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.

α | 0.2 | 0.5 | 0.8 |

Coherent gain | 0.91 | 0.79 | 0.69 |

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 |

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