Multifactor algorithms for noncyclic digital convolution

The computation of the (noncyclic) convolution of two sequences occurs in a variety of digital signal processing applications, e.g., correlation and linear filtering. Direct evaluation involving sequences of length N requires N multiplications and (N-1)²additions. A std. approach for large N uses the FFT to reduce the number of computations. This requires that one append a guard band of N zeros. Recently new convolution algorithms have been proposed based on rectangular transform techniques [1]. While these algorithms are highly efficient they are not easily implemented. The objective of this work is the development of a unified theory for multifactor convolution algorithms which are efficient and easily implemented. An algorithm has been developed which is a composition of a number of prime factor algorithms. If (where the niare prime and the kiare integers) is a product of many factors, this new algorithm can result in a significant savings. Computational comparisons are given for these new algorithms against the most efficient FFT methods as described in [2]. It is shown that the algorithms in [3] for N=2kare a special case of the multifactor algorithms.