scipy.signal.windows.general_hamming

scipy.signal.windows.general_hamming(M, alpha, sym=True)

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:

Mint

Number of points in the output window. If zero, an empty array is returned. An exception is thrown when it is negative.

alphafloat

The window coefficient, \(\alpha\)

symbool, 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:

wndarray

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

hamming, hann

Notes

The generalized Hamming window is defined as

\[w(n) = \alpha - \left(1 - \alpha\right) \cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1\]

Both the common Hamming window and Hann window are special cases of the generalized Hamming window with \(\alpha\) = 0.54 and \(\alpha\) = 0.5, respectively [2].

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

Examples

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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 \(\alpha\) parameter based on operating mode of the SAR instrument. Some common \(\alpha\) values include 0.75, 0.7 and 0.52 [4]. As an example, we plot these different windows.

>>> import numpy as np
>>> 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")

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