Mathematical Preliminaries

  • Jan W. M. Bergmans


Digital transmission and recording systems all convey discrete-time data sequences across channels with a continuous-time nature. Conversions between the discrete-time and continuous-time domains are key operations in such systems, as are linear filtering operations in both domains. It is, therefore, appropriate that we set out with an introduction to some of the mathematical concepts and tools that are central to all of these transformations. In doing so we will assume that the reader has a basic working knowledge of the theory of signal analysis in both domains. For this reason we recapitulate this theory only cursorily, aiming in part to familiarize the reader with our notation. The borderline between both domains, on the other hand, is relatively under-exposed in most textbooks on signal analysis and will therefore receive more detailed attention. Section 1.2 is concerned with deterministic signals and sequences, with an emphasis on the spectral effects of linear filtering, sampling and linear modulation. A similar account is given in Section 1.3 for stochastic sequences and signals. The remaining sections are concerned with various other subjects that are of interest to later chapters. These are probability density functions (Section 1.4), the arithmetic, geometric and harmonic averages of spectra (Section 1.5), properties of discrete-time minimum-phase functions (Section 1.6), and the method of Lagrange multipliers (Section 1.7). Appendices 1A and 1B provide a brief survey of properties of the Fourier transforms of continuous-time signals and discrete-time sequences, along with a summary of basic Fourier transform pairs.


Impulse Response Power Spectral Density Autocorrelation Function Decision Feedback Equalizer Spectral Effect 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W.B. Davenport and W.L. Root, Random Signals and Noise. New York: McGraw-Hill, 1958.MATHGoogle Scholar
  2. [2]
    J.L. Doob, Stochastic Processes. New York: Wiley, 1967.Google Scholar
  3. [3]
    A.W.M. van den Enden and N.A.M. Verhoeckx, Discrete-Time Signal Processing — An Introduction. Prentice-Hall International, U.K.: 1989.Google Scholar
  4. [4]
    A.J. Jerri, ‘The Shannon Sampling Theorem — Its Various Extensions and Applications: A Tutorial Review’, Proc IEEE, Vol. 65, pp. 1565–1596, 1977.MATHCrossRefGoogle Scholar
  5. [5]
    H. Nyquist, ‘Certain Topics in Telegraph Transmission Theory’, AIEE Trans., Vol. 47, pp. 617–644, Apr. 1928.Google Scholar
  6. [6]
    A.V. Oppenheim and R.W. Schafer, Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall, 1975.MATHGoogle Scholar
  7. [7]
    A.V. Oppenheim, A.S. Willsky and I.T. Young, Signals and Systems. Englewood Cliffs, N.J.: Prentice-Hall, 1983.MATHGoogle Scholar
  8. [8]
    J.G. Proakis, Digital Communications. New York: McGraw-Hill, 1983 (2nd ed. 1989 ).Google Scholar
  9. [9]
    A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill, 1965.MATHGoogle Scholar
  10. [10]
    A. Papoulis, Signal Analysis. New York: McGraw-Hill, 1984.Google Scholar
  11. [11]
    L. Rabiner and B. Gold, Theory and Application of Digital Signal Processing. Englewood Cliffs, N.J.: Prentice-Hall, 1975.Google Scholar
  12. [12]
    I.S. Sokolnikoff and R.M. Redheffer, Mathematics of Physics and Modern Engineering. New York: McGraw-Hill, 1958.MATHGoogle Scholar
  13. [13]
    Y.P. Tsividis and J.O. Voorman (Eds.), Integrated Continuous-Time Filters. New York: IEEE Press, 1993.Google Scholar
  14. [14]
    P. van der Wurf, ‘Statistical Analysis of Synchronous Random Pulse Trains by Means of Hybrid Correlation Functions’, in Proc. 11th Symposium on Information Theory in the Benelux, Noordwijkerhout, The Netherlands, Oct. 25/26, 1990, pp. 148–154.Google Scholar
  15. [15]
    J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering. New York: Wiley, 1965.Google Scholar
  16. [16]
    R.E. Ziemer, W.H. Tranter and D.R. Fannin, Signals and Systems: Continuous and Discrete. New York: MacMillan Publishing Co., 1983.Google Scholar
  17. [17]
    A.I. Zverev, Handbook of Filter Synthesis. New York: Wiley, 1967.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Jan W. M. Bergmans
    • 1
  1. 1.Philips ResearchEindhovenThe Netherlands

Personalised recommendations