Hann Poisson window

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The Hann-Poisson window coefficients are given by the following formula

Hann-Poisson window 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.

Hann-Poisson window

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.

Impulse response of a low pass filter with and without the Hann-Poisson window

The magnitude response of the same filter is shown on the graph below.

Magnitude response of a low pass filter with and without the Hann-Poisson window

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).

Hann-Poisson window at three different alphas

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.

Magnitude response of the Hann-Poisson window at three different alphas

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.

Discrete Fourier transform of the Hann-Poisson window

The Hann-Poisson window measures are as follows.


α0.30.50.7
Coherent gain0.460.430.41
Equivalent noise bandwidth1.571.611.66
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 dB1.50 bins1.54 bins1.58 bins
Main lobe is -6 dB2.08 bins2.14 bins2.20 bins
Overlap correlation at 50% overlap0.1400.1240.110
Amplitude flatness at 50% overlap0.8610.7790.705
Overlap correlation at 75% overlap0.6310.6120.591
Amplitude flatness at 75% overlap0.9770.9600.942


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mic
I've just fixed this topic. The previous version of the topic was wrong - it was missing a minus before the alpha parameter. the window still worked, but it wasn't the Hann-Poisson window.
Posted At 20-04-2014 14:16:48



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