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.
Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Poisson window with α = 0.5.
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.
|Equivalent noise bandwidth||1.00||1.02||1.05|
|Processing gain||-0.01 dB||-0.09 dB||-0.22 dB|
|Scalloping loss||-3.69 dB||-3.36 dB||-3.05 dB|
|Worst case processing loss||-3.70 dB||-3.45 dB||-3.27 dB|
|Highest sidelobe level||-14.7 dB||-17.5 dB||-19.9 dB|
|Sidelobe falloff||-6.1 dB / octave, -20.3 dB / decade||-6.3 dB / octave, -20.8 dB / decade||-6.4 dB / octave, -21.4 dB / decade|
|Main lobe is -3 dB||0.90 bins||0.94 bins||1.00 bins|
|Main lobe is -6 dB||1.24 bins||1.30 bins||1.36 bins|
|Overlap correlation at 50% overlap||0.497||0.480||0.450|
|Amplitude flatness at 50% overlap||0.995||0.970||0.925|
|Overlap correlation at 75% overlap||0.773||0.794||0.799|
|Amplitude flatness at 75% overlap||0.999||0.992||0.980|