The Discrete Fourier Transform

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The Fourier transform is among the most widely used tools for transforming data sequences and functions, from the time domain to their representation in the frequency domain. Analysis of sequences in the frequency domain, can uncover important properties which are not readily observable in the time domain.

For the purpose of learning dynamical systems, based on discrete sequences of observations, we are particularly interested in the Discrete Fourier Transform. The DFT maps a discrete sequence in the time domain (observations) to a discrete sequence in the frequency domain (frequency coefficients).

The mini tutorial on the Fourier transform provides a quick introduction to the continuous and the discrete transforms, sampling, the FFT (Fast Fourier Transform), and some of the applications in which the DFT is used.

For further information on the Fourier transform, you may wish to consult any of the following books:

  
   The Fourier Transform and Its Applications, by R. Bracewell,
   McGraw-Hill, 1965.

   Digital Signal Processing, by A.V. Oppenheim and R.W. Schafer, 
   Prentice-Hall, 1975.

   Applications of Discrete and Continuous Fourier Analysis, 
   by H.J. Weaver, John Wiley & sons, 1983.

   Discrete Time Signal Processing, by A.V. Oppenheim and 
   R.W. Schafer, Prentice-Hall, 1989.
       
   Signals, Systems and Transforms, by L.B. Jackson,
   Addison-Wesley, 1990.

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