Symmetric Convolution - Convolution Theorem

Convolution Theorem

The convolution theorem states that a convolution in the real domain can be represented as a pointwise multiplication across the frequency domain of a Fourier transform. Since sine and cosine transforms are related transforms a modified version of the convolution theorem can be applied, in which the concept of circular convolution is replaced with symmetric convolution. Using these transforms to compute discrete symmetric convolutions is non-trivial since discrete sine transforms (DSTs) and discrete cosine transforms (DCTs) can be counter-intuitively incompatible for computing symmetric convolution, i.e. symmetric convolution can only be computed between a fixed set of compatible transforms.