Poisson window

The coefficients of the Poisson window, also called the exponential window, are given by the following formula

$$a(k)=e^{-\alpha \frac{|k-M|}{M}}$$

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.

See also:
Window