Return the 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 for window.
|
|---|
| Returns : | out : array
The window, normalized to one (the value one
appears only if the number of samples is odd).
|
|---|
Notes
The Kaiser window is defined as
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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 many other windows by varying the beta
parameter.
| beta |
Window shape |
|---|
| 0 |
Rectangular |
| 5 |
Similar to a Hamming |
| 6 |
Similar to a Hanning |
| 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
get 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
| [R33] | 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). |
| [R34] | E.R. Kanasewich, “Time Sequence Analysis in Geophysics”, The
University of Alberta Press, 1975, pp. 177-178. |
Examples
>>> from numpy import kaiser
>>> kaiser(12, 14)
array([ 7.72686684e-06, 3.46009194e-03, 4.65200189e-02,
2.29737120e-01, 5.99885316e-01, 9.45674898e-01,
9.45674898e-01, 5.99885316e-01, 2.29737120e-01,
4.65200189e-02, 3.46009194e-03, 7.72686684e-06])
Plot the window and the frequency response:
>>> from numpy import clip, log10, array, kaiser, linspace
>>> from numpy.fft import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> window = kaiser(51, 14)
>>> plt.plot(window)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Kaiser window")
<matplotlib.text.Text object at 0x...>
>>> plt.ylabel("Amplitude")
<matplotlib.text.Text object at 0x...>
>>> plt.xlabel("Sample")
<matplotlib.text.Text object at 0x...>
>>> plt.show()
(Source code, png, pdf)
>>> plt.figure()
<matplotlib.figure.Figure object at 0x...>
>>> A = fft(window, 2048) / 25.5
>>> mag = abs(fftshift(A))
>>> freq = linspace(-0.5,0.5,len(A))
>>> response = 20*log10(mag)
>>> response = clip(response,-100,100)
>>> plt.plot(freq, response)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Frequency response of Kaiser window")
<matplotlib.text.Text object at 0x...>
>>> plt.ylabel("Magnitude [dB]")
<matplotlib.text.Text object at 0x...>
>>> plt.xlabel("Normalized frequency [cycles per sample]")
<matplotlib.text.Text object at 0x...>
>>> plt.axis('tight')
(-0.5, 0.5, -100.0, ...)
>>> plt.show()
(png, pdf)
