Reducing the complexity of discrete convolutions by a linear transformation and modulo arithmetic

Abstract

For the discrete convolution with a Toeplitz coefficient matrix, a general algorithm with minimum number of multiplications is derived by means of a linear transformation. In order to keep the results applicable to long convolutions with limited wordlength, modulo arithmetic and block-partitioning is introduced. The resulting algorithms reveal small complexity and generate no roundoff noise. The same holds for linear and cyclic convolution algorithms derived from the presented algorithms of the more general Toeplitz convolution. The main advantages of the new algorithms compared to similar algorithms based on number theoretic transforms are a simpler and more general derivation and far less restrictions for the convolution length.