cupyx.scipy.signal.windows.general_hamming#

cupyx.scipy.signal.windows.general_hamming(M, alpha, sym=True)[source]#

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, \(\alpha\)
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 – The window, with the maximum value normalized to 1 (though the value 1 does not appear if M is even and sym is True).

Return type

ndarray

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.

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 \(\alpha\) 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 cupyx.scipy.signal.windows
>>> import cupy as cp
>>> from cupy.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 = cupyx.scipy.signal.windows.general_hamming(41, alpha)
...     spatial_plot.plot(cupy.asnumpy(window), label="{:.2f}".format(alpha))
...     A = fft(window, 2048) / (len(window)/2.0)
...     freq = cupy.linspace(-0.5, 0.5, len(A))
...     response = 20 * cupy.log10(cupy.abs(fftshift(A / cupy.abs(A).max())))
...     freq_plot.plot(
...         cupy.asnumpy(freq), cupy.asnumpy(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