papyri
{ } Raw JSON

bundles / scipy 1.17.1 / scipy / signal / windows / _windows / kaiser

function

scipy.signal.windows._windows:kaiser

source: /scipy/signal/windows/_windows.py :1202

Signature

def   kaiser ( M ,  beta ,  sym = True ,  * ,  xp = None ,  device = None )

Summary

Return a Kaiser window.

Extended Summary

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

Parameters

M : int

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

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.

xp : array_namespace, optional

Optional array namespace. Should be compatible with the array API standard, or supported by array-api-compat. Default: numpy

device: any

optional device specification for output. Should match one of the supported device specification in xp.

Returns

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).

Notes

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 discrete 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.

Array API Standard Support

kaiser has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ✅                     ✅                   
Dask                  ✅                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

Plot the window and its frequency response: 
import numpy as np
from scipy import signal
from scipy.fft import fft, fftshift
import matplotlib.pyplot as plt
✓
window = signal.windows.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]")
✗

Aliases

  • scipy.signal.windows.kaiser

Referenced by

This package