Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction

TL;DR

This paper introduces a new algebraic approach to develop fast, general-radix algorithms for polynomial transforms, including Fourier and cosine transforms, using induction on polynomial algebra modules.

Contribution

It presents a novel technique based on polynomial algebra representation theory to derive efficient $O(n\log n)$ algorithms for polynomial transforms.

Findings

Derived $O(n\log n)$ algorithms for DFT and DCT-4

Unified algebraic framework for polynomial transform algorithms

Enhanced computational efficiency for signal processing applications

Abstract

A polynomial transform is the multiplication of an input vector $x\in\C^n$ by a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list $b=\{p_0(x),\dots,p_{n-1}(x)\}$ and sample points $\alpha_k\in\C$ from a list $\alpha=\{\alpha_0,\dots,\alpha_{n-1}\}$. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. Important examples include the discrete Fourier and cosine transforms. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel $O(n\log{n})$ general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.