The coefficients of the Poisson window, also called the exponential window, are given by the following formula
where N is the length of the filter, M = (N – 1) / 2, k = 0, 1, …, N – 1, and, typically, 0 α < 1. α is often set to 2 D / 8.69, were D is the desired decay of the window in decibels over half of the window length.
The multiplication of the Poisson window with the Hann window produces the hybrid Hann-Poisson window.
Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Poisson window with α = 0.5.
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 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 filter with the Poisson window becomes wider and the Gibbs phenomenon ripples in the pass band become smaller. At α = 0, the Poisson window becomes the rectangular window.
The following is the Poisson window with N = 201 and at three different values of α (0.2, 0.5, and 0.8).
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 Poisson window
The following is a comparison of the discrete Fourier transform of the Poisson window (α = 0.4) and the rectangular window.
The Poisson window measures are as follows.