Digital filters with poles via the FFT

A method is presented that uses the fast Fourier transform (FFT) to compute the output of an infinite-impulse-response digital filter. This method uses the summability of infinite-length geometric sequences to account for the aliasing that is inherent in using the discrete Fourier transform (DFT) to calculate convolutions. Previous procedures that use the FFT to realize recursive digital filters require that the filter have a large number of poles and zeros before the FFT method offers a computational advantage over the direct implementation of the filter. The technique presented here is competitive with direct filter implementation.