Spectral Representation of Deterministic Signals: Fourier Series and Transforms


In this chapter we will take a closer look at the spectral, or frequency domain, representation of deterministic (nonrandom) signals which was already mentioned in Chapter 1. The tools introduced below, usually called Fourier or harmonic analysis, will play a fundamental role later on in our study of random signals. Almost all of the calculations will be conducted in the complex form. Compared with working in the real domain, manipulation of formulas written in the complex form turns out to be simpler and all the tedium of remembering various trigonometric formulas is avoided. All of the results written in the complex form can be translated quickly into results for real trigonometric series expressed in terms of sines and cosines via de Moivre’s formula e jt = cos t + j sin t, familiar from Chapter 1.


Fourier Series Discrete Fourier Transform Spectral Representation Periodic Signal Fourier Expansion 
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  1. 3.
    See, e.g., A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1959.MATHGoogle Scholar
  2. 4.
    Proofs of these two mathematical theorems and other results quoted in this section can be found in, e.g., T. W. Körner, Fourier Analysis, Cambridge University Press, Cambridge, UK, 1988.MATHGoogle Scholar
  3. 7.
    For a more complete exposition of the theory and applications of the Dirac delta and related “distributions”, see A. I. Saichev and W. A. Woyczyński, Distributions in the Physical and Engineering Sciences, Vol. 1: Distributional Calculus, Integral Transforms, and Wavelets, Birkhäuser Boston, Cambridge, MA, 1998.Google Scholar
  4. 8.
    J. W. Cooley and O. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comput., 19 (1965), 297–301.MATHCrossRefMathSciNetGoogle Scholar

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