Discrete Fourier transform (DFT) is a frequency domain representation of finite-length discrete-time signals. It is also used to represent FIR discrete-time systems in the frequency domain. As the name implies, DFT is a discrete set of frequency samples uniformly distributed around the unit circle in the complex frequency plane that characterizes a discrete-time sequence of finite duration. DFT is also intrinsically related to the DTFT, as we will see in this chapter. Because DFT is a finite set of frequency samples, it is a computational tool to perform filtering and related operations. There is an efficient algorithm known as the fast Fourier transform (FFT) to perform filtering of long sequences, power spectrum estimation, and related tasks. We will learn about the FFT in this chapter as well.

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  1. 1.
    Ahmed N, Natarajan T, Rao KR (1974) Discrete cosine transform. IEEE Trans on Computers C-23:90–93MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ansari R (1985) An extension of the discrete Fourier transform. IEEE Trans Circuits Sys CAS32:618–619CrossRefGoogle Scholar
  3. 3.
    Bagchi S, Mitra SK (1998) Nonuniform discrete Fourier transform and its signal processing applications. Kluwer Academic Publishers, NorwellGoogle Scholar
  4. 4.
    Bellanger M (2000) Digital processing of signals: theory and practice, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  5. 5.
    Blahut RE (1985) Fast algorithms for digital signal processing. Addison-Wesley, ReadingzbMATHGoogle Scholar
  6. 6.
    Cochran WT et al (1967) What is the fast Fourier transform. Proc IEEE 55(10):164–174CrossRefGoogle Scholar
  7. 7.
    Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19:297–301MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hadamard J (1893) Resolution d’une question relative aux determinants. Bull Sci Math Ser 2(17) Part I:240–246Google Scholar
  9. 9.
    Narasimha MJ, Peterson AM (1978) On the computation of the discrete cosine transform. IEEE Trans Comm COM-26(6):934–936CrossRefGoogle Scholar
  10. 10.
    Oraintara S, Chen Y-J, Nguyen TQ (2002) Integer fast Fourier transform. IEEE Trans Signal Process 50:607–618MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  • K. S. Thyagarajan
    • 1
  1. 1.Extension ProgramUniversity of California, San DiegoSan DiegoUSA

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