Advanced DSP

Fast Convolution

A Fourier basis is an eigenvector of any LTI system. Furthermore, the eigenvalue corresponding to that eigenvector, is just the dot product of the LTI filter function and the same Fourier basis. This means that, if we project the signal and the filter onto the Argand plane of any Fourier basis, we just have to complex-multiply those two projections to get the projection of the output on the same plane.

The above fact, called the convolution theorem, is used to analyse LTI systems, as also to design them. Since the Fourier transform can be computed very fast using the FFT, it can also be used as a fast convolution methodology.

The only computable Fourier transform, the DFT, corresponds to circular convolution. If we want to implement linear convolution using this circular convolution, we have to pad the signal such that there is no wrap-around distortion.

If we have a huge signal and a modest filter applied onto it, we can make chunks of the signal and apply the filter to each of them. We have to remember to add the overlapping output segments together (overlap-add). Alternatively, we can overlap the input segments and throw away the wrap-around distorted parts of the output (ovelap-save).