Sampled Gaussian

In practical discrete-signal-processing applications, where the Gaussian is used as a convolution filter, it must also be represented at discrete intervals $g_k$. The Gaussian is normally sampled at uniform intervals; however, since it has infinite support, samples are taken over only 3 or 4 times the Gaussian's standard deviation:

$$g_k = \left\{ \begin{array}{ll} c e^{ - \frac{k^2}{ 2\sigm... ...\vert < 3 \sigma \\ 0 & \text{otherwise} \end{array}\right.$$ (2.19)

with $c$ the normalization factor chosen so that $\sum_k g_k = 1$. The Gaussian can be extended to the multidimensional case very simply as follows:

$$g_{k_1,k_2,\ldots,k_n} = g_{k_1} \cdot g_{k_2} \ldots g_{k_n}$$ (2.20)