An algebraic approach to multidimensional convolution (abstract only)

Abstract

Bidimensional acyclic convolution arises naturally by considering polynomials whose elements are themselves polynomials.Convolving two polynomials of length L on the ring of the polynomials with M coefficients takes a total number of operations of ML(log M + log L) + ML = Nlog N + N. The inverse transform takes again Nlog N element operations. This gives 2Nlog N + N element operations. If we operate in a finite ring such that o, the root of the unity of order N is a power of 2, then all the multiplications by powers of o become shifts.Transforming a one dimensional convolution into a bidimensional one permits to increase the sequence length without increasing the word length. If we perform acyclic convolutions then we must partially overlap and add the resulting sequences in order to obtain the final sequence(polynomial). We overlap always the second dimension minus 1. When the length of the sequences is 2M-1(padding zeros to be able to perform cyclic convolution), the overlapping is M-1. This can be shown easily by considering that the total length 2N-1 = 2ML-1 = (2L-2)(2M-1-(M-1))+2M-1.This procedure can be generalized for more dimensions in the same way.Let us suppose that N = 2P=4M.2s. We can apply M dimensional convolution of length 4, followed by one or zero convolution of length 2.