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.
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 |