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.
The Hann-Poisson window is the product of the Hann window and the Poisson window.
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 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.