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7.1: Introduction

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    • To study the algebraic description of the DFT and on the algebraic derivation of the general-radix Cooley-Tukey FFT from Factoring the Signal Processing Operators.

    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). As one may expect, a similar connection exists for the DFT but bears surprises. Namely, it turns out that the proper framework for the DFT requires modulo operations of polynomials, which means working with so-called polynomial algebras. Associated with polynomial algebras is the Chinese remainder theorem, which describes the DFT algebraically and can be used as a tool to concisely derive various FFTs as well as convolution algorithms (see also Winograd’s Short DFT Algorithms). The polynomial algebra framework was fully developed for signal processing as part of the algebraic signal processing theory (ASP). ASP identifies the structure underlying many transforms used in signal processing, provides deep insight into their properties, and enables the derivation of their fast algorithms. Here we focus on the algebraic description of the DFT and on the algebraic derivation 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 appearance of modulo operations.

    The \(z\)-transform associates with infinite discrete signals \(X+(\ldots,x(-1),x(0),x(1),\dots)\) a Laurent series:

    \[X \mapsto X(s) = \sum_{n \in \mathbb{Z}} x(n)s^n \]

    Here we used \(s=z^{-1}\) to simplify the notation in the following. The DTFT of \(X\) is the evaluation of \(X(s)\) on the unit circle

    \[X(e^{-jw}), -\pi\lt\omega\le\pi\]

    Finally, filtering or (linear) convolution is simply the multiplication of Laurent series,

    \[X \mapsto X(s) = \sum_{n \in \mathbb{Z}}^{N-1} x(n)s^n \]

    and that the DFT is an evaluation of these polynomials. Indeed, the definition of the DFT in Winograd’s Short DFT Algorithms shows that

    \[ C(k) = X(W_N^k)=X\left(e^{-j\dfrac{2\pi k}{N}}\right),\; 0\le k \lt N\]

    i.e., the DFT computes the evaluations of the polynomial \(X(s)\) at the \(n\)th roots of unity.
    The problem arises with the equivalent of Equation, since the multiplication \(H(s)X(s)\) 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 solution to both problems is to reduce the product modulo \(s^n-1\):

    \[H*_{\text{circ}}X \leftrightarrow H(s)X(s)\,\text{mod}\,(s^n-1)\]

    Concept Infinite Time Finite Time
    Signal \(X(s)=\sum_{n\in\mathbb{z}}x(n)s^n\) \(\sum_{n=0}^{N-1}x(n)s^n\)
    Filter \(H(s)=\sum_{n\in\mathbb{z}}h(n)s^n\) \(\sum_{n=0}^{N-1}h(n)s^n\)
    Convolution \(H(s)X(s)\) \(H(s)X(s)\text{mod}(s^n-1)\)
    Fourier transform \(\text{DTFT:} X(e^{-j\omega}), -\pi\lt\omega\le\pi\) \(\text{DFT:} X\left(e^{-j\dfrac{2\pi k}{n}}\right), 0\le k\lt n\)

    Infinite and finite discrete time signal processing

    The resulting polynomial then has again degree \(N−1\) and this form of convolution becomes equivalent to circular convolution of the polynomial coefficients. We also observe that the evaluation points in Equation are precisely the roots of \(s^n-1\). This connection will become clear in this chapter.

    The discussion is summarized in Table.

    The proper framework to describe the multiplication of polynomials modulo a fixed polynomial are polynomial algebras. Together with the Chinese remainder theorem, they provide the theoretical underpinning for the DFT and the Cooley-Tukey FFT.

    In this chapter, the DFT will naturally arise as a linear mapping with respect to chosen bases, i.e., as a matrix. Indeed, the definition shows that if all input and outputs are collected into vectors \(X=(X(0),\ldots,X(N−1))\) and \(C=(C(0),\ldots C(N−1))\), then Winograd’s Short DFT Algorithms is equivalent to



    \[DFT_N=\left[W_N^{kn}\right]_{0\le k,n \lt N^.}\]

    The matrix point of view is adopted in the FFT books.


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