The Discrete Fourier Transform

K. Deergha Rao () and M. N. S. Swamy ()
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K. Deergha Rao: Vasavi College of Engineering (affiliated to Osmania University), Department of Electronics and Communication Engineering
M. N. S. Swamy: Concordia University, Department of Electrical and Computer Engineering

Chapter Chapter 4 in Digital Signal Processing, 2018, pp 163-240 from Springer

Abstract: Abstract The DTFT of a discrete-time signal is a continuous function of the frequency ( $$ \omega $$ ), and hence, the relation between $$ X\left( {\text{e}^{{j}\omega } } \right) $$ and $$ x(n) $$ is not a computationally convenient representation. However, it is possible to develop an alternative frequency representation called the discrete Fourier transform (DFT) for finite duration sequences. The DFT is a discrete-time sequence with equal spacing in frequency. We first obtain the discrete-time Fourier series (DTFS) expansion of a periodic sequence. Next, we define the DFT of a finite length sequence and consider its properties in detail. We also show that the DTFS represents the DFT of a finite length sequence. Further, evaluation of linear convolution using the DFT is discussed. Finally, some fast Fourier transform (FFT) algorithms for efficient computation of DFT are described.

Keywords: Discrete-time Fourier Series (DTFS); Linear Convolution; Finite Length Sequence; Circular Convolution; Inverse Discrete Fourier Transform (IDFT) (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-8081-4_4

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DOI: 10.1007/978-981-10-8081-4_4

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