Kaiser-Bessel window

The Kaiser-Bessel window coefficients are given by the following formula

Kaiser-Bessel window formula

where N is the length of the filter and k = 0, 1, …, N – 1.

Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Kaiser-Bessel window.

Kaiser-Bessel 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-Bessel window is as follows.

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

Measures for the Kaiser-Bessel window

The following is a comparison of the discrete Fourier transform of the Kaiser-Bessel window and the rectangular window.

Discrete Fourier transform of the Kaiser-Bessel window

The Kaiser-Bessel window measures are as follows.

Coherent gain 0.40
Equivalent noise bandwidth 1.80
Processing gain -2.55 dB
Scalloping loss -1.02 dB
Worst case processing loss -3.57 dB
Highest sidelobe level -65.3 dB
Sidelobe falloff -11.8 dB / octave, -39.3 dB / decade
Main lobe is -3 dB 1.70 bins
Main lobe is -6 dB 2.40 bins
Overlap correlation at 50% overlap 0.072
Amplitude flatness at 50% overlap 0.608
Overlap correlation at 75% overlap 0.537
Amplitude flatness at 75% overlap 1.000

See also:
Window

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