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2.1: Introduction

  • Page ID
    1965
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    Learning Objectives

    • A change of index variable or an index mapping is used to uncouple the calculations of the discrete Fourier transform (DFT).
    • This can result is a significant reduction in the required arithmetic and the resulting algorithm is called the fast Fourier transform (FFT).

    A powerful approach to the development of efficient algorithms is to break a large problem into multiple small ones. One method for doing this with both the DFT and convolution uses a linear change of index variables to map the original one-dimensional problem into a multi-dimensional problem. This approach provides a unified derivation of the Cooley-Tukey FFT, the prime factor algorithm (PFA) FFT, and the Winograd Fourier transform algorithm (WFTA) FFT. It can also be applied directly to convolution to break it down into multiple short convolutions that can be executed faster than a direct implementation. It is often easy to translate an algorithm using index mapping into an efficient program.

    Definition

    The basic definition of the discrete Fourier transform (DFT) is

    \[C(k)=\sum_{n=0}^{N-1}x(n)W_{N}^{nk} \nonumber \]

    where nn" role="presentation" style="position:relative;" tabindex="0">


    This page titled 2.1: Introduction is shared under a CC BY license and was authored, remixed, and/or curated by C. Sidney Burrus.

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