Multidimensional discrete convolution: Difference between revisions

Revision as of 09:05, 12 September 2017

Revision as of 09:07, 12 September 2017

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[[File:Wiki gauss.png|thumb|300px|2D Gaussian Visualization where <math>\mu_1=\mu_2=0</math> and <math>\sigma_1=\sigma_2=1</math>]]

The Gaussian distribution sampled at discrete values in one dimension is given by the following (assuming <math>\mu=0</math>):<math display="block">G(n)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{n^2}{2\sigma^2}}</math>This is readily extended to a signal of ''M'' dimensions (assuming <math>\sigma</math> stays constant for all dimensions and <math>\mu_1=\mu_2=...=\mu_M=0</math>):<math display="block">G(n_1,n_2,...,n_M)=\frac{1}{(2\pi\sigma^M)^{M/2}\sigma^M}e^{-\frac{({n_1}^2+{n_2}^2+...+{n_M}^2)}{2\sigma^2}}</math>One important property to recognize is that the ''M'' dimensional signal is separable such that:<math display="block">G(n_1,n_2,...,n_M)=G(n_1)G(n_2)...G(n_M)</math>Then, Gaussian convolution with discrete-valued signals can be expressed as the following:

<math>y(n)=x(n)*G(n)</math>