The Hann window coefficients are given by the following formula

$$a(k)=0.5 * (1-\cos(\frac{2\pi\,k}{N-1}))$$

where N is the length of the filter and k = 0, 1, …, N – 1.

The Hann window belongs to the family of Hamming windows. The derivation of the Hann window is shown in the topic Hamming window. The Hann window is also a poiwer of cosine window (α = 2). When the Hann window is multiplied by the Poisson window, the result is the Hann-Poisson window.

## An example Hann window

Consider a finite impulse response (FIR) low pass filter of length N = 201. The following is the Hann 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 window is as follows.

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

## Measures for the Hann window

The following is a comparison of the discrete Fourier transform of the Hann window and the rectangular window.

The Hann window measures are as follows.

Coherent gain | 0.50 |

Equivalent noise bandwidth | 1.50 |

Processing gain | -1.77 dB |

Scalloping loss | -1.42 dB |

Worst case processing loss | -3.19 dB |

Highest sidelobe level | -31.5 dB |

Sidelobe falloff | -20.7 dB / octave, -68.9 dB / decade |

Main lobe is -3 dB | 1.44 bins |

Main lobe is -6 dB | 2.00 bins |

Overlap correlation at 50% overlap | 0.165 |

Amplitude flatness at 50% overlap | 1.000 |

Overlap correlation at 75% overlap | 0.658 |

Amplitude flatness at 75% overlap | 1.000 |

See also:

Window

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