Many common convolution-based processes such as Gaussian blur and taking the derivative of a signal in frequency-space are symmetric and this property can be exploited to make these convolutions easier to evaluate.
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.
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.
, 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.
For instance using a DCT-II, a symmetric signal need only have the positive half DCT-II transformed, since the frequency domain will implicitly construct the mirrored data comprising the other half.
Also the boundary conditions implicit in DSTs and DCTs create edge effects that are often more in keeping with neighbouring data than the periodic effects introduced by using the Fourier transform.