Agarwal-Cooley Convolution Algorithm

Abstract

The cyclic convolution algorithms of chapter 6 are efficient for special small block lengths, but as the size of the block length increases, other methods are required. First, as discussed in chapter 6, these algorithms keep the number of required multiplications small, but they can require many additions. Also, each size requires a different algorithm. There is no uniform tructure that can be repeatedly called upon. In this chapter, a technique similar to the Good-Thomas PFA will be developed to decompose a large size cyclic convolution into several small size cyclic convolutions that in turn can be evaluated using the Winograd cyclic convolution algorithm. These ideas were introduced by Agarwal and Cooley [1] in 1977. As in the Good-Thomas PFA, the CRT is used to define an indexing of data. This indexing changes a one-dimensional cyclic convolution into a two-dimensional one. We will see how to compute a two-dimensional cyclic convolution by ‘nesting’ a fast algorithm for a one-dimensional case inside another fast algorithm for a one-dimensional cyclic convolution. There are several two-dimensional cyclic convolution algorithms that, although important, will not be discussed. These can be found in [2].