Kaiser window

The Kaiser window coefficients are given by the following formula

Kaiser window formula

where N is the length of the filter, k = 0, 1, …, N – 1, α is any real number, and I0 is the zero order modified Bessel function of the first kind.

Computing the Kaiser window

The modified Bessel function of the first kind (the hyperbolic Bessel function) is defined as follows.

Modified Bessel function of the first kind

where β is the order of the function and Γ is the generalized factorial.

Note that for any positive integer n, Γ(n) = (n – 1)!. Then at β = 0, Γ(n + β + 1) = (n + 0 + 1 – 1)! = n!. This means that the zero order modified Bessel function of the first kind is simply

Zero order modified Bessel function of the first kind

The sum converges and we can be approximate for any x, if we take the first few terms, say

Approximation to the zero order modified Bessel function of the first kind

The Kaiser window itself, if we rewrite the numerator and denominator as per the discussion above, can be written as follows.

Simplified formula for the Kaiser window

As before, both sums converge and can be approximated by taking the first few terms.

An example Kaiser window

Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Kaiser window with α = 3 and M = 4.

Kaiser window

Given a sampling frequency of 2000 Hz and a filter cutoff frequency of 40 Hz, the impulse response of the filter with a rectangular window (with no window) and with the Kaiser window is as follows.

Impulse response of a low pass filter with and without the Kaiser window

The magnitude response of the same filter is shown on the graph below.

Magnitude response of a low pass filter with and without the Kaiser window

As the absolute value of α increases, the windowed filter produces a larger transition band, but better stop band attenuation. As |α| decreases, the filter has a smaller transition band and worse stop band attenuation. At α = 0 the Kaiser window becomes a rectangular window.

The Kaiser window with N = 201 for three different values of α (0.5, 1, and 5) is shown below.

Kaiser window for three different values of alpha

The following are the magnitude responses of these windows, applied to a filter with a cutoff frequency of 40 Hz, given a sampling frequency of 2000 Hz.

Magnitude response of the Kaiser window for three different values of alpha

Measures for the Kaiser window

The following is a comparison of the discrete Fourier transform of the Kaiser window (α = 1.0) and the rectangular window.

Discrete Fourier transform of the Kaiser window

The Kaiser window measures are as follows.

α 0.5 1.0 5.0
Coherent gain 0.85 0.67 0.42
Equivalent noise bandwidth 1.02 1.15 1.76
Processing gain -0.10 dB -0.62 dB -2.44 dB
Scalloping loss -3.31 dB -2.42 dB -1.05 dB
Worst case processing loss -3.41 dB -3.04 dB -3.49 dB
Highest sidelobe level -16.6 dB -24.6 dB -38.6 dB
Sidelobe falloff -6.4 dB / octave, -21.2 dB / decade -7.3 dB / octave, -24.2 dB / decade -15.5 dB / octave, -51.6 dB / decade
Main lobe is -3 dB 0.96 bins 1.10 bins 1.68 bins
Main lobe is -6 dB 1.30 bins 1.52 bins 2.34 bins
Overlap correlation at 50% overlap 0.479 0.370 0.080
Amplitude flatness at 50% overlap 0.895 0.829 0.691
Overlap correlation at 75% overlap 0.794 0.783 0.559
Amplitude flatness at 75% overlap 0.974 0.958 0.991

See also:
Window

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