Automatic generation of fast discrete signal transforms

This paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically. The class includes but is not limited to the discrete Fourier and the discrete trigonometric transforms. This is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms. Unlike previous methods, the algorithm is entirely automatic and uses the defining matrix as its sole input. The sparse matrix factorization algorithm consists of two steps: first, the "symmetry" of the matrix is computed in the form of a pair of group representations; second, the representations are stepwise decomposed, giving rise to a sparse factorization of the original transform matrix. We have successfully demonstrated the method by computing automatically efficient transforms in several important cases: for the DFT, we obtain the Cooley-Tukey (1965) FFT; for a class of transforms including the DCT, type II, the number of arithmetic operations for our fast transforms is the same as for the best-known algorithms. Our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist. The sparse matrix factorization algorithm is implemented within the software package AREP.