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Aliaksei Sandryhaila, Jelena Kovacevic and Markus Püschel (SIAM Journal on Matrix Analysis and Applications, Vol. 32, No. 2, pp. 364-384, 2011)
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction
Preprint (232 KB)
Bibtex

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.

Keywords:
Algebraic signal processing theory: Current status, Theory of transform algorithms