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The Discrete Fourier Transform

  • K. Deergha Rao
  • M. N. S. Swamy
Chapter

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.

References

  1. 1.
    J.W. Cooley, P.A.W. Lewis, P.D. Welch, Historical notes on the fast Fourier transform. IEEE Trans. Audio Electroacoust. 55(10), 1675–1677 (1967)Google Scholar
  2. 2.
    J.W. Cooley, P.A.W. Lewis, P.D. Welch, Historical notes on the fast Fourier transform. Proc. IEEE 55(10), 1675–1677 (1967)CrossRefGoogle Scholar
  3. 3.
    G. Goertzel, An algorithm for the evaluation of finite trignometric series. Am. Math Monthly 65, 34–35 (1958)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, 2nd edn. (Prentice Hall, 1999)Google Scholar
  5. 5.
    L. Rabiner, R. Schafer, C. Rader, IEEE Trans. Audio Electroacoust. 17(2) (1969)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Electronics and Communication EngineeringVasavi College of Engineering (affiliated to Osmania University)HyderabadIndia
  2. 2.Department of Electrical and Computer EngineeringConcordia UniversityMontrealCanada

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