Hann window

The Hann window coefficients are given by the following formula

Hann window formula

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.

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.

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

Measures for the Hann window

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

Discrete Fourier transform of the Hann 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

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