Symmetric Convolution - Mutually Compatible Transforms

Mutually Compatible Transforms

In order to compute symmetric convolution effectively, one must know which particular frequency domains (which are reachable by transforming real data through DSTs or DCTs) the inputs and outputs to the convolution can be and then tailor the symmetries of the transforms to the required symmetries of the convolution.

The following table documents which combinations of the domains from the main eight commonly used DST I-IV and DCT I-IV satisfy where represents the symmetric convolution operator. Convolution is a commutative operator, and so and are interchangeable.

Forward transforms of, and, through the transforms specified should allow the symmetric convolution to be computed as a pointwise multiplication, with any excess undefined frequency amplitudes set to zero. Possibilities for symmetric convolutions involving DSTs and DCTs V-VIII derived from the discrete Fourier transforms (DFTs) of odd logical order can be determined by adding four to each type in the above tables.