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Fast Computation of the DFT

  • D. SundararajanEmail author
Chapter
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Abstract

The frequency-domain analysis of signals and systems is efficient, in practice, due to the availability of fast algorithms for the computation of the DFT. While the DFT is defined for any length, practically efficient algorithms are available only for fast computation of the N-point DFT with N an integral power of 2. The algorithms are based on the classical divide-and-conquer strategy of developing fast algorithms. A N-point DFT is recursively decomposed into half-length DFTs, until the DFT becomes trivial. The DFTs of the smaller transforms are combined to form the DFT of the input data. The decomposition can start from the time-domain end or the frequency-domain end. The first type is called decimation-in-time (DIT) algorithms. The second type is called decimation-in-frequency (DIF) algorithms. These algorithms are developed assuming that the data is complex-valued. They can be tailored to suit real-valued data. A set of algorithms are derived and examples of computing the DFT are given.

Keywords

Sinusoids Complex exponentials Discrete Fourier transform Half-wave symmetry Butterfly computation Twiddle factor Decimation-in-frequency Decimation-in-time 

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Formerly at Department of Electrical and Computer EngineeringConcordia UniversityMontrealCanada

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