m-adic (polyadic) invariance and its applications

The concept of m-adic invariance allows approximation of a linear time-invariant (LTI) system by a linear m-adic invariant (LMI) system or, equivalently, approximation of a circulant matrix by a super-circulant matrix. The approximation reduces the number of multiplies required for computing N-point cyclic convolution to
2(\log_{m}N - 1)N
, where N = m
ⁿ
. The error introduced by the approximation can be removed, if desired, by subsequent processing. In one concrete case, determination of a small number of noncontiguous frequencies, this approach-approximation and subsequent correction-can effect substantial savings in a number of multiplies compared to both fast Fourier transform (FFT) algorithm and direct discrete Fourier transform (DDFT). These applications are preceded by a tutorial presentation of concepts which are basic to m-adic invariant systems.