Convolution

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Convolution

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Convolution



Definition: The convolution of two signals $x$ and $y$ in ${\ is denoted “$x\” and defined by

\

Note that this is cyclic or “circular” convolution.8.2 The importance of convolution in linear systems theory is discussed in §8.7

Convolution is commutative, i.e.,

\

Proof:

\

where in the first step we made the change of summation variable $l\, and in the second step, we made use of the fact that any sum over all $N$ terms is equivalent to a sum from $0$ to $N-1$.



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