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

TL;DR

This paper introduces an algebraic framework for deriving fast algorithms for DCTs and DSTs, generalizing the Cooley-Tukey approach, leading to new efficient computational methods.

Contribution

It develops a systematic algebraic methodology to derive fast algorithms for DCTs and DSTs, extending the Cooley-Tukey paradigm to these transforms.

Findings

Derived new fast algorithms for 16 DCTs and DSTs

Generalized Cooley-Tukey FFT principles to other transforms

Produced a large class of previously unknown algorithms

Abstract

This paper presents a systematic methodology based on the algebraic theory of signal processing to classify and derive fast algorithms for linear transforms. Instead of manipulating the entries of transform matrices, our approach derives the algorithms by 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 FFT and make the algorithms' derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast algorithms, many of which have not been found before.