The Hann-Poisson window coefficients are given by the following formula
where N is the length of the filter, M = (N – 1) / 2, k = 0, 1, …, N – 1, and usually 0 ≤ α < 1.
An example Hann-Poisson window
Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Hann-Poisson window with α = 0.3.
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 Hann-Poisson window is as follows.
The magnitude response of the same filter is shown on the graph below.
As the parameter α becomes larger, the transition band of the Hann-Poisson window increases and its Gibbs phenomenon ripples become smaller.
The following is the Hann-Poisson window with N = 201 and at three different values of α (0.3, 0.5, and 0.7).
The magnitude response of a filter at these values of α, given a sampling frequency of 2000 Hz and a cutoff frequency of 40 Hz, is shown below.
Measures for the Hann-Poisson window
The following is a comparison of the discrete Fourier transform of the Hann-Poisson window and the rectangular window.
The Hann-Poisson window measures are as follows.
|Equivalent noise bandwidth||1.57||1.61||1.66|
|Processing gain||-1.95 dB||-2.07 dB||-2.20 dB|
|Scalloping loss||-1.32 dB||-1.25 dB||-1.19 dB|
|Worst case processing loss||-3.27 dB||-3.33 dB||-3.40 dB|
|Highest sidelobe level||-37.6 dB||-35.2 dB||-33.0 dB|
|Sidelobe falloff||-17.4 dB / octave, -57.7 dB / decade||-16.8 dB / octave, -55.9 dB / decade||-16.4 dB / octave, -54.6 dB / decade|
|Main lobe is -3 dB||1.50 bins||1.54 bins||1.58 bins|
|Main lobe is -6 dB||2.08 bins||2.14 bins||2.20 bins|
|Overlap correlation at 50% overlap||0.140||0.124||0.110|
|Amplitude flatness at 50% overlap||0.861||0.779||0.705|
|Overlap correlation at 75% overlap||0.631||0.612||0.591|
|Amplitude flatness at 75% overlap||0.977||0.960||0.942|