Convolution (cyclic or acyclic) is commutative, i.e.,

Proof:

In the first step we made the change of summation variable
, and in the second step, we made use of the fact
that any sum over all terms is equivalent to a sum from 0 to
.
Next Section: Convolution as a Filtering Operation Previous Section: Examples

Detailed derivation of the Discrete Fourier Transform (DFT) and its associated mathematics, including elementary audio signal processing applications and matlab programming examples.