18.5 Convolution

Convolution is a mathematical operation commonly used in signal processing.

Convolution is often denoted with an asterisk, *, as in (f * g), the convolution of functions f and g. Typically in signal processing, f is an input signal and g is the impulse response of a system under consideration. Their discrete convolution is defined as:

$(f*g)_m=\sum _nf(n)g(m-n) \,\!$

Both f and g are assumed to extend infinitely in both directions, while in fact the inputs are always finite sequences. If we view all the data points outside the input range as zeros, then the convolution is said to be a linear convolution. Alternatively, if the data points outside the input range are taken as the periodic repetition of the data points within the input range (the periods of the signal and the impulse response are the same), the convolution will be a circular convolution. If we let $M_f$ and $M_g$ denote the lengths of f and g, for a linear convolution, the length of the result sequence in Origin is