A family of discrete Fourier transforms with pseudo-cyclic convolution properties

An extended family of Discrete Fourier transforms is introduced. These transforms, which may be implemented by using FFTs, allow the computation of pseudo-cyclic convolutions by multiplication in the transform domain. The choice of a suitable transform (DFT1/4) or the combined use of two complementary transforms allows a fast and efficient computation of aperiodic convolutions of waveforms of duration N by using N-point transforms that require no zero padding. Finally, all members of this family are shown to be equivalent asymptotically to the Karhunen-Loève transform of an arbitrary wide sense stationary process.