scipy.signal.chebwin

scipy.signal.chebwin(M, at, sym=True)

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:

\[W(k) = \frac {\cos\{M \cos^{-1}[\beta \cos(\frac{\pi k}{M})]\}} {\cosh[M \cosh^{-1}(\beta)]}\]

where

\[\beta = \cosh \left [\frac{1}{M} \cosh^{-1}(10^\frac{A}{20}) \right ]\]

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

[R163]

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

[R164]

Peter Lynch, “The Dolph-Chebyshev Window: A Simple Optimal Filter”, American Meteorological Society (April 1997) http://mathsci.ucd.ie/~plynch/Publications/Dolph.pdf

[R165]

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.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]")

(Source code)