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The Discrete Fourier Transform

In: Digital Signal Processing

Author

Listed:
  • 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)

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.

Suggested Citation

  • K. Deergha Rao & M. N. S. Swamy, 2018. "The Discrete Fourier Transform," Springer Books, in: Digital Signal Processing, chapter 0, pages 163-240, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-8081-4_4
    DOI: 10.1007/978-981-10-8081-4_4
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