## Gaussian Window and Transform

The Gaussian ``bell curve'' is possibly the only smooth, nonzero function, known in closed form, that transforms to itself.^{4.15}

(4.55) |

It also achieves the

*minimum time-bandwidth product*

(4.56) |

when ``width'' is defined as the square root of its second central moment. For even functions ,

(4.57) |

Since the true Gaussian function has infinite duration, in practice we must window it with some usual finite window, or truncate it. Depalle [58] suggests using a

*triangular window*raised to some power for this purpose, which

*preserves the absence of side lobes*for sufficiently large . It also preserves

*non-negativity*of the transform.

### Matlab for the Gaussian Window

In matlab,`w = gausswin(M,alpha)`returns a length window with parameter where is defined, as in Harris [101], so that the window shape is invariant with respect to window length :

function [w] = gausswin(M,alpha) n = -(M-1)/2 : (M-1)/2; w = exp((-1/2) * (alpha * n/((M-1)/2)) .^ 2)';An implementation in terms of unnormalized standard deviation (

`sigma`in samples) is as follows:

function [w] = gaussianwin(M,sigma) n= -(M-1)/2 : (M-1)/2; w = exp(-n .* n / (2 * sigma * sigma))';In this case,

`sigma`would normally be specified as a fraction of the window length (

`sigma = M/8`in the sample below). Note that, on a dB scale, Gaussians are

*quadratic*. This means that

*parabolic interpolation*of a sampled Gaussian transform is

*exact*. This can be a useful fact to remember when estimating sinusoidal peak frequencies in spectra. For example, one suggested implication is that, for typical windows, quadratic interpolation of spectral peaks may be more accurate on a

*log-magnitude scale*(

*e.g.*, dB) than on a linear magnitude scale (this has been observed empirically for a variety of cases).

### Gaussian Window and Transform

Figure 3.36 shows an example length Gaussian window and its transform. The`sigma`parameter was set to so that simple truncation of the Gaussian yields a side-lobe level better than dB. Also overlaid on the window transform is a parabola; we see that the main lobe is well fit by the parabola until the side lobes begin. Since the transform of a Gaussian is a Gaussian (exactly), the side lobes are entirely caused by truncating the window. More properties and applications of the Gaussian function can be found in Appendix D.

### Exact Discrete Gaussian Window

It can be shown [44] that(4.58) |

where is the time index, and is the frequency index for a length (even)

*normalized*DFT (DFT divided by ). In other words, the Normalized DFT (NDFT) of this particular sampled Gaussian pulse is exactly the complex-conjugate of the same Gaussian pulse. (The proof is nontrivial.)

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Optimized Windows

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Dolph-Chebyshev Window