Welch window

The Welch window coefficients are given by the following formula

Welch window formula

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

The Welch window is a quadratic polynomial and a parabola.

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

Welch 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 Welch window is as follows.

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

Measures for the Welch window

The following graph compares the discrete Fourier transform of the Welch window with that of the rectangular window.

Discrete Fourier transform of the Welch window

The Welch window measures are as follows.

Coherent gain 0.67
Equivalent noise bandwidth 1.20
Processing gain -0.78 dB
Scalloping loss -2.23 dB
Worst case processing loss -3.02 dB
Highest sidelobe level -21.3 dB
Sidelobe falloff -11.0 dB / octave, -36.5 dB / decade
Main lobe is -3 dB 1.16 bins
Main lobe is -6 dB 1.58 bins
Overlap correlation at 50% overlap 0.345
Amplitude flatness at 50% overlap 0.670
Overlap correlation at 75% overlap 0.765
Amplitude flatness at 75% overlap 0.910

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