# Gaussian window

The Gaussian window coefficients are given by the following formula

where N is the length of the filter, M = (N – 1) / 2, k = 0, 1, …, N – 1, and σ > 0.

## An example Gaussian window

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

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

As the parameter σ becomes larger, the transition band of the window decreases and its stop band attenuation becomes worse. As σ approaches infinity, the Gaussian window approaches a rectangular window. If, on the other hand, σ approaches zero, the windowed Gaussian filter approaches an all pass filter.

The following is the Gaussian window with N = 201 and at three different values of σ (0.3, 0.5, and 0.7).

The magnitude response of a filter at these values of σ, given a sampling frequency of 2000 Hz and a cutoff frequency of 40 Hz, is shown below.

## Measures for the Gaussian window

The following graph shows the discrete Fourier transform of the Gaussian window with σ = 0.3 against the discrete Fourier transform of the rectangular window.

The measures of the Gaussian window are as follows.

 σ 0.3 0.5 0.7 Coherent gain 0.37 0.60 0.74 Equivalent noise bandwidth 1.89 1.23 1.08 Processing gain -2.76 dB -0.91 dB -0.32 dB Scalloping loss -0.95 dB -2.12 dB -2.84 dB Worst case processing loss -3.71 dB -3.04 dB -3.16 dB Highest sidelobe level -64.3 dB -32.0 dB -20.9 dB Sidelobe falloff -10.5 dB / octave, -34.8 dB / decade -7.5 dB / octave, -24.8 dB / decade -6.7 dB / octave, -22.3 dB / decade Main lobe is -3 dB 1.78 bins 1.18 bins 1.02 bins Main lobe is -6 dB 2.50 bins 1.64 bins 1.42 bins Overlap correlation at 50% overlap 0.060 0.311 0.431 Amplitude flatness at 50% overlap 0.497 0.936 0.878 Overlap correlation at 75% overlap 0.498 0.755 0.801 Amplitude flatness at 75% overlap 0.998 0.973 0.969

## Approximate confined Gaussian window

The Gaussian window above is actually an approximate confined Gaussian window. A more general approximation of the latter is defined by

A comparison of the Gaussian and confined Gaussian windows for N = 201 and σ = 0.5 is shown below.

Note that the confined Gaussian window does not necessarily peak at 1 in the middle.

The corresponding magnitude responses, with the same low pass filter used above, are shown on the graph below.

## Measures for the approximate confined Gaussian window

The measures of the approximate confined Gaussian window at three values of σ (0.3, 0.5, and 0.7) are as follows.

 σ 0.3 0.5 0.7 Coherent gain 0.37 0.57 0.61 Equivalent noise bandwidth 1.89 1.32 1.24 Processing gain -2.77 dB -1.19 dB -0.94 dB Scalloping loss -0.95 dB -1.85 dB -2.07 dB Worst case processing loss -3.71 dB -3.04 dB -3.01 dB Highest sidelobe level -63.6 dB -30.3 dB -23.5 dB Sidelobe falloff -18.4 dB / octave, -61.1 dB / decade -16.3 dB / octave, -54.3 dB / decade -16.1 dB / octave, -53.5 dB / decade Main lobe is -3 dB 1.78 bins 1.26 bins 1.20 bins Main lobe is -6 dB 2.50 bins 1.76 bins 1.64 bins Overlap correlation at 50% overlap 0.059 0.261 0.313 Amplitude flatness at 50% overlap 0.498 0.841 0.720 Overlap correlation at 75% overlap 0.497 0.728 0.754 Amplitude flatness at 75% overlap 0.998 0.944 0.926

## Generalized normal window

Another generalization of the Gaussian window is the generalized normal window given by the following formula.

where α > 0 (usually α ≥ 1). At α = 2, the generalized normal window becomes the Gaussian window. As α increases, the top of the generalized normal window becomes flatter, the transition band of the corresponding filter becomes narrower, and its stop band attenuation becomes worse. As α approaches infinity, the generalized normal window approaches the rectangular window.

The following is the generalized normal window at three different values of α (2, 4, and 6) and with σ = 0.5.

The corresponding magnitude responses of the same low pass filter used above with these windows are as follows.

## Measures for the generalized normal window

The measures of the generalized normal window with σ = 0.5 and three values of α are as follows.

 α 2 4 6 Coherent gain 0.60 0.54 0.52 Equivalent noise bandwidth 1.23 1.56 1.71 Processing gain -0.91 dB -1.94 dB -2.34 dB Scalloping loss -2.12 dB -1.30 dB -1.09 dB Worst case processing loss -3.04 dB -3.24 dB -3.43 dB Highest sidelobe level -32.0 dB -19.7 dB -16.3 dB Sidelobe falloff -7.5 dB / octave, -24.8 dB / decade -13.2 dB / octave, -44.0 dB / decade -30.8 dB / octave, -102.4 dB / decade Main lobe is -3 dB 1.18 bins 1.50 bins 1.62 bins Main lobe is -6 dB 1.64 bins 2.06 bins 2.22 bins Overlap correlation at 50% overlap 0.311 0.143 0.086 Amplitude flatness at 50% overlap 0.936 0.825 0.821 Overlap correlation at 75% overlap 0.755 0.635 0.582 Amplitude flatness at 75% overlap 0.973 0.948 0.900

Rating