Bohman window

The Bohman window coefficients are given by the following formula

$$w(k)=(1-|\frac{k}{M}-1|)\,cos(\pi |\frac{k}{M}-1|)+\frac{1}{\pi}\,sin(\pi |\frac{k}{M}-1|)$$

where N is the length of the filter, M = (N – 1) / 2, and k = 0, 1, …, N – 1.

The Bohman window is the convolution of the sine window with itself.

An example Bohman window

Take a finite impulse response (FIR) low pass filter of length N = 201. The following is the Bohman window.

Bohman 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 Bohman window is as follows.

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

Measures for the Bohman window

The following is a plot of the discrete Fourier transform of the Bohman window against the discrete Fourier transform of the rectangular window.

Discrete Fourier transform of the Bohman window

The measures of the Bohman window are as follows.

Coherent gain 0.40
Equivalent noise bandwidth 1.79
Processing gain -2.53 dB
Scalloping loss -1.02 dB
Worst case processing loss -3.55 dB
Highest sidelobe level -46.0 dB
Sidelobe falloff -21.4 dB / octave, -71.0 dB / decade
Main lobe is -3 dB 1.70 bins
Main lobe is -6 dB 2.38 bins
Overlap correlation at 50% overlap 0.073
Amplitude flatness at 50% overlap 0.637
Overlap correlation at 75% overlap 0.544
Amplitude flatness at 75% overlap 0.982

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