Hann-Poisson window

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

$$a(k)=0.5 * (1-cos(\frac{\pi\,k}{M}))\,e^{-\alpha\frac{|k-M|}{M}}$$

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 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.

α 0.3 0.5 0.7
Coherent gain 0.46 0.43 0.41
Equivalent noise bandwidth 1.57 1.61 1.66
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 dB 1.50 bins 1.54 bins 1.58 bins
Main lobe is -6 dB 2.08 bins 2.14 bins 2.20 bins
Overlap correlation at 50% overlap 0.140 0.124 0.110
Amplitude flatness at 50% overlap 0.861 0.779 0.705
Overlap correlation at 75% overlap 0.631 0.612 0.591
Amplitude flatness at 75% overlap 0.977 0.960 0.942

See also:
Window

Comments

admin: First posted on 2014 04 20

mic, 2014 04 20: 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.

admin: First posted on 2011 05 31

Sándor Bránya, 2011 05 31: I have found here an reassuring answer of my question: What is time division exactly? Thank you!

anonymous, 2016 07 31: Is the sentence "The remaining 15 bits are the number of MIDI ticks per bit." supposed to be "The remaining 15 bits are the number of MIDI ticks per beat."?

mic, 2016 08 11: Yes on the "MIDI ticks per beat". I fixed it above. On the question about time division, there is also a topic on MIDI time division in this site. There is a link to it above.

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