np.pi*t*f2)
+    >>> plt.plot(t, x)
+    >>> plt.axis([0, 1, -1.1, 1.1])
+    >>> plt.show()
+
+    the DFT will have spectral leakage:
+
+    >>> plt.plot(rfftfreq(fs, 1/fs), abs(rfft(x)))
+    >>> plt.margins(0, 0.1)
+    >>> plt.show()
+
+    While the DFT always samples the z-transform around the unit circle, the
+    chirp z-transform allows us to sample the Z-transform along any
+    logarithmic spiral, such as a circle with radius smaller than unity:
+
+    >>> M = fs // 2  # Just positive frequencies, like rfft
+    >>> a = np.exp(-f1/fs)  # Starting point of the circle, radius < 1
+    >>> w = np.exp(-1j*np.pi/M)  # "Step size" of circle
+    >>> points = czt_points(M + 1, w, a)  # M + 1 to include Nyquist
+    >>> plt.plot(points.real, points.imag, '.')
+    >>> plt.gca().add_patch(plt.Circle((0,0), radius=1, fill=False, alpha=.3))
+    >>> plt.axis('equal'); plt.axis([-1.05, 1.05, -0.05, 1.05])
+    >>> plt.show()
+
+    With the correct radius, this transforms the decaying sinusoid (and others
+    with the same decay rate) without spectral leakage:
+
+    >>> z_vals = czt(x, M + 1, w, a)  # Include Nyquist for comparison to rfft
+    >>> freqs = np.angle(points)*fs/(2*np.pi)  # angle = omega, radius = sigma
+    >>> plt.plot(freqs, abs(z_vals))
+    >>> plt.margins(0, 0.1)
+    >>> plt.show()
+    """
+    x = np.asarray(x)
+    transform = CZT(x.shape[axis], m=m, w=w, a=a)
+    return transform(x, axis=axis)
+
+
+def zoom_fft(x, fn, m=None, *, fs=2, endpoint=False, axis=-1):
+    """
+    Compute the DFT of `x` only for frequencies in range `fn`.
+
+    Parameters
+    ----------
+    x : array
+        The signal to transform.
+    fn : array_like
+        A length-2 sequence [`f1`, `f2`] giving the frequency range, or a
+        scalar, for which the range [0, `fn`] is assumed.
+    m : int, optional
+        The number of points to evaluate.  The default is the length of `x`.
+    fs : float, optional
+        The sampling frequency.  If ``fs=10`` represented 10 kHz, for example,
+        then `f1` and `f2` would also be given in kHz.
+        The default sampling frequency is 2, so `f1` and `f2` should be
+        in the range [0, 1] to keep the transform below the Nyquist
+        frequency.
+    endpoint : bool, optional
+        If True, `f2` is the last sample. Otherwise, it is not included.
+        Default is False.
+    axis : int, optional
+        Axis over which to compute the FFT. If not given, the last axis is
+        used.
+
+    Returns
+    -------
+    out : ndarray
+        The transformed signal.  The Fourier transform will be calculated
+        at the points f1, f1+df, f1+2df, ..., f2, where df=(f2-f1)/m.
+
+    See Also
+    --------
+    ZoomFFT : Class that creates a callable partial FFT function.
+
+    Notes
+    -----
+    The defaults are chosen such that ``signal.zoom_fft(x, 2)`` is equivalent
+    to ``fft.fft(x)`` and, if ``m > len(x)``, that ``signal.zoom_fft(x, 2, m)``
+    is equivalent to ``fft.fft(x, m)``.
+
+    To graph the magnitude of the resulting transform, use::
+
+        plot(linspace(f1, f2, m, endpoint=False), abs(zoom_fft(x, [f1, f2], m)))
+
+    If the transform needs to be repeated, use `ZoomFFT` to construct
+    a specialized transform function which can be reused without
+    recomputing constants.
+
+    Examples
+    --------
+    To plot the transform results use something like the following:
+
+    >>> import numpy as np
+    >>> from scipy.signal import zoom_fft
+    >>> t = np.linspace(0, 1, 1021)
+    >>> x = np.cos(2*np.pi*15*t) + np.sin(2*np.pi*17*t)
+    >>> f1, f2 = 5, 27
+    >>> X = zoom_fft(x, [f1, f2], len(x), fs=1021)
+    >>> f = np.linspace(f1, f2, len(x))
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(f, 20*np.log10(np.abs(X)))
+    >>> plt.show()
+    """
+    x = np.asarray(x)
+    transform = ZoomFFT(x.shape[axis], fn, m=m, fs=fs, endpoint=endpoint)
+    return transform(x, axis=axis)

venv/lib/python3.12/site-packages/scipy/signal/_delegators.py

@@ -0,0 +1,573 @@
+"""Delegators for alternative backends in scipy.signal.
+
+The signature of `func_signature` must match the signature of signal.func.
+The job of a `func_signature` is to know which arguments of `signal.func`
+are arrays.
+
+* signatures are generated by
+
+--------------
+ import inspect
+ from scipy import signal
+
+ names = [x for x in dir(signal) if not x.startswith('_')]
+ objs = [getattr(signal, name) for name in names]
+ funcs = [obj for obj in objs if inspect.isroutine(obj)]
+
+ for func in funcs:
+     try:
+        sig = inspect.signature(func)
+     except ValueError:
+         sig = "( FIXME )"
+     print(f"def {func.__name__}_signature{sig}:\n\treturn array_namespace(...
+ )\n\n")
+---------------
+
+* which arguments to delegate on: manually trawled the documentation for
+  array-like and array arguments
+
+"""
+import numpy as np
+from scipy._lib._array_api import array_namespace, np_compat, is_jax
+
+
+def _skip_if_lti(arg):
+    """Handle `system` arg overloads.
+
+    ATM, only pass tuples through. Consider updating when cupyx.lti class
+    is supported.
+    """
+    if isinstance(arg, tuple):
+        return arg
+    else:
+        return (None,)
+
+
+def _skip_if_str_or_tuple(window):
+    """Handle `window` being a str or a tuple or an array-like.
+    """
+    if isinstance(window, str) or isinstance(window, tuple) or callable(window):
+        return None
+    else:
+        return window
+
+
+def _skip_if_poly1d(arg):
+    return None if isinstance(arg, np.poly1d) else arg
+
+
+###################
+
+def abcd_normalize_signature(A=None, B=None, C=None, D=None):
+    return array_namespace(A, B, C, D)
+
+def argrelextrema_signature(data, *args, **kwds):
+    return array_namespace(data)
+
+argrelmax_signature = argrelextrema_signature
+argrelmin_signature = argrelextrema_signature
+
+
+def band_stop_obj_signature(wp, ind, passb, stopb, gpass, gstop, type):
+    return array_namespace(passb, stopb)
+
+
+def bessel_signature(N, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+butter_signature = bessel_signature
+
+
+def cheby2_signature(N, rs, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+def cheby1_signature(N, rp, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+def ellip_signature(N, rp, rs, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+########################## XXX: no arrays in, arrays out
+def besselap_signature(N, norm='phase', *, xp=None, device=None):
+    return np if xp is None else xp
+
+
+def buttap_signature(N, *, xp=None, device=None):
+    return np if xp is None else xp
+
+
+def cheb1ap_signature(N, rp, *, xp=None, device=None):
+    return np if xp is None else xp
+
+
+def cheb2ap_signature(N, rs, *, xp=None, device=None):
+    return np if xp is None else xp
+
+def ellipap_signature(N, rp, rs, *, xp=None, device=None):
+    return np if xp is None else xp
+
+def correlation_lags_signature(in1_len, in2_len, mode='full'):
+    return np
+
+
+def czt_points_signature(m, w=None, a=(1+0j)):
+    return np
+
+
+def gammatone_signature(
+    freq, ftype, order=None, numtaps=None, fs=None, *, xp=None, device=None
+    ):
+    return np_compat if xp is None else xp
+
+
+def iircomb_signature(
+    w0, Q, ftype='notch', fs=2.0, *, pass_zero=False, xp=None, device=None
+):
+    return np_compat if xp is None else xp
+
+
+def iirnotch_signature(w0, Q, fs=2.0, *, xp=None, device=None):
+    return np if xp is None else xp
+
+iirpeak_signature = iirnotch_signature
+
+
+def savgol_coeffs_signature(
+    window_length, polyorder, deriv=0, delta=1.0, pos=None, use='conv',
+    *, xp=None, device=None
+):
+    return np if xp is None else xp
+
+
+
+def unit_impulse_signature(shape, idx=None, dtype=float):
+    return np
+############################
+
+
+def buttord_signature(wp, ws, gpass, gstop, analog=False, fs=None):
+    return array_namespace(wp, ws)
+
+cheb1ord_signature = buttord_signature
+cheb2ord_signature = buttord_signature
+ellipord_signature = buttord_signature
+
+
+########### NB: scalars in, scalars out
+def kaiser_atten_signature(numtaps, width):
+    return np
+
+def kaiser_beta_signature(a):
+    return np
+
+def kaiserord_signature(ripple, width):
+    return np
+
+def get_window_signature(window, Nx, fftbins=True, *, xp=None, device=None):
+    return np if xp is None else xp
+#################################
+
+
+def bode_signature(system, w=None, n=100):
+    return array_namespace(*_skip_if_lti(system), w)
+
+dbode_signature = bode_signature
+
+
+def freqresp_signature(system, w=None, n=10000):
+    return array_namespace(*_skip_if_lti(system), w)
+
+dfreqresp_signature = freqresp_signature
+
+
+def impulse_signature(system, X0=None, T=None, N=None):
+    return array_namespace(*_skip_if_lti(system), X0, T)
+
+
+def dimpulse_signature(system, x0=None, t=None, n=None):
+    return array_namespace(*_skip_if_lti(system), x0, t)
+
+
+def lsim_signature(system, U, T, X0=None, interp=True):
+    return array_namespace(*_skip_if_lti(system), U, T, X0)
+
+
+def dlsim_signature(system, u, t=None, x0=None):
+    return array_namespace(*_skip_if_lti(system), u, t, x0)
+
+
+def step_signature(system, X0=None, T=None, N=None):
+    return array_namespace(*_skip_if_lti(system), X0, T)
+
+def dstep_signature(system, x0=None, t=None, n=None):
+    return array_namespace(*_skip_if_lti(system), x0, t)
+
+
+def cont2discrete_signature(system, dt, method='zoh', alpha=None):
+    return array_namespace(*_skip_if_lti(system))
+
+
+def bilinear_signature(b, a, fs=1.0):
+    return array_namespace(b, a)
+
+
+def bilinear_zpk_signature(z, p, k, fs):
+    return array_namespace(z, p)
+
+
+def chirp_signature(t,*args, **kwds):
+    return array_namespace(t)
+
+
+############## XXX: array-likes in, str out
+def choose_conv_method_signature(in1, in2, *args, **kwds):
+    return array_namespace(in1, in2)
+############################################
+
+
+def convolve_signature(in1, in2, *args, **kwds):
+    return array_namespace(in1, in2)
+
+fftconvolve_signature = convolve_signature
+oaconvolve_signature = convolve_signature
+correlate_signature = convolve_signature
+correlate_signature = convolve_signature
+convolve2d_signature = convolve_signature
+correlate2d_signature = convolve_signature
+
+
+def coherence_signature(x, y, fs=1.0, window='hann_periodic', *args, **kwds):
+    return array_namespace(x, y, _skip_if_str_or_tuple(window))
+
+
+def csd_signature(x, y, fs=1.0, window='hann_periodic', *args, **kwds):
+    return array_namespace(x, y, _skip_if_str_or_tuple(window))
+
+
+def periodogram_signature(x, fs=1.0, window='boxcar', *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def welch_signature(x, fs=1.0, window='hann_periodic', *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def spectrogram_signature(x, fs=1.0, window=('tukey_periodic', 0.25), *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def stft_signature(x, fs=1.0, window='hann_periodic', *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def istft_signature(Zxx, fs=1.0, window='hann_periodic', *args, **kwds):
+    return array_namespace(Zxx, _skip_if_str_or_tuple(window))
+
+
+def resample_signature(x, num, t=None, axis=0, window=None, domain='time'):
+    return array_namespace(x, t, _skip_if_str_or_tuple(window))
+
+
+def resample_poly_signature(x, up, down, axis=0, window=('kaiser', 5.0), *args, **kwds):
+    return array_namespace(x, _skip_if_str_or_tuple(window))
+
+
+def check_COLA_signature(window, nperseg, noverlap, tol=1e-10):
+    return array_namespace(_skip_if_str_or_tuple(window))
+
+
+def check_NOLA_signature(window, nperseg, noverlap, tol=1e-10):
+    return array_namespace(_skip_if_str_or_tuple(window))
+
+
+def czt_signature(x, *args, **kwds):
+    return array_namespace(x)
+
+decimate_signature = czt_signature
+gauss_spline_signature = czt_signature
+
+
+def deconvolve_signature(signal, divisor):
+    return array_namespace(signal, divisor)
+
+
+def detrend_signature(data, axis=1, type='linear', bp=0, *args, **kwds):
+    xp = array_namespace(data)
+    # JAX doesn't accept JAX arrays for bp, only ints, lists and NumPy
+    # arrays.
+    return xp if is_jax(xp) else array_namespace(data, bp)
+
+
+def filtfilt_signature(b, a, x, *args, **kwds):
+    return array_namespace(b, a, x)
+
+
+def lfilter_signature(b, a, x, axis=-1, zi=None):
+    return array_namespace(b, a, x, zi)
+
+
+def envelope_signature(z, *args, **kwds):
+    return array_namespace(z)
+
+
+def find_peaks_signature(
+    x, height=None, threshold=None, distance=None, prominence=None, width=None,
+    wlen=None, rel_height=0.5, plateau_size=None
+):
+    # TODO: fix me - `prominence` is not necessarily an array.
+    # return array_namespace(x, height, threshold, prominence, width, plateau_size)
+    # See https://github.com/scipy/scipy/pull/22644#issuecomment-3568443768. For now:
+    return np_compat
+
+def find_peaks_cwt_signature(
+    vector, widths, wavelet=None, max_distances=None, *args, **kwds
+):
+    return array_namespace(vector, widths, max_distances)
+
+
+def findfreqs_signature(num, den, N, kind='ba'):
+    return array_namespace(num, den)
+
+
+def firls_signature(numtaps, bands, desired, *, weight=None, fs=None):
+    return array_namespace(bands, desired, weight)
+
+
+def firwin_signature(numtaps, cutoff, *args, **kwds):
+    if isinstance(cutoff, int | float):
+        xp = np_compat
+    else:
+        xp = array_namespace(cutoff)
+    return xp
+
+
+def firwin2_signature(numtaps, freq, gain, *args, **kwds):
+    return array_namespace(freq, gain)
+
+
+def freqs_zpk_signature(z, p, k, worN=200, *args, **kwds):
+    return array_namespace(z, p, worN)
+
+freqz_zpk_signature = freqs_zpk_signature
+
+
+def freqs_signature(b, a, worN=200, *args, **kwds):
+    return array_namespace(b, a, worN)
+
+
+def freqz_signature(b, a=1, worN=512, *args, **kwds):
+    # differs from freqs: `a` has a default value
+    return array_namespace(b, a, worN)
+
+
+def freqz_sos_signature(sos, worN=512, *args, **kwds):
+    return array_namespace(sos, worN)
+
+sosfreqz_signature = freqz_sos_signature
+
+
+def gausspulse_signature(t, *args, **kwds):
+    arr_t = None if isinstance(t, str) else t
+    return array_namespace(arr_t)
+
+
+def group_delay_signature(system, w=512, whole=False, fs=6.283185307179586):
+    return array_namespace(*system, w)
+
+
+def hilbert_signature(x, *args, **kwds):
+    return array_namespace(x)
+
+hilbert2_signature = hilbert_signature
+
+
+def iirdesign_signature(wp, ws, *args, **kwds):
+    return array_namespace(wp, ws)
+
+
+def iirfilter_signature(N, Wn, *args, **kwds):
+    return array_namespace(Wn)
+
+
+def invres_signature(r, p, k, tol=0.001, rtype='avg'):
+    return array_namespace(r, p, k)
+
+invresz_signature = invres_signature
+
+
+############################### XXX: excluded, blacklisted on CuPy (mismatched API)
+def lfilter_zi_signature(b, a):
+    return array_namespace(b, a)
+
+def sosfilt_zi_signature(sos):
+    return array_namespace(sos)
+
+# needs to be blacklisted on CuPy (is not implemented)
+def remez_signature(numtaps, bands, desired, *, weight=None, **kwds):
+    return array_namespace(bands, desired, weight)
+#############################################
+
+def lfiltic_signature(b, a, y, x=None):
+    return array_namespace(b, a, y, x)
+
+
+def lombscargle_signature(
+    x, y, freqs, precenter=False, normalize=False, *,
+    weights=None, floating_mean=False
+):
+    return array_namespace(x, y, freqs, weights)
+
+
+def lp2bp_signature(b, a, *args, **kwds):
+    return array_namespace(b, a)
+
+lp2bs_signature = lp2bp_signature
+lp2hp_signature = lp2bp_signature
+lp2lp_signature = lp2bp_signature
+
+tf2zpk_signature = lp2bp_signature
+tf2sos_signature = lp2bp_signature
+
+normalize_signature = lp2bp_signature
+residue_signature = lp2bp_signature
+residuez_signature = residue_signature
+
+
+def lp2bp_zpk_signature(z, p, k, *args, **kwds):
+    return array_namespace(z, p)
+
+lp2bs_zpk_signature = lp2bp_zpk_signature
+lp2hp_zpk_signature = lp2bs_zpk_signature
+lp2lp_zpk_signature = lp2bs_zpk_signature
+
+
+def zpk2sos_signature(z, p, k, *args, **kwds):
+    return array_namespace(z, p)
+
+zpk2ss_signature = zpk2sos_signature
+zpk2tf_signature = zpk2sos_signature
+
+
+def max_len_seq_signature(nbits, state=None, length=None, taps=None):
+    return array_namespace(state, taps)
+
+
+def medfilt_signature(volume, kernel_size=None):
+    return array_namespace(volume)
+
+
+def medfilt2d_signature(input, kernel_size=3):
+    return array_namespace(input)
+
+
+def minimum_phase_signature(h, *args, **kwds):
+    return array_namespace(h)
+
+
+def order_filter_signature(a, domain, rank):
+    return array_namespace(a, domain)
+
+
+def peak_prominences_signature(x, peaks, *args, **kwds):
+    return array_namespace(x, peaks)
+
+
+peak_widths_signature = peak_prominences_signature
+
+
+def place_poles_signature(A, B, poles, method='YT', rtol=0.001, maxiter=30):
+    return array_namespace(A, B, poles)
+
+
+def savgol_filter_signature(x, *args, **kwds):
+    return array_namespace(x)
+
+
+def sawtooth_signature(t, width=1):
+    return array_namespace(t)
+
+
+def sepfir2d_signature(input, hrow, hcol):
+    return array_namespace(input, hrow, hcol)
+
+
+def sos2tf_signature(sos):
+    return array_namespace(sos)
+
+
+sos2zpk_signature = sos2tf_signature
+
+
+def sosfilt_signature(sos, x, axis=-1, zi=None):
+    return array_namespace(sos, x, zi)
+
+
+def sosfiltfilt_signature(sos, x, *args, **kwds):
+    return array_namespace(sos, x)
+
+
+def spline_filter_signature(Iin, lmbda=5.0):
+    return array_namespace(Iin)
+
+
+def square_signature(t, duty=0.5):
+    return array_namespace(t, duty)
+
+
+def ss2tf_signature(A, B, C, D, input=0):
+    return array_namespace(A, B, C, D)
+
+ss2zpk_signature = ss2tf_signature
+
+
+def sweep_poly_signature(t, poly, phi=0):
+    return array_namespace(t, _skip_if_poly1d(poly))
+
+
+def symiirorder1_signature(signal, c0, z1, precision=-1.0):
+    return array_namespace(signal)
+
+
+def symiirorder2_signature(input, r, omega, precision=-1.0):
+    return array_namespace(input, r, omega)
+
+
+def cspline1d_signature(signal, *args, **kwds):
+    return array_namespace(signal)
+
+qspline1d_signature = cspline1d_signature
+cspline2d_signature = cspline1d_signature
+qspline2d_signature = qspline1d_signature
+
+
+def cspline1d_eval_signature(cj, newx, *args, **kwds):
+    return array_namespace(cj, newx)
+
+qspline1d_eval_signature = cspline1d_eval_signature
+
+
+def tf2ss_signature(num, den):
+    return array_namespace(num, den)
+
+
+def unique_roots_signature(p, tol=0.001, rtype='min'):
+    return array_namespace(p)
+
+
+def upfirdn_signature(h, x, up=1, down=1, axis=-1, mode='constant', cval=0):
+    return array_namespace(h, x)
+
+
+def vectorstrength_signature(events, period):
+    return array_namespace(events, period)
+
+
+def wiener_signature(im, mysize=None, noise=None):
+    return array_namespace(im)
+
+
+def zoom_fft_signature(x, fn, m=None, *, fs=2, endpoint=False, axis=-1):
+    return array_namespace(x, fn)

venv/lib/python3.12/site-packages/scipy/signal/_fir_filter_design.py

@@ -0,0 +1,1553 @@
+"""Functions for FIR filter design."""
+
+from math import ceil, log, log2
+import warnings
+from typing import Literal
+
+import numpy as np
+from scipy.fft import irfft, fft, ifft
+from scipy.linalg import (toeplitz, hankel, solve, LinAlgError, LinAlgWarning,
+                          lstsq)
+from scipy.signal._arraytools import _validate_fs
+from .windows import get_window
+from . import _sigtools
+
+from scipy._lib._array_api import array_namespace, xp_size, xp_default_dtype
+import scipy._external.array_api_extra as xpx
+
+
+__all__ = ['kaiser_beta', 'kaiser_atten', 'kaiserord',
+           'firwin', 'firwin2', 'firwin_2d', 'remez', 'firls', 'minimum_phase']
+
+
+# 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"{A:f} is too small for the Kaiser formula.")
+    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
+
+
+def firwin(numtaps, cutoff, *, width=None, window='hamming', pass_zero=True,
+           scale=True, fs=None):
+    r"""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
+        former case, as a float, the cutoff frequency should correspond
+        with the half-amplitude point, where the attenuation will be -6 dB.
+        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`. It should be noted
+        that this is different from the behavior of `~scipy.signal.iirdesign`,
+        where the cutoff is the half-power point (-3 dB).
+    width : float or None, optional
+        If not ``None``, then a `~scipy.signal.windows.kaiser` window is calculated
+        where `width` specifies the approximate width of the transition region
+        (expressed in the same unit as `fs`). This is achieved by utilizing
+        `~scipy.signal.kaiser_atten` to calculate an attenuation which is passed to
+        `~scipy.signal.kaiser_beta` for determining the β parameter for the kaiser
+        window. In this case, the `window` argument is ignored.
+    window : str or tuple of str and parameter values, optional
+        Desired window to use. Default is ``'hamming'``. The window will be symmetric,
+        unless a suffix ``'_periodic'`` is appended to the window name (e.g.,
+        ``'hamming_perodic'``) Consult `~scipy.signal.get_window` for a list of windows
+        and required parameters.
+    pass_zero : {True, False, 'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional
+        Toggles the zero frequency bin (or DC gain) to be in the passband (``True``) or
+        in the stopband (``False``).  ``'bandstop'``, ``'lowpass'`` are synonyms for
+        ``True`` and ``'bandpass'``, ``'highpass'`` are synonyms for ``False``.
+        ``'lowpass'``, ``'highpass'`` additionally require `cutoff` to be a scalar
+        value or a length-one array. Default: ``True``.
+
+        .. 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
+
+    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 : ndarray
+        FIR filter coefficients as 1d array with `numtaps` entries.
+
+    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:  Window method FIR filter design specifying gain-frequency pairs.
+    firwin_2d: 2D FIR filter design using the window method.
+    firls: FIR filter design using least-squares error minimization.
+    minimum_phase: Convert a FIR filter to minimum phase
+    remez: Calculate the minimax optimal filter using the Remez exchange algorithm.
+
+    Examples
+    --------
+    The following example calculates frequency responses of a 30 Hz low-pass filter
+    with various numbers of taps. The transition region width is 20 Hz. The upper plot
+    shows the gain and the lower plot the phase. The vertical dashed line marks the
+    corner frequency and the gray background the transition region.
+
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy as np
+    >>> import scipy.signal as signal
+    ...
+    >>> fs = 200  # sampling frequency and number of taps
+    >>> f_c, width = 30, 20  # corner frequency (-6 dB gain)
+    >>> taps = [20, 40, 60]  # number of taps
+    ...
+    >>> fg, (ax0, ax1) = plt.subplots(2, 1, sharex='all', layout="constrained",
+    ...                               figsize=(5, 4))
+    >>> ax0.set_title(rf"Response of ${f_c}\,$Hz low-pass Filter with " +
+    ...               rf"${width}\,$Hz transition region")
+    >>> ax0.set(ylabel="Gain in dB", xlim=(0, fs/2))
+    >>> ax1.set(xlabel=rf"Frequency $f\,$ in hertz (sampling frequency $f_S={fs}\,$Hz)",
+    ...         ylabel="Phase in Degrees")
+    ...
+    >>> for n in taps:  # calculate filter and plot response:
+    ...     bb = signal.firwin(n, f_c, width=width, fs=fs)
+    ...     f, H = signal.freqz(bb, fs=fs)  # calculate frequency response
+    ...     H_dB, H_ph = 20 * np.log10(abs(H)), np.rad2deg(np.unwrap(np.angle(H)))
+    ...     H_ph[H_dB<-150] = np.nan
+    ...     ax0.plot(f, H_dB, alpha=.5, label=rf"{n} taps")
+    ...     ax1.plot(f, H_ph, alpha=.5, label=rf"{n} taps")
+    >>> for ax_ in (ax0, ax1):
+    ...     ax_.axvspan(f_c-width/2, f_c+width/2,color='gray', alpha=.25)
+    ...     ax_.axvline(f_c, color='gray', linestyle='--', alpha=.5)
+    ...     ax_.grid()
+    ...     ax_.legend()
+    >>> plt.show()
+
+    The plots show that with increasing number of taps, the suppression in the stopband
+    increases and the (negative) slope of the phase steepens, which signifies a longer
+    signal delay in the filter. Note that the plots contain numeric artifacts caused by
+    the limited frequency resolution: The phase jumps are not real---in reality the
+    phase is a straight line with a constant negative slope. Furthermore, the gains
+    contain zero values (i.e., -∞ dB), which are also not depicted.
+
+    The second example determines the frequency responses of a 30 Hz low-pass filter
+    with 40 taps. This time the width of the transition region varies:
+
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy as np
+    >>> import scipy.signal as signal
+    ...
+    >>> fs = 200  # sampling frequency and number of taps
+    >>> n, f_c = 40, 30  # number of taps and corner frequency (-6 dB gain)
+    >>> widths = (2, 10, 25)  # width of transition region
+    ...
+    >>> fg, ax = plt.subplots(1, 1, layout="constrained",)
+    >>> ax.set(title=rf"Response of {n}-tap ${f_c}\,$Hz low-pass Filter",
+    ...        xlabel=rf"Frequency $f\,$ in hertz (sampling frequency $f_S={fs}\,$Hz)",
+    ...        ylabel="Gain in dB", xlim=(0, fs/2))
+    >>> ax.axvline(f_c, color='gray', linestyle='--', alpha=.7)  # mark corner frequency
+    ...
+    >>> for width in widths:  # calculate filter and plot response:
+    ...     bb = signal.firwin(n, f_c, width=width, fs=fs)
+    ...     f, H = signal.freqz(bb, fs=fs)  # calculate frequency response
+    ...     H_dB= 20 * np.log10(abs(H))  # convert to dB
+    ...     ax.plot(f, H_dB, alpha=.5, label=rf"width$={width}\,$Hz")
+    ...
+    >>> ax.grid()
+    >>> ax.legend()
+    >>> plt.show()
+
+    It can be seen in the plot above that with increasing width of the transition
+    region the suppression in the stopband increases. Since the phase does not vary, it
+    is not depicted.
+
+    Instead of defining a transition region width, a window function can also be
+    specified. The plot below depicts the response of an 80 tap band-pass filter having
+    a passband ranging form 40 Hz to 60 Hz (denoted by a gray background) with
+    different windows:
+
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy as np
+    >>> import scipy.signal as signal
+    ...
+    >>> fs, n = 200, 80  # sampling frequency and number of taps
+    >>> cutoff = [40, 60]  # corner frequencies (-6 dB gain)
+    >>> windows = ('boxcar', 'hamming', 'hann', 'blackman')
+    ...
+    >>> fg, ax = plt.subplots(1, 1, layout="constrained")  # set up plotting
+    >>> ax.set(title=rf"Response of {n}-tap Filter with ${cutoff}\,$Hz passband",
+    ...        xlabel=rf"Frequency $f\,$ in hertz (sampling frequency $f_S={fs}\,$Hz)",
+    ...        ylabel="Gain in dB", xlim=(0, fs/2))
+    >>> ax.axvspan(*cutoff, color='gray', alpha=.25)  # mark passband
+    ...
+    >>> for win in windows:  # calculate filter and plot response:
+    ...     bb = signal.firwin(n, cutoff, window=win, pass_zero=False, fs=fs)
+    ...     f, H = signal.freqz(bb, fs=fs)  # calculate frequency response
+    ...     H_dB = 20 * np.log10(abs(H))  # convert to dB
+    ...     ax.plot(f, H_dB, alpha=.5, label=win)
+    ...
+    >>> ax.grid()
+    >>> ax.legend()
+    >>> plt.show()
+
+    The plot shows that the choice of window mainly influences the balance of the
+    suppression in the stopband against the width of the transition region. Note that
+    utilizing the `~scipy.signal.windows.boxcar` window corresponds to just truncating
+    the ideal infinite impulse response to the length of `numtaps` samples.
+
+    The last example illustrates  how to use the `cutoff` and `pass_zero` parameters to
+    create a low-pass, a high-pass, a band-stop and a band-pass filter. The desired
+    ideal frequency gain is drawn as a gray dashed line whereas the response of the FIR
+    filter is depicted as a blue continuous line:
+
+    >>> from itertools import product
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy as np
+    >>> import scipy.signal as signal
+    ...
+    >>> cutoffs = [0.5, (.25, .75)]  # cutoff parameters
+    >>> fg, axx = plt.subplots(4, 1, sharex='all', layout="constrained",
+    ...                        figsize=(5, 4))
+    >>> for ax, (cutoff, pass_zero) in zip(axx, product(cutoffs, (True, False))):
+    ...     ax.set(title=f"firwin(41, {cutoff=}, {pass_zero=}, fs=2)", ylabel="Gain")
+    ...     ax.set_yticks([0.5], minor=True)  # mark gain of 0.5 (= -6 dB)
+    ...     ax.grid(which='minor', axis='y')
+    ...
+    ...     bb = signal.firwin(41, cutoff, pass_zero=pass_zero, fs=2)
+    ...     ff, HH = signal.freqz(bb, fs=2)
+    ...     ax.plot(ff, abs(HH), 'C0-', label="FIR Response")
+    ...
+    ...     f_d = np.hstack(([0], np.atleast_1d(cutoff), [1]))
+    ...     H_d = np.tile([1, 0] if pass_zero else [0, 1], 2)[:len(f_d)]
+    ...     H_d[-1] = H_d[-2] # account for symmetry at Nyquist frequency
+    ...     ax.step(f_d, H_d, 'k--', where='post', alpha=.3, label="Desired Response")
+    >>> axx[-1].set(xlabel=r"Frequency $f\,$ in hertz (sampling frequency $f_S=2\,$Hz)",
+    ...             xlim=(0, 1))
+    >>> plt.show()
+    """
+    # NB: scipy's version of array_namespace returns `np_compat` for int or floats
+    xp = array_namespace(cutoff)
+
+    # 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)
+    fs = 2 if fs is None else fs
+
+    nyq = 0.5 * fs
+
+    cutoff = xp.asarray(cutoff, dtype=xp_default_dtype(xp))
+    cutoff = xpx.atleast_nd(cutoff, ndim=1, xp=xp) / float(nyq)
+
+    # Check for invalid input.
+    if cutoff.ndim > 1:
+        raise ValueError("The cutoff argument must be at most "
+                         "one-dimensional.")
+    if xp_size(cutoff) == 0:
+        raise ValueError("At least one cutoff frequency must be given.")
+    if xp.min(cutoff) <= 0 or xp.max(cutoff) >= 1:
+        raise ValueError("Invalid cutoff frequency: frequencies must be "
+                         "greater than 0 and less than fs/2.")
+    if xp.any(cutoff[1:] - cutoff[:-1] <= 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 pass_zero in ('bandstop', 'lowpass'):
+        if pass_zero == 'lowpass':
+            if xp_size(cutoff) != 1:
+                raise ValueError('cutoff must have one element if '
+                                 f'pass_zero=="lowpass", got {cutoff.shape}')
+        elif xp_size(cutoff) <= 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 xp_size(cutoff) != 1:
+                raise ValueError('cutoff must have one element if '
+                                 f'pass_zero=="highpass", got {cutoff.shape}')
+        elif xp_size(cutoff) <= 1:
+            raise ValueError('cutoff must have at least two elements if '
+                             f'pass_zero=="bandpass", got {cutoff.shape}')
+        pass_zero = False
+    elif not (pass_zero is True or pass_zero is False):
+        raise ValueError(f"Parameter {pass_zero=} not in (True, False, 'bandpass', " +
+                         "'lowpass', 'highpass', 'bandstop')")
+
+    pass_nyquist = (xp_size(cutoff) % 2 == 0) == 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 = xp.concat((xp.zeros(int(pass_zero)), cutoff, xp.ones(int(pass_nyquist))))
+
+
+    # `bands` is a 2-D array; each row gives the left and right edges of
+    # a passband.
+    bands = xp.reshape(cutoff, (-1, 2))
+
+    # Build up the coefficients.
+    alpha = 0.5 * (numtaps - 1)
+    m = xp.arange(0, numtaps, dtype=cutoff.dtype) - alpha
+    h = 0
+    for j in range(bands.shape[0]):
+        left, right = bands[j, 0], bands[j, 1]
+        h += right * xpx.sinc(right * m, xp=xp)
+        h -= left * xpx.sinc(left * m, xp=xp)
+
+    # Get and apply the window function.
+    win = get_window(window, numtaps, fftbins=False, xp=xp)
+    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 = xp.cos(xp.pi * m * scale_frequency)
+        s = xp.sum(h * c)
+        h /= s
+
+    return h
+
+
+# Original version of firwin2 from scipy ticket #457, submitted by "tash".
+#
+# Rewritten by Warren Weckesser, 2010.
+def firwin2(numtaps, freq, gain, *, nfreqs=None, window='hamming',
+            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 : str or (str, float) or float, or None, optional
+        Desired window to use. Default is ``'hamming'``. The window will be symmetric,
+        unless a suffix ``'_periodic'`` is appended to the window name (e.g.,
+        ``'hamming_perodic'``) Consult `~scipy.signal.get_window` for a list of windows
+        and required parameters. If ``None``, no window function is applied.
+    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]
+
+    """
+    xp = array_namespace(freq, gain)
+    freq, gain = xp.asarray(freq), xp.asarray(gain)
+
+    fs = _validate_fs(fs, allow_none=True)
+    fs = 2 if fs is None else fs
+    nyq = 0.5 * fs
+
+    if freq.shape[0] != gain.shape[0]:
+        raise ValueError('freq and gain must be of same length.')
+
+    if nfreqs is not None and numtaps >= nfreqs:
+        raise ValueError(
+            f'ntaps must be less than nfreqs, but firwin2 was called with '
+            f'ntaps={numtaps} and nfreqs={nfreqs}'
+        )
+
+    if freq[0] != 0 or freq[-1] != nyq:
+        raise ValueError('freq must start with 0 and end with fs/2.')
+    d = freq[1:] - freq[:-1]
+    if xp.any(d < 0):
+        raise ValueError('The values in freq must be nondecreasing.')
+    d2 = d[:-1] + d[1:]
+    if xp.any(d2 == 0):
+        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 xp.any(d == 0):
+        # Tweak any repeated values in freq so that interp works.
+        freq = xp.asarray(freq, copy=True)
+        eps = xp.finfo(xp_default_dtype(xp)).eps * nyq
+        for k in range(freq.shape[0] - 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 = freq[1:] - freq[:-1]
+        if xp.any(d <= 0):
+            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, np.asarray(freq), np.asarray(gain))  # XXX array-api-extra#193
+    x = xp.asarray(x)
+    fx = xp.asarray(fx)
+
+    # Adjust the phases of the coefficients so that the first `ntaps` of the
+    # inverse FFT are the desired filter coefficients.
+    shift = xp.exp(-(numtaps - 1) / 2. * 1j * xp.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.
+        wind = get_window(window, numtaps, fftbins=False, xp=xp)
+    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[xp_size(out) // 2] = 0.0
+
+    return out
+
+
+def remez(numtaps, bands, desired, *, weight=None, 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`.
+    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()
+
+    """
+    xp = array_namespace(bands, desired, weight)
+    bands = np.asarray(bands)
+    desired = np.asarray(desired)
+    if weight is not None:
+        weight = np.asarray(weight)
+
+    fs = _validate_fs(fs, allow_none=True)
+    fs = 1.0 if fs is None else fs
+
+    # 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()
+    result = _sigtools._remez(numtaps, bands, desired, weight, tnum, fs,
+                              maxiter, grid_density)
+    return xp.asarray(result)
+
+
+def firls(numtaps, bands, desired, *, weight=None, 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`.
+    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.
+           https://eeweb.engineering.nyu.edu/iselesni/EL713/firls/firls.pdf
+
+    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()
+
+    """
+    xp = array_namespace(bands, desired)
+    bands = np.asarray(bands)
+    desired = np.asarray(desired)
+
+    fs = _validate_fs(fs, allow_none=True)
+    fs = 2 if fs is None else fs
+    nyq = 0.5 * fs
+
+    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(f'nyq must be positive, got {nyq} <= 0.')
+    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 = bands.reshape((-1, 2))
+
+    # check remaining params
+    desired = np.asarray(desired).flatten()
+    if bands.size != desired.size:
+        raise ValueError(
+            f"desired must have one entry per frequency, got {desired.size} "
+            f"gains for {bands.size} frequencies."
+        )
+    desired = desired.reshape((-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 xp.asarray(coeffs)
+
+
+def _dhtm(mag, xp):
+    """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 = xp.zeros(mag.shape[0])
+    # Leave Nyquist and DC at 0, knowing np.abs(fftfreq(N)[midpt]) == 0.5
+    midpt = mag.shape[0] // 2
+    sig = xpx.at(sig)[1:midpt].set(1.0)
+    sig = xpx.at(sig)[midpt+1:].set(-1.0)
+    # 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 = xp.real(ifft(mag * xp.exp(fft(sig * ifft(xp.log(mag))))))
+    return recon
+
+
+def minimum_phase(h,
+                  method: Literal['homomorphic', 'hilbert'] = 'homomorphic',
+                  n_fft: int | None = None, *, half: bool = True):
+    """Convert a linear-phase FIR filter to minimum phase.
+
+    Parameters
+    ----------
+    h : array
+        Linear-phase FIR filter coefficients.
+    method : {'hilbert', 'homomorphic'}
+        The provided methods are:
+
+        - '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 using half
+          the number of taps when ``half=True`` (default), or the
+          original magnitude spectrum using the same number of taps
+          when ``half=False``.
+        - '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).
+    half : bool
+        If ``True``, create a filter that is half the length of the original, with a
+        magnitude spectrum that is the square root of the original. If ``False``,
+        create a filter that is the same length as the original, with a magnitude
+        spectrum that is designed to match the original (only supported when
+        ``method='homomorphic'``).
+
+        .. versionadded:: 1.14.0
+
+    Returns
+    -------
+    h_minimum : array
+        The minimum-phase version of the filter, with length
+        ``(len(h) + 1) // 2`` when ``half is True`` or ``len(h)`` otherwise.
+
+    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 `this DSPGuru page
+    <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," 3rd edition.
+           Upper Saddle River, N.J.: Pearson, 2009.
+
+    Examples
+    --------
+    Create an optimal linear-phase low-pass filter `h` with a transition band of
+    [0.2, 0.3] (assuming a Nyquist frequency of 1):
+
+    >>> 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_hil = minimum_phase(h_linear, method='hilbert')
+    >>> h_hom = minimum_phase(h_linear, method='homomorphic')
+    >>> h_hom_full = minimum_phase(h_linear, method='homomorphic', half=False)
+
+    Compare the impulse and frequency response of the four filters:
+
+    >>> fig0, ax0 = plt.subplots(figsize=(6, 3), tight_layout=True)
+    >>> fig1, axs = plt.subplots(3, sharex='all', figsize=(6, 6), tight_layout=True)
+    >>> ax0.set_title("Impulse response")
+    >>> ax0.set(xlabel='Samples', ylabel='Amplitude', xlim=(0, len(h_linear) - 1))
+    >>> axs[0].set_title("Frequency Response")
+    >>> axs[0].set(xlim=(0, .65), ylabel="Magnitude / dB")
+    >>> axs[1].set(ylabel="Phase / rad")
+    >>> axs[2].set(ylabel="Group Delay / samples", ylim=(-31, 81),
+    ...             xlabel='Normalized Frequency (Nyqist frequency: 1)')
+    >>> for h, lb in ((h_linear,   f'Linear ({len(h_linear)})'),
+    ...               (h_hil,      f'Min-Hilbert ({len(h_hil)})'),
+    ...               (h_hom,      f'Min-Homomorphic ({len(h_hom)})'),
+    ...               (h_hom_full, f'Min-Homom. Full ({len(h_hom_full)})')):
+    ...     w_H, H = freqz(h, fs=2)
+    ...     w_gd, gd = group_delay((h, 1), fs=2)
+    ...
+    ...     alpha = 1.0 if lb == 'linear' else 0.5  # full opacity for 'linear' line
+    ...     ax0.plot(h, '.-', alpha=alpha, label=lb)
+    ...     axs[0].plot(w_H, 20 * np.log10(np.abs(H)), alpha=alpha)
+    ...     axs[1].plot(w_H, np.unwrap(np.angle(H)), alpha=alpha, label=lb)
+    ...     axs[2].plot(w_gd, gd, alpha=alpha)
+    >>> ax0.grid(True)
+    >>> ax0.legend(title='Filter Phase (Order)')
+    >>> axs[1].legend(title='Filter Phase (Order)', loc='lower right')
+    >>> for ax_ in axs:  # shade transition band:
+    ...     ax_.axvspan(freq[1], freq[2], color='y', alpha=.25)
+    ...     ax_.grid(True)
+    >>> plt.show()
+
+    The impulse response and group delay plot depict the 75 sample delay of the linear
+    phase filter `h`. The phase should also be linear in the stop band--due to the small
+    magnitude, numeric noise dominates there. Furthermore, the plots show that the
+    minimum phase filters clearly show a reduced (negative) phase slope in the pass and
+    transition band. The plots also illustrate that the filter with parameters
+    ``method='homomorphic', half=False`` has same order and magnitude response as the
+    linear filter `h` whereas the other minimum phase filters have only half the order
+    and the square root  of the magnitude response.
+    """
+    xp = array_namespace(h)
+
+    h = xp.asarray(h)
+    if xp.isdtype(h.dtype, "complex floating"):
+        raise ValueError('Complex filters not supported')
+    if h.ndim != 1 or h.shape[0] <= 2:
+        raise ValueError('h must be 1-D and at least 2 samples long')
+    n_half = h.shape[0] // 2
+
+    if not xp.any(xp.flip(h[-n_half:]) - h[:n_half] <= 1e-8 + 1e-6*abs(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 method == "hilbert" and not half:
+        raise ValueError("`half=False` is only supported when `method='homomorphic'`")
+    if n_fft is None:
+        n_fft = 2 ** int(ceil(log2(2 * (h.shape[0] - 1) / 0.01)))
+    n_fft = int(n_fft)
+    if n_fft < h.shape[0]:
+        raise ValueError(f'n_fft must be at least len(h)=={len(h)}')
+
+    if method == 'hilbert':
+        w = xp.arange(n_fft, dtype=xp.float64) * (2 * xp.pi / n_fft * n_half)
+        H = xp.real(fft(h, n_fft) * xp.exp(1j * w))
+        dp = max(H) - 1
+        ds = 0 - min(H)
+        S = 4. / (xp.sqrt(1+dp+ds) + xp.sqrt(1-dp+ds)) ** 2
+        H += ds
+        H *= S
+        H = xp.sqrt(H)
+        H += 1e-10  # ensure that the log does not explode
+        h_minimum = _dhtm(H, xp)
+    else:  # method == 'homomorphic'
+        # zero-pad; calculate the DFT
+        h_temp = xp.abs(fft(h, n_fft))
+        # take 0.25*log(|H|**2) = 0.5*log(|H|)
+        h_temp += 1e-7 * xp.min(h_temp[h_temp > 0])  # don't let log blow up
+        h_temp = xp.log(h_temp)
+        if half:  # halving of magnitude spectrum optional
+            h_temp *= 0.5
+        # IDFT
+        h_temp = xp.real(ifft(h_temp))
+        # multiply pointwise by the homomorphic filter
+        # lmin[n] = 2u[n] - d[n]
+        # i.e., double the positive frequencies and zero out the negative ones;
+        # Oppenheim+Shafer 3rd ed p991 eq13.42b and p1004 fig13.7
+        win = xp.zeros(n_fft, dtype=h_temp.dtype)
+        win = xpx.at(win)[0].set(1)
+        stop = n_fft // 2
+        win = xpx.at(win)[1:stop].set(2)
+        # Nyquist freq: odd use 2, even use 1
+        win = xpx.at(win)[stop].set(1 + (n_fft % 2))
+        h_temp *= win
+        h_temp = ifft(xp.exp(fft(h_temp)))
+        h_minimum = xp.real(h_temp)
+    n_out = (n_half + h.shape[0] % 2) if half else h.shape[0]
+    return h_minimum[:n_out]
+
+
+def firwin_2d(hsize, window, *, fc=None, fs=2, circular=False,
+              pass_zero=True, scale=True):
+    """
+    2D FIR filter design using the window method.
+
+    This function computes the coefficients of a 2D finite impulse response
+    filter. The filter is separable with linear phase; it will be designed
+    as a product of two 1D filters with dimensions defined by `hsize`.
+    Additionally, it can create approximately circularly symmetric 2-D windows.
+
+    Parameters
+    ----------
+    hsize : tuple or list of length 2
+        Lengths of the filter in each dimension. `hsize[0]` specifies the
+        number of coefficients in the row direction and `hsize[1]` specifies
+        the number of coefficients in the column direction.
+    window : tuple or list of length 2 or str
+        Desired window to use for each 1D filter or a single window type for creating
+        circularly symmetric 2-D windows. Each element should be a string or tuple of
+        string and parameter values. The generated windows will be symmetric, unless a
+        suffix ``'_periodic'`` is appended to the window name (e.g.,
+        ``'hamming_perodic'``). Consult `~scipy.signal.get_window` for a list of windows
+        and required parameters.
+    fc : float or 1-D array_like, optional
+        Cutoff frequency of the filter in the same units as `fs`. This defines
+        the frequency at which the filter's gain drops to approximately -6 dB
+        (half power) in a low-pass or high-pass filter. For multi-band filters,
+        `fc` can be an array of cutoff frequencies (i.e., band edges) in the
+        range [0, fs/2], with each band specified in pairs. Required if
+        `circular` is False.
+    fs : float, optional
+        The sampling frequency of the signal. Default is 2.
+    circular : bool, optional
+        Whether to create a circularly symmetric 2-D window. Default is ``False``.
+    pass_zero : {True, False, 'bandpass', 'lowpass', 'highpass', 'bandstop'}, optional
+        This parameter is directly passed to `firwin` for each scalar frequency axis.
+        Hence, if ``True``, the DC gain, i.e., the gain at frequency (0, 0), is 1.
+        If ``False``, the DC gain is 0 at frequency (0, 0) if `circular` is ``True``.
+        If `circular` is ``False`` the frequencies (0, f1) and (f0, 0) will
+        have gain 0.
+        It can also be a string argument for the desired filter type
+        (equivalent to ``btype`` in IIR design functions).
+    scale : bool, optional
+        This parameter is directly passed to `firwin` for each scalar frequency axis.
+        Set to ``True`` to scale the coefficients so that the frequency
+        response is exactly unity at a certain frequency on one frequency axis.
+        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
+
+    Returns
+    -------
+    filter_2d : (hsize[0], hsize[1]) ndarray
+        Coefficients of 2D FIR filter.
+
+    Raises
+    ------
+    ValueError
+        - If `hsize` and `window` are not 2-element tuples or lists.
+        - If `cutoff` is None when `circular` is True.
+        - If `cutoff` is outside the range [0, `fs`/2] and `circular` is ``False``.
+        - If any of the elements in `window` are not recognized.
+    RuntimeError
+        If `firwin` fails to converge when designing the filter.
+
+    See Also
+    --------
+    firwin: FIR filter design using the window method for 1d arrays.
+    get_window: Return a window of a given length and type.
+
+    Examples
+    --------
+    Generate a 5x5 low-pass filter with cutoff frequency 0.1:
+
+    >>> import numpy as np
+    >>> from scipy.signal import get_window
+    >>> from scipy.signal import firwin_2d
+    >>> hsize = (5, 5)
+    >>> window = (("kaiser", 5.0), ("kaiser", 5.0))
+    >>> fc = 0.1
+    >>> filter_2d = firwin_2d(hsize, window, fc=fc)
+    >>> filter_2d
+    array([[0.00025366, 0.00401662, 0.00738617, 0.00401662, 0.00025366],
+           [0.00401662, 0.06360159, 0.11695714, 0.06360159, 0.00401662],
+           [0.00738617, 0.11695714, 0.21507283, 0.11695714, 0.00738617],
+           [0.00401662, 0.06360159, 0.11695714, 0.06360159, 0.00401662],
+           [0.00025366, 0.00401662, 0.00738617, 0.00401662, 0.00025366]])
+
+    Generate a circularly symmetric 5x5 low-pass filter with Hamming window:
+
+    >>> filter_2d = firwin_2d((5, 5), 'hamming', fc=fc, circular=True)
+    >>> filter_2d
+    array([[-0.00020354, -0.00020354, -0.00020354, -0.00020354, -0.00020354],
+           [-0.00020354,  0.01506844,  0.09907658,  0.01506844, -0.00020354],
+           [-0.00020354,  0.09907658, -0.00020354,  0.09907658, -0.00020354],
+           [-0.00020354,  0.01506844,  0.09907658,  0.01506844, -0.00020354],
+           [-0.00020354, -0.00020354, -0.00020354, -0.00020354, -0.00020354]])
+
+    Generate Plots comparing the product of two 1d filters with a circular
+    symmetric filter:
+
+    >>> import matplotlib.pyplot as plt
+    >>> hsize, fc = (50, 50), 0.05
+    >>> window = (("kaiser", 5.0), ("kaiser", 5.0))
+    >>> filter0_2d = firwin_2d(hsize, window, fc=fc)
+    >>> filter1_2d = firwin_2d((50, 50), 'hamming', fc=fc, circular=True)
+    ...
+    >>> fg, (ax0, ax1) = plt.subplots(1, 2, tight_layout=True, figsize=(6.5, 3.5))
+    >>> ax0.set_title("Product of 2 Windows")
+    >>> im0 = ax0.imshow(filter0_2d, cmap='viridis', origin='lower', aspect='equal')
+    >>> fg.colorbar(im0, ax=ax0, shrink=0.7)
+    >>> ax1.set_title("Circular Window")
+    >>> im1 = ax1.imshow(filter1_2d, cmap='plasma', origin='lower', aspect='equal')
+    >>> fg.colorbar(im1, ax=ax1, shrink=0.7)
+    >>> plt.show()
+    """
+    if len(hsize) != 2:
+            raise ValueError("hsize must be a 2-element tuple or list")
+
+    if circular:
+        if fc is None:
+            raise ValueError("Cutoff frequency `fc` must be "
+                             "provided when `circular` is True")
+
+        n_r = max(hsize[0], hsize[1]) * 8  # oversample 1d window by factor 8
+
+        win_r = firwin(n_r, cutoff=fc, window=window, fs=fs)
+
+        f1, f2 = np.meshgrid(np.linspace(-1, 1, hsize[0]), np.linspace(-1, 1, hsize[1]))
+        r = np.sqrt(f1**2 + f2**2)
+
+        win_2d = np.interp(r, np.linspace(0, 1, n_r), win_r)
+        return win_2d
+
+    if len(window) != 2:
+        raise ValueError("window must be a 2-element tuple or list")
+
+    row_filter = firwin(hsize[0], cutoff=fc, window=window[0], fs=fs)
+    col_filter = firwin(hsize[1], cutoff=fc, window=window[1], fs=fs)
+
+    return np.outer(row_filter, col_filter)

venv/lib/python3.12/site-packages/scipy/signal/_lti_conversion.py

@@ -0,0 +1,583 @@
+"""
+ltisys -- a collection of functions to convert linear time invariant systems
+from one representation to another.
+"""
+
+import numpy as np
+from numpy import (r_, eye, atleast_2d, poly, dot,
+                   asarray, zeros, array, outer)
+from scipy import linalg
+
+from scipy._lib._array_api import (array_namespace, xp_size, xp_promote,
+                                   xp_result_type)
+import scipy._external.array_api_extra as xpx
+from ._filter_design import tf2zpk, zpk2tf, normalize
+
+
+__all__ = ['tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk',
+           'cont2discrete']
+
+
+def tf2ss(num, den):
+    r"""Transfer function to state-space representation.
+
+    Parameters
+    ----------
+    num, den : array_like
+        Sequences representing the coefficients of the numerator and
+        denominator polynomials, in order of descending degree. The
+        denominator needs to be at least as long as the numerator.
+
+    Returns
+    -------
+    A, B, C, D : ndarray
+        State space representation of the system, in controller canonical
+        form.
+
+    Examples
+    --------
+    Convert the transfer function:
+
+    .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
+
+    >>> num = [1, 3, 3]
+    >>> den = [1, 2, 1]
+
+    to the state-space representation:
+
+    .. math::
+
+        \dot{\textbf{x}}(t) =
+        \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
+
+        \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
+
+    >>> from scipy.signal import tf2ss
+    >>> A, B, C, D = tf2ss(num, den)
+    >>> A
+    array([[-2., -1.],
+           [ 1.,  0.]])
+    >>> B
+    array([[ 1.],
+           [ 0.]])
+    >>> C
+    array([[ 1.,  2.]])
+    >>> D
+    array([[ 1.]])
+    """
+    # Controller canonical state-space representation.
+    #  if M+1 = len(num) and K+1 = len(den) then we must have M <= K
+    #  states are found by asserting that X(s) = U(s) / D(s)
+    #  then Y(s) = N(s) * X(s)
+    #
+    #   A, B, C, and D follow quite naturally.
+    #
+    num, den = normalize(num, den)   # Strips zeros, checks arrays
+    nn = len(num.shape)
+    if nn == 1:
+        num = asarray([num], num.dtype)
+    M = num.shape[1]
+    K = len(den)
+    if M > K:
+        msg = "Improper transfer function. `num` is longer than `den`."
+        raise ValueError(msg)
+    if M == 0 or K == 0:  # Null system
+        return (array([], float), array([], float), array([], float),
+                array([], float))
+
+    # pad numerator to have same number of columns has denominator
+    num = np.hstack((np.zeros((num.shape[0], K - M), dtype=num.dtype), num))
+
+    if num.shape[-1] > 0:
+        D = atleast_2d(num[:, 0])
+
+    else:
+        # We don't assign it an empty array because this system
+        # is not 'null'. It just doesn't have a non-zero D
+        # matrix. Thus, it should have a non-zero shape so that
+        # it can be operated on by functions like 'ss2tf'
+        D = array([[0]], float)
+
+    if K == 1:
+        D = D.reshape(num.shape)
+
+        return (zeros((1, 1)), zeros((1, D.shape[1])),
+                zeros((D.shape[0], 1)), D)
+
+    frow = -array([den[1:]])
+    A = r_[frow, eye(K - 2, K - 1)]
+    B = eye(K - 1, 1)
+    C = num[:, 1:] - outer(num[:, 0], den[1:])
+    D = D.reshape((C.shape[0], B.shape[1]))
+
+    return A, B, C, D
+
+
+def abcd_normalize(A=None, B=None, C=None, D=None):
+    r"""Check state-space matrices compatibility and ensure they are 2d arrays.
+
+    First, the input matrices are converted into two-dimensional arrays with
+    appropriate dtype as needed. Then the dimensions n, q, p are determined by
+    investigating the array shapes. If an input is ``None``, or has size zero, it is
+    set to an array of zeros of compatible shape. Finally, it is verified that all
+    parameter shapes are compatible with each other. If that fails, a ``ValueError`` is
+    raised. Note that the dimensions n, q, p are allowed to be zero.
+
+    Parameters
+    ----------
+    A : array_like, optional
+        Two-dimensional array of shape (n, n).
+    B : array_like, optional
+        Two-dimensional array of shape (n, p).
+    C : array_like, optional
+        Two-dimensional array of shape (q, n).
+    D : array_like, optional
+        Two-dimensional array of shape (q, p).
+
+    Returns
+    -------
+    A, B, C, D : array
+        State-space matrices as two-dimensional arrays with identical dtype.
+        The result dtype is determined based on the standard
+        `dtype promotion rules <https://numpy.org/doc/2.3/reference/arrays.promotion.html>`_
+        except for when the inputs are all of integer dtype, in which case the returned
+        arrays will have the default floating point dtype of ``float64``.
+
+    Raises
+    ------
+    ValueError
+        If the dimensions n, q, or p could not be determined or if the shapes are
+        incompatible with each other.
+
+    Notes
+    -----
+    If a matrix is not modified, the original matrix (not a copy) is returned.
+
+    The :ref:`tutorial_signal_state_space_representation` section of the
+    :ref:`user_guide` presents the corresponding definitions of continuous-time and
+    disrcete time state space systems.
+
+    See Also
+    --------
+    StateSpace: Linear Time Invariant system in state-space form.
+    dlti: Discrete-time linear time invariant system base class.
+    tf2ss: Transfer function to state-space representation.
+    ss2tf: State-space to transfer function.
+    ss2zpk: State-space representation to zero-pole-gain representation.
+    cont2discrete: Transform a continuous to a discrete state-space system.
+
+    Examples
+    --------
+    The following example demonstrates that the passed lists are converted into
+    two-dimensional arrays:
+
+    >>> from scipy.signal import abcd_normalize
+    >>> AA, BB, CC, DD = abcd_normalize(A=[[1, 2], [3, 4]], B=[[-1], [5]],
+    ...                                 C=[[4, 5]], D=2.5)
+    >>> AA.shape, BB.shape, CC.shape, DD.shape
+    ((2, 2), (2, 1), (1, 2), (1, 1))
+
+    In the following, the missing parameter C is assumed to be an array of zeros
+    with shape (1, 2):
+
+    >>> from scipy.signal import abcd_normalize
+    >>> AA, BB, CC, DD = abcd_normalize(A=[[1, 2], [3, 4]], B=[[-1], [5]], D=2.5)
+    >>> AA.shape, BB.shape, CC.shape, DD.shape
+    ((2, 2), (2, 1), (1, 2), (1, 1))
+    >>> CC
+    array([[0., 0.]])
+
+    """
+    if A is None and B is None and C is None:
+        raise ValueError("Dimension n is undefined for parameters A = B = C = None!")
+    if B is None and D is None:
+        raise ValueError("Dimension p is undefined for parameters B = D = None!")
+    if C is None and D is None:
+        raise ValueError("Dimension q is undefined for parameters C = D = None!")
+
+    xp = array_namespace(A, B, C, D)
+    A, B, C, D = xp_promote(A, B, C, D, xp=xp, force_floating=True)
+    dtype = xp_result_type(A, B, C, D, xp=xp)
+
+    # convert inputs into 2d arrays (zero-size 2d array if None):
+    A, B, C, D = (
+        xpx.atleast_nd(xp.asarray(M_), ndim=2, xp=xp)
+        if M_ is not None else xp.zeros((0, 0), dtype=dtype)
+        for M_ in (A, B, C, D)
+    )
+
+    n = A.shape[0] or B.shape[0] or C.shape[1] # try finding non-zero dimensions
+    p = B.shape[1] or D.shape[1]
+    q = C.shape[0] or D.shape[0]
+
+    # Create zero matrices as needed:
+    A = xp.zeros((n, n), dtype=dtype) if xp_size(A) == 0 else A
+    B = xp.zeros((n, p), dtype=dtype) if xp_size(B) == 0 else B
+    C = xp.zeros((q, n), dtype=dtype) if xp_size(C) == 0 else C
+    D = xp.zeros((q, p), dtype=dtype) if xp_size(D) == 0 else D
+
+    if A.shape != (n, n):
+        raise ValueError(f"Parameter A has shape {A.shape} but should be ({n}, {n})!")
+    if B.shape != (n, p):
+        raise ValueError(f"Parameter B has shape {B.shape} but should be ({n}, {p})!")
+    if C.shape != (q, n):
+        raise ValueError(f"Parameter C has shape {C.shape} but should be ({q}, {n})!")
+    if D.shape != (q, p):
+        raise ValueError(f"Parameter D has shape {D.shape} but should be ({q}, {p})!")
+
+    return A, B, C, D
+
+
+def ss2tf(A, B, C, D, input=0):
+    r"""State-space to transfer function.
+
+    A, B, C, D defines a linear state-space system with `p` inputs,
+    `q` outputs, and `n` state variables.
+
+    Parameters
+    ----------
+    A : array_like
+        State (or system) matrix of shape ``(n, n)``
+    B : array_like
+        Input matrix of shape ``(n, p)``
+    C : array_like
+        Output matrix of shape ``(q, n)``
+    D : array_like
+        Feedthrough (or feedforward) matrix of shape ``(q, p)``
+    input : int, optional
+        For multiple-input systems, the index of the input to use.
+
+    Returns
+    -------
+    num : 2-D ndarray
+        Numerator(s) of the resulting transfer function(s). `num` has one row
+        for each of the system's outputs. Each row is a sequence representation
+        of the numerator polynomial.
+    den : 1-D ndarray
+        Denominator of the resulting transfer function(s). `den` is a sequence
+        representation of the denominator polynomial.
+
+    Notes
+    -----
+    Before calculating `num` and `den`, the function `abcd_normalize` is called to
+    convert the parameters `A`, `B`, `C`, `D` into two-dimesional arrays of the
+    same dtype. The resulting dtype will be based on NumPy's dtype promotion rules,
+    except in the case where each of `A`, `B`, `C`, and `D` has integer dtype, in which
+    case the resulting dtype will be the default floating point dtype of ``float64``.
+
+    The :ref:`tutorial_signal_state_space_representation` section of the
+    :ref:`user_guide` presents the corresponding definitions of continuous-time and
+    disrcete time state space systems.
+
+    Examples
+    --------
+    Convert the state-space representation:
+
+    .. math::
+
+        \dot{\textbf{x}}(t) =
+        \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
+
+        \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
+        \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
+
+    >>> A = [[-2, -1], [1, 0]]
+    >>> B = [[1], [0]]  # 2-D column vector
+    >>> C = [[1, 2]]    # 2-D row vector
+    >>> D = 1
+
+    to the transfer function:
+
+    .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
+
+    >>> from scipy.signal import ss2tf
+    >>> ss2tf(A, B, C, D)
+    (array([[1., 3., 3.]]), array([ 1.,  2.,  1.]))
+    """
+    # transfer function is C (sI - A)**(-1) B + D
+
+    # Check consistency and make them all floating point 2d arrays:
+    A, B, C, D = abcd_normalize(A, B, C, D)
+
+    nout, nin = D.shape
+    if input >= nin:
+        raise ValueError("System does not have the input specified.")
+
+    # make SIMO from possibly MIMO system.
+    B = B[:, input:input + 1]
+    D = D[:, input:input + 1]
+
+    try:
+        den = poly(A)
+    except ValueError:
+        den = 1
+
+    if (B.size == 0) and (C.size == 0):
+        num = np.ravel(D)
+        if (D.size == 0) and (A.size == 0):
+            den = []
+        return num, den
+
+    num_states = A.shape[0]
+    type_test = A[:, 0] + B[:, 0] + C[0, :] + D + 0.0
+    num = np.empty((nout, num_states + 1), type_test.dtype)
+    for k in range(nout):
+        Ck = atleast_2d(C[k, :])
+        num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den
+
+    return num, den
+
+
+def zpk2ss(z, p, k):
+    """Zero-pole-gain representation to state-space representation.
+
+    Parameters
+    ----------
+    z, p : sequence
+        Zeros and poles.
+    k : float
+        System gain.
+
+    Returns
+    -------
+    A, B, C, D : ndarray
+        State space representation of the system, in controller canonical
+        form.
+
+    """
+    return tf2ss(*zpk2tf(z, p, k))
+
+
+def ss2zpk(A, B, C, D, input=0):
+    """State-space representation to zero-pole-gain representation.
+
+    A, B, C, D defines a linear state-space system with `p` inputs,
+    `q` outputs, and `n` state variables.
+
+    Parameters
+    ----------
+    A : array_like
+        State (or system) matrix of shape ``(n, n)``
+    B : array_like
+        Input matrix of shape ``(n, p)``
+    C : array_like
+        Output matrix of shape ``(q, n)``
+    D : array_like
+        Feedthrough (or feedforward) matrix of shape ``(q, p)``
+    input : int, optional
+        For multiple-input systems, the index of the input to use.
+
+    Returns
+    -------
+    z, p : sequence
+        Zeros and poles.
+    k : float
+        System gain.
+
+    """
+    return tf2zpk(*ss2tf(A, B, C, D, input=input))
+
+
+def cont2discrete(system, dt, method="zoh", alpha=None):
+    """
+    Transform a continuous to a discrete state-space system.
+
+    Parameters
+    ----------
+    system : a tuple describing the system or an instance of `lti`
+        The following gives the number of elements in the tuple and
+        the interpretation:
+
+        * 1: (instance of `lti`)
+        * 2: (num, den)
+        * 3: (zeros, poles, gain)
+        * 4: (A, B, C, D)
+
+    dt : float
+        The discretization time step.
+    method : str, optional
+        Which method to use:
+
+        * gbt: generalized bilinear transformation
+        * bilinear: Tustin's approximation ("gbt" with alpha=0.5)
+        * euler: Euler (or forward differencing) method ("gbt" with alpha=0)
+        * backward_diff: Backwards differencing ("gbt" with alpha=1.0)
+        * zoh: zero-order hold (default)
+        * foh: first-order hold (*versionadded: 1.3.0*)
+        * impulse: equivalent impulse response (*versionadded: 1.3.0*)
+
+    alpha : float within [0, 1], optional
+        The generalized bilinear transformation weighting parameter, which
+        should only be specified with method="gbt", and is ignored otherwise
+
+    Returns
+    -------
+    sysd : tuple containing the discrete system
+        Based on the input type, the output will be of the form
+
+        * (num, den, dt)   for transfer function input
+        * (zeros, poles, gain, dt)   for zeros-poles-gain input
+        * (A, B, C, D, dt) for state-space system input
+
+    Notes
+    -----
+    By default, the routine uses a Zero-Order Hold (zoh) method to perform
+    the transformation. Alternatively, a generalized bilinear transformation
+    may be used, which includes the common Tustin's bilinear approximation,
+    an Euler's method technique, or a backwards differencing technique.
+
+    The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear
+    approximation is based on [2]_ and [3]_, the First-Order Hold (foh) method
+    is based on [4]_.
+
+    References
+    ----------
+    .. [1] https://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models
+
+    .. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf
+
+    .. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized
+        bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754,
+        2009.
+        (https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf)
+
+    .. [4] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control
+        of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley,
+        pp. 204-206, 1998.
+
+    Examples
+    --------
+    We can transform a continuous state-space system to a discrete one:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.signal import cont2discrete, lti, dlti, dstep
+
+    Define a continuous state-space system.
+
+    >>> A = np.array([[0, 1],[-10., -3]])
+    >>> B = np.array([[0],[10.]])
+    >>> C = np.array([[1., 0]])
+    >>> D = np.array([[0.]])
+    >>> l_system = lti(A, B, C, D)
+    >>> t, x = l_system.step(T=np.linspace(0, 5, 100))
+    >>> fig, ax = plt.subplots()
+    >>> ax.plot(t, x, label='Continuous', linewidth=3)
+
+    Transform it to a discrete state-space system using several methods.
+
+    >>> dt = 0.1
+    >>> for method in ['zoh', 'bilinear', 'euler', 'backward_diff', 'foh', 'impulse']:
+    ...    d_system = cont2discrete((A, B, C, D), dt, method=method)
+    ...    s, x_d = dstep(d_system)
+    ...    ax.step(s, np.squeeze(x_d), label=method, where='post')
+    >>> ax.axis([t[0], t[-1], x[0], 1.4])
+    >>> ax.legend(loc='best')
+    >>> fig.tight_layout()
+    >>> plt.show()
+
+    """
+    if hasattr(system, 'to_discrete') and callable(system.to_discrete):
+        return system.to_discrete(dt=dt, method=method, alpha=alpha)
+
+    if len(system) == 2:
+        sysd = cont2discrete(tf2ss(system[0], system[1]), dt, method=method,
+                             alpha=alpha)
+        return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
+    elif len(system) == 3:
+        sysd = cont2discrete(zpk2ss(system[0], system[1], system[2]), dt,
+                             method=method, alpha=alpha)
+        return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
+    elif len(system) == 4:
+        a, b, c, d = system
+    else:
+        raise ValueError("First argument must either be a tuple of 2 (tf), "
+                         "3 (zpk), or 4 (ss) arrays.")
+
+    if method == 'gbt':
+        if alpha is None:
+            raise ValueError("Alpha parameter must be specified for the "
+                             "generalized bilinear transform (gbt) method")
+        elif alpha < 0 or alpha > 1:
+            raise ValueError("Alpha parameter must be within the interval "
+                             "[0,1] for the gbt method")
+
+    if method == 'gbt':
+        # This parameter is used repeatedly - compute once here
+        ima = np.eye(a.shape[0]) - alpha*dt*a
+        ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a)
+        bd = linalg.solve(ima, dt*b)
+
+        # Similarly solve for the output equation matrices
+        cd = linalg.solve(ima.transpose(), c.transpose())
+        cd = cd.transpose()
+        dd = d + alpha*np.dot(c, bd)
+
+    elif method == 'bilinear' or method == 'tustin':
+        return cont2discrete(system, dt, method="gbt", alpha=0.5)
+
+    elif method == 'euler' or method == 'forward_diff':
+        return cont2discrete(system, dt, method="gbt", alpha=0.0)
+
+    elif method == 'backward_diff':
+        return cont2discrete(system, dt, method="gbt", alpha=1.0)
+
+    elif method == 'zoh':
+        # Build an exponential matrix
+        em_upper = np.hstack((a, b))
+
+        # Need to stack zeros under the a and b matrices
+        em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])),
+                              np.zeros((b.shape[1], b.shape[1]))))
+
+        em = np.vstack((em_upper, em_lower))
+        ms = linalg.expm(dt * em)
+
+        # Dispose of the lower rows
+        ms = ms[:a.shape[0], :]
+
+        ad = ms[:, 0:a.shape[1]]
+        bd = ms[:, a.shape[1]:]
+
+        cd = c
+        dd = d
+
+    elif method == 'foh':
+        # Size parameters for convenience
+        n = a.shape[0]
+        m = b.shape[1]
+
+        # Build an exponential matrix similar to 'zoh' method
+        em_upper = linalg.block_diag(np.block([a, b]) * dt, np.eye(m))
+        em_lower = zeros((m, n + 2 * m))
+        em = np.block([[em_upper], [em_lower]])
+
+        ms = linalg.expm(em)
+
+        # Get the three blocks from upper rows
+        ms11 = ms[:n, 0:n]
+        ms12 = ms[:n, n:n + m]
+        ms13 = ms[:n, n + m:]
+
+        ad = ms11
+        bd = ms12 - ms13 + ms11 @ ms13
+        cd = c
+        dd = d + c @ ms13
+
+    elif method == 'impulse':
+        if not np.allclose(d, 0):
+            raise ValueError("Impulse method is only applicable "
+                             "to strictly proper systems")
+
+        ad = linalg.expm(a * dt)
+        bd = ad @ b * dt
+        cd = c
+        dd = c @ b * dt
+
+    else:
+        raise ValueError(f"Unknown transformation method '{method}'")
+
+    return ad, bd, cd, dd, dt

venv/lib/python3.12/site-packages/scipy/signal/_max_len_seq.py

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

venv/lib/python3.12/site-packages/scipy/signal/_peak_finding.py

@@ -0,0 +1,1307 @@
+"""
+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 a 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.
+        * 'widths', '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.argsort(line[0])
+        rows = np.array(line[0])[sortargs]
+        cols = np.array(line[1])[sortargs]
+        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
+    --------
+    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`.
+     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.
+
+    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