Sampled Gaussian

In practical applications of discrete signal processing, where the Gaussian is used as a convolutional filter, it must also be represented at discrete steps $g_k$. The Gaussian is typically sampled at a uniform step, but since it has infinite support, a sufficient number of samples are taken for only 3 or 4 times the standard deviation of the Gaussian:

$$g_k = \left\{ \begin{array}{ll} c e^{ - \frac{k^2}{ 2\sigm... ...\vert < 3 \sigma \\ 0 & \text{otherwise} \end{array}\right.$$ (2.17)

with $c$ being a normalization factor chosen such that $\sum_k g_k = 1$.

It is possible to extend the Gaussian to the multidimensional case in a very straightforward manner as follows:

$$g_{k_1,k_2,\ldots,k_n} = g_{k_1} \cdot g_{k_2} \ldots g_{k_n}$$ (2.18)