package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Return a modified Bartlett-Hann window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.barthann(51) >>> plt.plot(window) >>> plt.title('Bartlett-Hann window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Bartlett-Hann window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Bartlett window.

The Bartlett window is very similar to a triangular window, except that the end points are at zero. It is often used in signal processing for tapering a signal, without generating too much ripple in the frequency domain.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The triangular window, with the first and last samples equal to zero and the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

See Also -------- triang : A triangular window that does not touch zero at the ends

Notes ----- The Bartlett window is defined as

.. math:: w(n) = \frac

M-1 \left( \fracM-1

• \left|n - \fracM-1

  \right| \right)

Most references to the Bartlett window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. Note that convolution with this window produces linear interpolation. It is also known as an apodization (which means'removing the foot', i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. The Fourier transform of the Bartlett is the product of two sinc functions. Note the excellent discussion in Kanasewich. 2_

References ---------- .. 1 M.S. Bartlett, 'Periodogram Analysis and Continuous Spectra', Biometrika 37, 1-16, 1950. .. 2 E.R. Kanasewich, 'Time Sequence Analysis in Geophysics', The University of Alberta Press, 1975, pp. 109-110. .. 3 A.V. Oppenheim and R.W. Schafer, 'Discrete-Time Signal Processing', Prentice-Hall, 1999, pp. 468-471. .. 4 Wikipedia, 'Window function', https://en.wikipedia.org/wiki/Window_function .. 5 W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 'Numerical Recipes', Cambridge University Press, 1986, page 429.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.bartlett(51) >>> plt.plot(window) >>> plt.title('Bartlett window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Bartlett window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Blackman window.

The Blackman window is a taper formed by using the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only slightly worse than a Kaiser window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The Blackman window is defined as

.. math:: w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M)

The 'exact Blackman' window was designed to null out the third and fourth sidelobes, but has discontinuities at the boundaries, resulting in a 6 dB/oct fall-off. This window is an approximation of the 'exact' window, which does not null the sidelobes as well, but is smooth at the edges, improving the fall-off rate to 18 dB/oct. 3_

Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means 'removing the foot', i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a 'near optimal' tapering function, almost as good (by some measures) as the Kaiser window.

References ---------- .. 1 Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. 2 Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471. .. 3 Harris, Fredric J. (Jan 1978). 'On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform'. Proceedings of the IEEE 66 (1): 51-83. :doi:`10.1109/PROC.1978.10837`.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.blackman(51) >>> plt.plot(window) >>> plt.title('Blackman window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = np.abs(fftshift(A / abs(A).max())) >>> response = 20 * np.log10(np.maximum(response, 1e-10)) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Blackman window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a minimum 4-term Blackman-Harris window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.blackmanharris(51) >>> plt.plot(window) >>> plt.title('Blackman-Harris window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Blackman-Harris window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Bohman window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.bohman(51) >>> plt.plot(window) >>> plt.title('Bohman window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Bohman window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a boxcar or rectangular window.

Also known as a rectangular window or Dirichlet window, this is equivalent to no window at all.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional Whether the window is symmetric. (Has no effect for boxcar.)

Returns ------- w : ndarray The window, with the maximum value normalized to 1.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.boxcar(51) >>> plt.plot(window) >>> plt.title('Boxcar window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the boxcar window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Dolph-Chebyshev window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. at : float Attenuation (in dB). sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value always normalized to 1

Notes ----- This window optimizes for the narrowest main lobe width for a given order `M` and sidelobe equiripple attenuation `at`, using Chebyshev polynomials. It was originally developed by Dolph to optimize the directionality of radio antenna arrays.

Unlike most windows, the Dolph-Chebyshev is defined in terms of its frequency response:

.. math:: W(k) = \frac \cos\{M \cos^{-1\beta \cos(\frac{\pi k}{M})}

}

\cosh[M \cosh^{-1(\beta)

where

.. math:: \beta = \cosh \left \frac{1}{M} \cosh^{-1}(10^\frac{A}{20}) \right

and 0 <= abs(k) <= M-1. A is the attenuation in decibels (`at`).

The time domain window is then generated using the IFFT, so power-of-two `M` are the fastest to generate, and prime number `M` are the slowest.

The equiripple condition in the frequency domain creates impulses in the time domain, which appear at the ends of the window.

References ---------- .. 1 C. Dolph, 'A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level', Proceedings of the IEEE, Vol. 34, Issue 6 .. 2 Peter Lynch, 'The Dolph-Chebyshev Window: A Simple Optimal Filter', American Meteorological Society (April 1997) http://mathsci.ucd.ie/~plynch/Publications/Dolph.pdf .. 3 F. J. Harris, 'On the use of windows for harmonic analysis with the discrete Fourier transforms', Proceedings of the IEEE, Vol. 66, No. 1, January 1978

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.chebwin(51, at=100) >>> plt.plot(window) >>> plt.title('Dolph-Chebyshev window (100 dB)') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Dolph-Chebyshev window (100 dB)') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a window with a simple cosine shape.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes -----

.. versionadded:: 0.13.0

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.cosine(51) >>> plt.plot(window) >>> plt.title('Cosine window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the cosine window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample') >>> plt.show()

Compute the Discrete Prolate Spheroidal Sequences (DPSS).

DPSS (or Slepian sequences) are often used in multitaper power spectral density estimation (see 1_). The first window in the sequence can be used to maximize the energy concentration in the main lobe, and is also called the Slepian window.

Parameters ---------- M : int Window length. NW : float Standardized half bandwidth corresponding to ``2*NW = BW/f0 = BW*N*dt`` where ``dt`` is taken as 1. Kmax : int | None, optional Number of DPSS windows to return (orders ``0`` through ``Kmax-1``). If None (default), return only a single window of shape ``(M,)`` instead of an array of windows of shape ``(Kmax, M)``. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis. norm :

, 'approximate', 'subsample'

| None, optional If 'approximate' or 'subsample', then the windows are normalized by the maximum, and a correction scale-factor for even-length windows is applied either using ``M**2/(M**2+NW)`` ('approximate') or a FFT-based subsample shift ('subsample'), see Notes for details. If None, then 'approximate' is used when ``Kmax=None`` and 2 otherwise (which uses the l2 norm). return_ratios : bool, optional If True, also return the concentration ratios in addition to the windows.

Returns ------- v : ndarray, shape (Kmax, N) or (N,) The DPSS windows. Will be 1D if `Kmax` is None. r : ndarray, shape (Kmax,) or float, optional The concentration ratios for the windows. Only returned if `return_ratios` evaluates to True. Will be 0D if `Kmax` is None.

Notes ----- This computation uses the tridiagonal eigenvector formulation given in 2_.

The default normalization for ``Kmax=None``, i.e. window-generation mode, simply using the l-infinity norm would create a window with two unity values, which creates slight normalization differences between even and odd orders. The approximate correction of ``M**2/float(M**2+NW)`` for even sample numbers is used to counteract this effect (see Examples below).

For very long signals (e.g., 1e6 elements), it can be useful to compute windows orders of magnitude shorter and use interpolation (e.g., `scipy.interpolate.interp1d`) to obtain tapers of length `M`, but this in general will not preserve orthogonality between the tapers.

.. versionadded:: 1.1

References ---------- .. 1 Percival DB, Walden WT. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press; 1993. .. 2 Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and uncertainty V: The discrete case. Bell System Technical Journal, Volume 57 (1978), 1371430. .. 3 Kaiser, JF, Schafer RW. On the Use of the I0-Sinh Window for Spectrum Analysis. IEEE Transactions on Acoustics, Speech and Signal Processing. ASSP-28 (1): 105-107; 1980.

Examples -------- We can compare the window to `kaiser`, which was invented as an alternative that was easier to calculate 3_ (example adapted from `here <https://ccrma.stanford.edu/~jos/sasp/Kaiser_DPSS_Windows_Compared.html>`_):

>>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.signal import windows, freqz >>> N = 51 >>> fig, axes = plt.subplots(3, 2, figsize=(5, 7)) >>> for ai, alpha in enumerate((1, 3, 5)): ... win_dpss = windows.dpss(N, alpha) ... beta = alpha*np.pi ... win_kaiser = windows.kaiser(N, beta) ... for win, c in ((win_dpss, 'k'), (win_kaiser, 'r')): ... win /= win.sum() ... axesai, 0.plot(win, color=c, lw=1.) ... axesai, 0.set(xlim=0, N-1, title=r'$\alpha$ = %s' % alpha, ... ylabel='Amplitude') ... w, h = freqz(win) ... axesai, 1.plot(w, 20 * np.log10(np.abs(h)), color=c, lw=1.) ... axesai, 1.set(xlim=0, np.pi, ... title=r'$\beta$ = %0.2f' % beta, ... ylabel='Magnitude (dB)') >>> for ax in axes.ravel(): ... ax.grid(True) >>> axes2, 1.legend('DPSS', 'Kaiser') >>> fig.tight_layout() >>> plt.show()

And here are examples of the first four windows, along with their concentration ratios:

>>> M = 512 >>> NW = 2.5 >>> win, eigvals = windows.dpss(M, NW, 4, return_ratios=True) >>> fig, ax = plt.subplots(1) >>> ax.plot(win.T, linewidth=1.) >>> ax.set(xlim=0, M-1, ylim=-0.1, 0.1, xlabel='Samples', ... title='DPSS, M=%d, NW=%0.1f' % (M, NW)) >>> ax.legend('win[%d] (%0.4f)' % (ii, ratio) ... for ii, ratio in enumerate(eigvals)) >>> fig.tight_layout() >>> plt.show()

Using a standard :math:`l_\infty` norm would produce two unity values for even `M`, but only one unity value for odd `M`. This produces uneven window power that can be counteracted by the approximate correction ``M**2/float(M**2+NW)``, which can be selected by using ``norm='approximate'`` (which is the same as ``norm=None`` when ``Kmax=None``, as is the case here). Alternatively, the slower ``norm='subsample'`` can be used, which uses subsample shifting in the frequency domain (FFT) to compute the correction:

>>> Ms = np.arange(1, 41) >>> factors = (50, 20, 10, 5, 2.0001) >>> energy = np.empty((3, len(Ms), len(factors))) >>> for mi, M in enumerate(Ms): ... for fi, factor in enumerate(factors): ... NW = M / float(factor) ... # Corrected using empirical approximation (default) ... win = windows.dpss(M, NW) ... energy0, mi, fi = np.sum(win ** 2) / np.sqrt(M) ... # Corrected using subsample shifting ... win = windows.dpss(M, NW, norm='subsample') ... energy1, mi, fi = np.sum(win ** 2) / np.sqrt(M) ... # Uncorrected (using l-infinity norm) ... win /= win.max() ... energy2, mi, fi = np.sum(win ** 2) / np.sqrt(M) >>> fig, ax = plt.subplots(1) >>> hs = ax.plot(Ms, energy2, '-o', markersize=4, ... markeredgecolor='none') >>> leg = hs[-1] >>> for hi, hh in enumerate(hs): ... h1 = ax.plot(Ms, energy0, :, hi, '-o', markersize=4, ... color=hh.get_color(), markeredgecolor='none', ... alpha=0.66) ... h2 = ax.plot(Ms, energy1, :, hi, '-o', markersize=4, ... color=hh.get_color(), markeredgecolor='none', ... alpha=0.33) ... if hi == len(hs) - 1: ... leg.insert(0, h10) ... leg.insert(0, h20) >>> ax.set(xlabel='M (samples)', ylabel=r'Power / $\sqrtM$') >>> ax.legend(leg, 'Uncorrected', r'Corrected: $\frac{M^2}{M^2+NW}$', ... 'Corrected (subsample)') >>> fig.tight_layout()

Return an exponential (or Poisson) window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. center : float, optional Parameter defining the center location of the window function. The default value if not given is ``center = (M-1) / 2``. This parameter must take its default value for symmetric windows. tau : float, optional Parameter defining the decay. For ``center = 0`` use ``tau = -(M-1) / ln(x)`` if ``x`` is the fraction of the window remaining at the end. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The Exponential window is defined as

.. math:: w(n) = e^

|n-center| / \tau

}

References ---------- S. Gade and H. Herlufsen, 'Windows to FFT analysis (Part I)', Technical Review 3, Bruel & Kjaer, 1987.

Examples -------- Plot the symmetric window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> M = 51 >>> tau = 3.0 >>> window = signal.exponential(M, tau=tau) >>> plt.plot(window) >>> plt.title('Exponential Window (tau=3.0)') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -35, 0) >>> plt.title('Frequency response of the Exponential window (tau=3.0)') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

This function can also generate non-symmetric windows:

>>> tau2 = -(M-1) / np.log(0.01) >>> window2 = signal.exponential(M, 0, tau2, False) >>> plt.figure() >>> plt.plot(window2) >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

Return a flat top window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- Flat top windows are used for taking accurate measurements of signal amplitude in the frequency domain, with minimal scalloping error from the center of a frequency bin to its edges, compared to others. This is a 5th-order cosine window, with the 5 terms optimized to make the main lobe maximally flat. 1_

References ---------- .. 1 D'Antona, Gabriele, and A. Ferrero, 'Digital Signal Processing for Measurement Systems', Springer Media, 2006, p. 70 :doi:`10.1007/0-387-28666-7`.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.flattop(51) >>> plt.plot(window) >>> plt.title('Flat top window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the flat top window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Gaussian window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. std : float The standard deviation, sigma. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The Gaussian window is defined as

.. math:: w(n) = e^ -\frac{1

\left(\fracn\sigma\right)^2

}

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.gaussian(51, std=7) >>> plt.plot(window) >>> plt.title(r'Gaussian window ($\sigma$=7)') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title(r'Frequency response of the Gaussian window ($\sigma$=7)') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Generic weighted sum of cosine terms window

Parameters ---------- M : int Number of points in the output window a : array_like Sequence of weighting coefficients. This uses the convention of being centered on the origin, so these will typically all be positive numbers, not alternating sign. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

References ---------- .. 1 A. Nuttall, 'Some windows with very good sidelobe behavior,' IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 1, pp. 84-91, Feb 1981. :doi:`10.1109/TASSP.1981.1163506`. .. 2 Heinzel G. et al., 'Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows', February 15, 2002 https://holometer.fnal.gov/GH_FFT.pdf

Examples -------- Heinzel describes a flat-top window named 'HFT90D' with formula: 2_

.. math:: w_j = 1 - 1.942604 \cos(z) + 1.340318 \cos(2z)

• 0.440811 \cos(3z) + 0.043097 \cos(4z)

where

.. math:: z = \frac

\pi j

N, j = 0...N - 1

Since this uses the convention of starting at the origin, to reproduce the window, we need to convert every other coefficient to a positive number:

>>> HFT90D = 1, 1.942604, 1.340318, 0.440811, 0.043097

The paper states that the highest sidelobe is at -90.2 dB. Reproduce Figure 42 by plotting the window and its frequency response, and confirm the sidelobe level in red:

>>> from scipy.signal.windows import general_cosine >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = general_cosine(1000, HFT90D, sym=False) >>> plt.plot(window) >>> plt.title('HFT90D window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 10000) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = np.abs(fftshift(A / abs(A).max())) >>> response = 20 * np.log10(np.maximum(response, 1e-10)) >>> plt.plot(freq, response) >>> plt.axis(-50/1000, 50/1000, -140, 0) >>> plt.title('Frequency response of the HFT90D window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample') >>> plt.axhline(-90.2, color='red') >>> plt.show()

Return a window with a generalized Gaussian shape.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. p : float Shape parameter. p = 1 is identical to `gaussian`, p = 0.5 is the same shape as the Laplace distribution. sig : float The standard deviation, sigma. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The generalized Gaussian window is defined as

.. math:: w(n) = e^ -\frac{1

\left|\fracn\sigma\right|^

p

}

the half-power point is at

.. math:: (2 \log(2))^

/(2 p)

\sigma

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.general_gaussian(51, p=1.5, sig=7) >>> plt.plot(window) >>> plt.title(r'Generalized Gaussian window (p=1.5, $\sigma$=7)') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title(r'Freq. resp. of the gen. Gaussian ' ... r'window (p=1.5, $\sigma$=7)') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a generalized Hamming window.

The generalized Hamming window is constructed by multiplying a rectangular window by one period of a cosine function 1_.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. alpha : float The window coefficient, :math:`\alpha` sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The generalized Hamming window is defined as

.. math:: w(n) = \alpha - \left(1 - \alpha\right) \cos\left(\frac

\pin

M-1\right) \qquad 0 \leq n \leq M-1

Both the common Hamming window and Hann window are special cases of the generalized Hamming window with :math:`\alpha` = 0.54 and :math:`\alpha` = 0.5, respectively 2_.

See Also -------- hamming, hann

Examples -------- The Sentinel-1A/B Instrument Processing Facility uses generalized Hamming windows in the processing of spaceborne Synthetic Aperture Radar (SAR) data 3_. The facility uses various values for the :math:`\alpha` parameter based on operating mode of the SAR instrument. Some common :math:`\alpha` values include 0.75, 0.7 and 0.52 4_. As an example, we plot these different windows.

>>> from scipy.signal.windows import general_hamming >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> fig1, spatial_plot = plt.subplots() >>> spatial_plot.set_title('Generalized Hamming Windows') >>> spatial_plot.set_ylabel('Amplitude') >>> spatial_plot.set_xlabel('Sample')

>>> fig2, freq_plot = plt.subplots() >>> freq_plot.set_title('Frequency Responses') >>> freq_plot.set_ylabel('Normalized magnitude dB') >>> freq_plot.set_xlabel('Normalized frequency cycles per sample')

>>> for alpha in 0.75, 0.7, 0.52: ... window = general_hamming(41, alpha) ... spatial_plot.plot(window, label='.2f'.format(alpha)) ... A = fft(window, 2048) / (len(window)/2.0) ... freq = np.linspace(-0.5, 0.5, len(A)) ... response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) ... freq_plot.plot(freq, response, label='.2f'.format(alpha)) >>> freq_plot.legend(loc='upper right') >>> spatial_plot.legend(loc='upper right')

References ---------- .. 1 DSPRelated, 'Generalized Hamming Window Family', https://www.dsprelated.com/freebooks/sasp/Generalized_Hamming_Window_Family.html .. 2 Wikipedia, 'Window function', https://en.wikipedia.org/wiki/Window_function .. 3 Riccardo Piantanida ESA, 'Sentinel-1 Level 1 Detailed Algorithm Definition', https://sentinel.esa.int/documents/247904/1877131/Sentinel-1-Level-1-Detailed-Algorithm-Definition .. 4 Matthieu Bourbigot ESA, 'Sentinel-1 Product Definition', https://sentinel.esa.int/documents/247904/1877131/Sentinel-1-Product-Definition

Return a window of a given length and type.

Parameters ---------- window : string, float, or tuple The type of window to create. See below for more details. Nx : int The number of samples in the window. fftbins : bool, optional If True (default), create a 'periodic' window, ready to use with `ifftshift` and be multiplied by the result of an FFT (see also :func:`~scipy.fft.fftfreq`). If False, create a 'symmetric' window, for use in filter design.

Returns ------- get_window : ndarray Returns a window of length `Nx` and type `window`

Notes ----- Window types:

• `~scipy.signal.windows.boxcar`
• `~scipy.signal.windows.triang`
• `~scipy.signal.windows.blackman`
• `~scipy.signal.windows.hamming`
• `~scipy.signal.windows.hann`
• `~scipy.signal.windows.bartlett`
• `~scipy.signal.windows.flattop`
• `~scipy.signal.windows.parzen`
• `~scipy.signal.windows.bohman`
• `~scipy.signal.windows.blackmanharris`
• `~scipy.signal.windows.nuttall`
• `~scipy.signal.windows.barthann`
• `~scipy.signal.windows.kaiser` (needs beta)
• `~scipy.signal.windows.gaussian` (needs standard deviation)
• `~scipy.signal.windows.general_gaussian` (needs power, width)
• `~scipy.signal.windows.slepian` (needs width)
• `~scipy.signal.windows.dpss` (needs normalized half-bandwidth)
• `~scipy.signal.windows.chebwin` (needs attenuation)
• `~scipy.signal.windows.exponential` (needs decay scale)
• `~scipy.signal.windows.tukey` (needs taper fraction)

If the window requires no parameters, then `window` can be a string.

If the window requires parameters, then `window` must be a tuple with the first argument the string name of the window, and the next arguments the needed parameters.

If `window` is a floating point number, it is interpreted as the beta parameter of the `~scipy.signal.windows.kaiser` window.

Each of the window types listed above is also the name of a function that can be called directly to create a window of that type.

Examples -------- >>> from scipy import signal >>> signal.get_window('triang', 7) array( 0.125, 0.375, 0.625, 0.875, 0.875, 0.625, 0.375) >>> signal.get_window(('kaiser', 4.0), 9) array( 0.08848053, 0.29425961, 0.56437221, 0.82160913, 0.97885093, 0.97885093, 0.82160913, 0.56437221, 0.29425961) >>> signal.get_window(4.0, 9) array( 0.08848053, 0.29425961, 0.56437221, 0.82160913, 0.97885093, 0.97885093, 0.82160913, 0.56437221, 0.29425961)

Return a Hamming window.

The Hamming window is a taper formed by using a raised cosine with non-zero endpoints, optimized to minimize the nearest side lobe.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The Hamming window is defined as

.. math:: w(n) = 0.54 - 0.46 \cos\left(\frac

\pin

M-1\right) \qquad 0 \leq n \leq M-1

The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and is described in Blackman and Tukey. It was recommended for smoothing the truncated autocovariance function in the time domain. Most references to the Hamming window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means 'removing the foot', i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function.

References ---------- .. 1 Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. 2 E.R. Kanasewich, 'Time Sequence Analysis in Geophysics', The University of Alberta Press, 1975, pp. 109-110. .. 3 Wikipedia, 'Window function', https://en.wikipedia.org/wiki/Window_function .. 4 W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 'Numerical Recipes', Cambridge University Press, 1986, page 425.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.hamming(51) >>> plt.plot(window) >>> plt.title('Hamming window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Hamming window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Hann window.

The Hann window is a taper formed by using a raised cosine or sine-squared with ends that touch zero.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The Hann window is defined as

.. math:: w(n) = 0.5 - 0.5 \cos\left(\frac

\pin

M-1\right) \qquad 0 \leq n \leq M-1

The window was named for Julius von Hann, an Austrian meteorologist. It is also known as the Cosine Bell. It is sometimes erroneously referred to as the 'Hanning' window, from the use of 'hann' as a verb in the original paper and confusion with the very similar Hamming window.

Most references to the Hann window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means 'removing the foot', i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function.

References ---------- .. 1 Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. 2 E.R. Kanasewich, 'Time Sequence Analysis in Geophysics', The University of Alberta Press, 1975, pp. 106-108. .. 3 Wikipedia, 'Window function', https://en.wikipedia.org/wiki/Window_function .. 4 W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 'Numerical Recipes', Cambridge University Press, 1986, page 425.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.hann(51) >>> plt.plot(window) >>> plt.title('Hann window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = np.abs(fftshift(A / abs(A).max())) >>> response = 20 * np.log10(np.maximum(response, 1e-10)) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Hann window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

`hanning` is deprecated, use `scipy.signal.windows.hann` instead!

Return a Kaiser window.

The Kaiser window is a taper formed by using a Bessel function.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. beta : float Shape parameter, determines trade-off between main-lobe width and side lobe level. As beta gets large, the window narrows. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

Notes ----- The Kaiser window is defined as

.. math:: w(n) = I_0\left( \beta \sqrt

-\frac

n^2

(M-1)^2

}

\right)/I_0(\beta)

with

.. math:: \quad -\fracM-1

\leq n \leq \fracM-1

,

where :math:`I_0` is the modified zeroth-order Bessel function.

The Kaiser was named for Jim Kaiser, who discovered a simple approximation to the DPSS window based on Bessel functions. The Kaiser window is a very good approximation to the Digital Prolate Spheroidal Sequence, or Slepian window, which is the transform which maximizes the energy in the main lobe of the window relative to total energy.

The Kaiser can approximate other windows by varying the beta parameter. (Some literature uses alpha = beta/pi.) 4_

==== ======================= beta Window shape ==== ======================= 0 Rectangular 5 Similar to a Hamming 6 Similar to a Hann 8.6 Similar to a Blackman ==== =======================

A beta value of 14 is probably a good starting point. Note that as beta gets large, the window narrows, and so the number of samples needs to be large enough to sample the increasingly narrow spike, otherwise NaNs will be returned.

Most references to the Kaiser window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means 'removing the foot', i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function.

References ---------- .. 1 J. F. Kaiser, 'Digital Filters' - Ch 7 in 'Systems analysis by digital computer', Editors: F.F. Kuo and J.F. Kaiser, p 218-285. John Wiley and Sons, New York, (1966). .. 2 E.R. Kanasewich, 'Time Sequence Analysis in Geophysics', The University of Alberta Press, 1975, pp. 177-178. .. 3 Wikipedia, 'Window function', https://en.wikipedia.org/wiki/Window_function .. 4 F. J. Harris, 'On the use of windows for harmonic analysis with the discrete Fourier transform,' Proceedings of the IEEE, vol. 66, no. 1, pp. 51-83, Jan. 1978. :doi:`10.1109/PROC.1978.10837`.

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.kaiser(51, beta=14) >>> plt.plot(window) >>> plt.title(r'Kaiser window ($\beta$=14)') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title(r'Frequency response of the Kaiser window ($\beta$=14)') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a minimum 4-term Blackman-Harris window according to Nuttall.

This variation is called 'Nuttall4c' by Heinzel. 2_

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

References ---------- .. 1 A. Nuttall, 'Some windows with very good sidelobe behavior,' IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 1, pp. 84-91, Feb 1981. :doi:`10.1109/TASSP.1981.1163506`. .. 2 Heinzel G. et al., 'Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new flat-top windows', February 15, 2002 https://holometer.fnal.gov/GH_FFT.pdf

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.nuttall(51) >>> plt.plot(window) >>> plt.title('Nuttall window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Nuttall window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Parzen window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

References ---------- .. 1 E. Parzen, 'Mathematical Considerations in the Estimation of Spectra', Technometrics, Vol. 3, No. 2 (May, 1961), pp. 167-190

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.parzen(51) >>> plt.plot(window) >>> plt.title('Parzen window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Parzen window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a digital Slepian (DPSS) window.

Used to maximize the energy concentration in the main lobe. Also called the digital prolate spheroidal sequence (DPSS).

.. note:: Deprecated in SciPy 1.1. `slepian` will be removed in a future version of SciPy, it is replaced by `dpss`, which uses the standard definition of a digital Slepian window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. width : float Bandwidth sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value always normalized to 1

See Also -------- dpss

References ---------- .. 1 D. Slepian & H. O. Pollak: 'Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,' Bell Syst. Tech. J., vol.40, pp.43-63, 1961. https://archive.org/details/bstj40-1-43 .. 2 H. J. Landau & H. O. Pollak: 'Prolate spheroidal wave functions, Fourier analysis and uncertainty-II,' Bell Syst. Tech. J. , vol.40, pp.65-83, 1961. https://archive.org/details/bstj40-1-65

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.slepian(51, width=0.3) >>> plt.plot(window) >>> plt.title('Slepian (DPSS) window (BW=0.3)') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Slepian window (BW=0.3)') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

============================================== Discrete Fourier transforms (:mod:`scipy.fft`) ==============================================

.. currentmodule:: scipy.fft

Fast Fourier Transforms (FFTs) ==============================

.. autosummary:: :toctree: generated/

fft - Fast (discrete) Fourier Transform (FFT) ifft - Inverse FFT fft2 - 2-D FFT ifft2 - 2-D inverse FFT fftn - N-D FFT ifftn - N-D inverse FFT rfft - FFT of strictly real-valued sequence irfft - Inverse of rfft rfft2 - 2-D FFT of real sequence irfft2 - Inverse of rfft2 rfftn - N-D FFT of real sequence irfftn - Inverse of rfftn hfft - FFT of a Hermitian sequence (real spectrum) ihfft - Inverse of hfft hfft2 - 2-D FFT of a Hermitian sequence ihfft2 - Inverse of hfft2 hfftn - N-D FFT of a Hermitian sequence ihfftn - Inverse of hfftn

Discrete Sin and Cosine Transforms (DST and DCT) ================================================ .. autosummary:: :toctree: generated/

dct - Discrete cosine transform idct - Inverse discrete cosine transform dctn - N-D Discrete cosine transform idctn - N-D Inverse discrete cosine transform dst - Discrete sine transform idst - Inverse discrete sine transform dstn - N-D Discrete sine transform idstn - N-D Inverse discrete sine transform

Helper functions ================

.. autosummary:: :toctree: generated/

fftshift - Shift the zero-frequency component to the center of the spectrum ifftshift - The inverse of `fftshift` fftfreq - Return the Discrete Fourier Transform sample frequencies rfftfreq - DFT sample frequencies (for usage with rfft, irfft) next_fast_len - Find the optimal length to zero-pad an FFT for speed set_workers - Context manager to set default number of workers get_workers - Get the current default number of workers

Backend control ===============

.. autosummary:: :toctree: generated/

set_backend - Context manager to set the backend within a fixed scope skip_backend - Context manager to skip a backend within a fixed scope set_global_backend - Sets the global fft backend register_backend - Register a backend for permanent use

Return a triangular window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

See Also -------- bartlett : A triangular window that touches zero

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.triang(51) >>> plt.plot(window) >>> plt.title('Triangular window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample')

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = np.abs(fftshift(A / abs(A).max())) >>> response = 20 * np.log10(np.maximum(response, 1e-10)) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the triangular window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')

Return a Tukey window, also known as a tapered cosine window.

Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. alpha : float, optional Shape parameter of the Tukey window, representing the fraction of the window inside the cosine tapered region. If zero, the Tukey window is equivalent to a rectangular window. If one, the Tukey window is equivalent to a Hann window. sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis.

Returns ------- w : ndarray The window, with the maximum value normalized to 1 (though the value 1 does not appear if `M` is even and `sym` is True).

References ---------- .. 1 Harris, Fredric J. (Jan 1978). 'On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform'. Proceedings of the IEEE 66 (1): 51-83. :doi:`10.1109/PROC.1978.10837` .. 2 Wikipedia, 'Window function', https://en.wikipedia.org/wiki/Window_function#Tukey_window

Examples -------- Plot the window and its frequency response:

>>> from scipy import signal >>> from scipy.fft import fft, fftshift >>> import matplotlib.pyplot as plt

>>> window = signal.tukey(51) >>> plt.plot(window) >>> plt.title('Tukey window') >>> plt.ylabel('Amplitude') >>> plt.xlabel('Sample') >>> plt.ylim(0, 1.1)

>>> plt.figure() >>> A = fft(window, 2048) / (len(window)/2.0) >>> freq = np.linspace(-0.5, 0.5, len(A)) >>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max()))) >>> plt.plot(freq, response) >>> plt.axis(-0.5, 0.5, -120, 0) >>> plt.title('Frequency response of the Tukey window') >>> plt.ylabel('Normalized magnitude dB') >>> plt.xlabel('Normalized frequency cycles per sample')