Algebraic Signal Processing Theory: Cooley–Tukey Type Algorithms for DCTs and DSTs

This paper presents a systematic methodology to derive and classify fast algorithms for linear transforms. The approach is based on the algebraic signal processing theory. This means that the algorithms are not derived by manipulating the entries of transform matrices, but by a stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebraic principles that generalize the well-known Cooley-Tukey fast Fourier transform (FFT) and make the algorithms' derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast general radix algorithms, many of which have not been found before.