# signal.windows¶

The suite of window functions.

## Module Contents¶

### Functions¶

 _len_guards(M) Handle small or incorrect window lengths _extend(M,sym) Extend window by 1 sample if needed for DFT-even symmetry _truncate(w,needed) Truncate window by 1 sample if needed for DFT-even symmetry _cos_win(M,a,sym=True) r boxcar(M,sym=True) Return a boxcar or rectangular window. triang(M,sym=True) Return a triangular window. parzen(M,sym=True) Return a Parzen window. bohman(M,sym=True) Return a Bohman window. blackman(M,sym=True) r nuttall(M,sym=True) Return a minimum 4-term Blackman-Harris window according to Nuttall. blackmanharris(M,sym=True) Return a minimum 4-term Blackman-Harris window. flattop(M,sym=True) Return a flat top window. bartlett(M,sym=True) r hann(M,sym=True) r tukey(M,alpha=0.5,sym=True) rReturn a Tukey window, also known as a tapered cosine window. barthann(M,sym=True) Return a modified Bartlett-Hann window. hamming(M,sym=True) rReturn a Hamming window. kaiser(M,beta,sym=True) rReturn a Kaiser window. gaussian(M,std,sym=True) rReturn a Gaussian window. general_gaussian(M,p,sig,sym=True) rReturn a window with a generalized Gaussian shape. chebwin(M,at,sym=True) rReturn a Dolph-Chebyshev window. slepian(M,width,sym=True) Return a digital Slepian (DPSS) window. cosine(M,sym=True) Return a window with a simple cosine shape. exponential(M,center=None,tau=1.0,sym=True) rReturn an exponential (or Poisson) window. get_window(window,Nx,fftbins=True) Return a window.
_len_guards(M)

Handle small or incorrect window lengths

_extend(M, sym)

Extend window by 1 sample if needed for DFT-even symmetry

_truncate(w, needed)

Truncate window by 1 sample if needed for DFT-even symmetry

_cos_win(M, a, sym=True)

r Generic weighted sum of cosine terms window

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.
 [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

Heinzel describes a flat-top window named “HFT90D” with formula: [2]_

where

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

>>> window = signal._cos_win(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 = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> 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')

boxcar(M, sym=True)

Return a boxcar or rectangular window.

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

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

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

triang(M, sym=True)

Return a triangular window.

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.
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).

bartlett : A triangular window that touches zero

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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 = 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 triangular window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")

parzen(M, sym=True)

Return a Parzen window.

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

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

bohman(M, sym=True)

Return a Bohman window.

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.
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).

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

blackman(M, sym=True)

r 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.

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.
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).

The Blackman window is defined as

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.

 [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.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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 = 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 window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")

nuttall(M, sym=True)

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

This variation is called “Nuttall4c” by Heinzel. [2]_

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.
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).
 [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

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

blackmanharris(M, sym=True)

Return a minimum 4-term Blackman-Harris window.

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.
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).

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

flattop(M, sym=True)

Return a flat top window.

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.
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).

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]_

 [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.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

bartlett(M, sym=True)

r 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.

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.
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).

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

The Bartlett window is defined as

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]_

 [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”, http://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.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

hann(M, sym=True)

r Return a Hann window.

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

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.
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).

The Hann window is defined as

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.

 [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”, http://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.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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 = 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 Hann window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")

tukey(M, alpha=0.5, sym=True)

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

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.
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).
 [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”, http://en.wikipedia.org/wiki/Window_function#Tukey_window

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

barthann(M, sym=True)

Return a modified Bartlett-Hann window.

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.
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).

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

hamming(M, sym=True)

rReturn 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.

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.
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).

The Hamming window is defined as

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.

 [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”, http://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.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

kaiser(M, beta, sym=True)

rReturn a Kaiser window.

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

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.
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).

The Kaiser window is defined as

with

where 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.

 [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”, http://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.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

gaussian(M, std, sym=True)

rReturn a Gaussian window.

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.
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).

The Gaussian window is defined as

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

general_gaussian(M, p, sig, sym=True)

rReturn a window with a generalized Gaussian shape.

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.
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).

The generalized Gaussian window is defined as

the half-power point is at

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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 "
...           "window (p=1.5, $\sigma$=7)")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")

chebwin(M, at, sym=True)

rReturn a Dolph-Chebyshev window.

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

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:

where

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.

 [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

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

slepian(M, width, sym=True)

Return a digital Slepian (DPSS) window.

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

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.
w : ndarray
The window, with the maximum value always normalized to 1
 [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

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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]")

cosine(M, sym=True)

Return a window with a simple cosine shape.

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.
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).

New in version 0.13.0.

Plot the window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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()

exponential(M, center=None, tau=1.0, sym=True)

rReturn an exponential (or Poisson) window.

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.
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).

The Exponential window is defined as

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

Plot the symmetric window and its frequency response:

>>> from scipy import signal
>>> from scipy.fftpack 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")

get_window(window, Nx, fftbins=True)

Return a window.

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 fftpack.fftfreq). If False, create a “symmetric” window, for use in filter design.
get_window : ndarray
Returns a window of length Nx and type window

Window types:

boxcar, triang, blackman, hamming, hann, bartlett, flattop, parzen, bohman, blackmanharris, nuttall, barthann, kaiser (needs beta), gaussian (needs standard deviation), general_gaussian (needs power, width), slepian (needs width), chebwin (needs attenuation), exponential (needs decay scale), 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 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.

>>> 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])