Automatic derivation and implementation of fast convolution algorithms

This paper surveys algorithms for computing linear and cyclic convolution. Algorithms are presented in a uniform mathematical notation that allows automatic derivation, optimization, and implementation. Using the tensor product and Chinese remainder theorem, a space of algorithms is defined and the task of finding the best algorithm is turned into an optimization problem over this space of algorithms. This formulation led to the discovery of new algorithms with reduced operation count. Symbolic tools are presented for deriving and implementing algorithms.