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by Markus Pueschel, Carnegie Mellon University
In infinite, or non-periodic, discrete-time signal processing, there is a strong connection between the $z$ -transform, Laurent series, convolution, and the discrete-time Fourier transform (DTFT) [link] . As one may expect, a similar connection exists for the DFT but bears surprises. Namely, it turns out that the properframework for the DFT requires modulo operations of polynomials, which means working with so-called polynomial algebras [link] . Associated with polynomial algebras is the Chinese remainder theorem, which describes the DFT algebraically and can beused as a tool to concisely derive various FFTs as well as convolution algorithms [link] , [link] , [link] , [link] (see also Winograd’s Short DFT Algorithms ). The polynomial algebra framework was fully developed for signal processing as part of thealgebraic signal processing theory (ASP). ASP identifies the structure underlying many transforms used in signal processing, provides deepinsight into their properties, and enables the derivation of their fast algorithms [link] , [link] , [link] , [link] . Here we focus on the algebraic description of the DFT and on the algebraicderivation of the general-radix Cooley-Tukey FFT from Factoring the Signal Processing Operators . The derivation will make use of and extend the Polynomial Description of Signals . We start with motivating the appearanceof modulo operations.
The $z$ -transform associates with infinite discrete signals $X=(\cdots ,x(-1),x(0),x(1),\cdots )$ a Laurent series:
Here we used $s={z}^{-1}$ to simplify the notation in the following. The DTFT of $X$ is the evaluation of $X\left(s\right)$ on the unit circle
Finally, filtering or (linear) convolution is simply the multiplication of Laurent series,
For finite signals $X=\left(x\right(0),\cdots ,x(N-1\left)\right)$ one expects that the equivalent of [link] becomes a mapping to polynomials of degree $N-1$ ,
and that the DFT is an evaluation of these polynomials. Indeed, the definition of the DFT in Winograd’s Short DFT Algorithms shows that
i.e., the DFT computes the evaluations of the polynomial $X\left(s\right)$ at the $n$ th roots of unity.
The problem arises with the equivalent of [link] , since the multiplication $H\left(s\right)X\left(s\right)$ of two polynomials of degree $N-1$ yields one of degree $2N-2$ . Also, it does not coincide with the circular convolution known to be associated with the DFT. The solutionto both problems is to reduce the product modulo ${s}^{n}-1$ :
Concept | Infinite Time | Finite Time |
Signal | $X\left(s\right)={\sum}_{n\in \mathbb{Z}}x\left(n\right){s}^{n}$ | ${\sum}_{n=0}^{N-1}x\left(n\right){s}^{n}$ |
Filter | $H\left(s\right)={\sum}_{n\in \mathbb{Z}}h\left(n\right){s}^{n}$ | ${\sum}_{n=0}^{N-1}h\left(n\right){s}^{n}$ |
Convolution | $H\left(s\right)X\left(s\right)$ | $H\left(s\right)X\left(s\right)\text{mod}({s}^{n}-1)$ |
Fourier transform | $\text{DTFT:}\phantom{\rule{4.pt}{0ex}}X\left({e}^{-j\omega}\right),\phantom{\rule{1.em}{0ex}}-\pi <\omega \le \pi $ | $\text{DFT:}\phantom{\rule{4.pt}{0ex}}X\left({e}^{-j\frac{2\pi k}{n}}\right),\phantom{\rule{1.em}{0ex}}0\le k<n$ |
The resulting polynomial then has again degree $N-1$ and this form of convolution becomes equivalent to circular convolution of thepolynomial coefficients. We also observe that the evaluation points in [link] are precisely the roots of ${s}^{n}-1$ . This connection will become clear in this chapter.
The discussion is summarized in [link] .
The proper framework to describe the multiplication of polynomials modulo a fixed polynomial are polynomial algebras. Together with theChinese remainder theorem, they provide the theoretical underpinning for the DFT and the Cooley-Tukey FFT.
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