Advertisement

Classical Methods of Magnitude Approximation

  • Belle A. Shenoi
Part of the Digital Signal Processing book series (DIGSIGNAL)

Abstract

Electric wave filters, called simply as filters in this book, are circuits that are used to selectively filter out some of the frequency components of an input signal and transmit the remaining components as the output signal in such a way that the quality of information contained in the signal is improved. We consider continuous-time signals, one-dimensional (1-D) and two-dimensional (2-D) discrete-time signals as the input and output of filters and characterize them in the frequency domain by their appropriate Fourier Transform. For example, if x (t) and y(t) are the continuous-time, input and output signals of a linear, time-invariant, continuous-time filter, their Fourier Transforms X () and Y() are related by the equation Y() = H()X() where H() is the Fourier Transform of the unit impulse response h(t) of the continuous-time filter. The relation can also be expressed in terms of these complex valued functions as
$$\left| {Y(j\omega )} \right|{e^{j\varphi }} = \left[ {\left| {H(j\omega )} \right|{e^{j\theta }}} \right]\left[ {|X(j\omega )|{e^{j\phi }}} \right] = \left| {H(j\omega )} \right|\left| {X(j\omega )} \right|{e^{j(\theta + \phi )}}$$
(1.1)
We see that the magnitude of the output as a function of frequency is |H (jω)| times the magnitude |X (jω)| of the input signal as a function of the frequency and the phase angle of the output signal is the phase angle of the input increased by that of H(). Therefore by properly shaping the magnitude of H() and the phase angle θ() as a function of frequency, we can design the filter to transmit some frequencies and block out other frequencies as we decide.

Keywords

Group Delay Digital Filter Magnitude Response Infinite Impulse Response Analog Filter 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, NY, 1965.Google Scholar
  2. [2]
    A. Antoniou, Digital Filters: Analysis, Design and Applications, McGraw-Hill, NY, 1993.Google Scholar
  3. [3]
    J.W. Sandler and B.L. Bardakjian, “LeastP th optimization of recursive digital filters,” IEEE Trans. on Audio and Electroacoustics, vol. 21, pp. 460–470, October 1973.CrossRefGoogle Scholar
  4. [4]
    J.W. Sandler and R.E. Seviora, “Current trends in network optimization,” IEEE Trans. on Microwave Theory and Techniques,MTT-18, pp. 1159–1170, December 1970.CrossRefGoogle Scholar
  5. [5]
    C.S. Burrus and T.W. Parks, “Time domain design of recursive digital filters,” IEEE Trans. on Audio and Electroacoustics, AU-18, pp. 137–141, 1970.CrossRefGoogle Scholar
  6. [6]
    C. Charalambous, “Acceleration of least p thalgorithm for minimax optimization with engineering applications,” Mathematical Programming, vol. 17, pp. 270–297, 1979.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Erich Christian and Egon Eisenmann, Design Tables and Graphs, John Wiley, NY, 1966.Google Scholar
  8. [8]
    A.G. Constantinides, “Spectral transformations for digital filters,” Proc. IEE, vol. 117 No. 8, pp. 1585–1590, August 1970.Google Scholar
  9. [9]
    S. Darlington, “A history of network synthesis and filter theory for circuits composed of resistors, inductors and capacitors,” IEEE Trans. on Circuits and Systems, CAS-31, pp. 3–12, January 1984.Google Scholar
  10. [10]
    R.W. Daniels, Approximation Methods for Electronic Filter Design, McGraw-Hill, 1974.Google Scholar
  11. [11]
    A.G. Evans and R. Fischl, “Optimal least squares time domain synthesis of recursive digital filters,” ibid, AU-21, pp. 61–65, 1973.Google Scholar
  12. [12]
    FILSYN Software for Filter Analysis and Design, DGS Associates, 1353 Santa Way, Santa Clara, CA 95051.Google Scholar
  13. [13]
    R.A. Friedenson et al, “RC-Active filters for the D-3 Channel Bank,” B.S.T.J. vol. 54, pp. 507–529, 1975.Google Scholar
  14. [14]
    A.H. Gray, Jr. and J.D. Markel, “A computer program for designing digital elliptic filters,” IEEE Trans. on Acoustics, Speech and Signal Processing, ASSP-24, pp. 529–538, June 1976.CrossRefGoogle Scholar
  15. [15]
    O. Heaviside, Electromagnetic Theory, 1893 (Reprinted by Dover Publications, New York, 1950).MATHGoogle Scholar
  16. [16]
    D.S. Humpherys,The Analysis, Design and Synthesis of Electrical Filters, Prentice-Hall, 1970.Google Scholar
  17. The Signal Processing Toolbox and MATLAB(R)-Software from The Mathworks Inc, Natick, MA.Google Scholar
  18. [18]
    S.K. Mitra and J.F. Kaiser (Eds.), Handbook for Digital Signal Processing, John Wiley&Sons, 1993.MATHGoogle Scholar
  19. [19]
    A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, Prentice-Hall, 1989.Google Scholar
  20. [20]
    A. Papoulis, “A new class of filters,” Proc.IRE, vol. 46, pp. 649–653, March 1958.CrossRefGoogle Scholar
  21. [21]
    T.W. Parks and C.S. Burrus, Digital Filter Design,John Wiley, NY, 1987.MATHGoogle Scholar
  22. [22]
    J.G. Proakis and D.G. Manolakis, Digital Signal Processing-Principles, Algorithms, and Applications, Prentice-Hall 1996.Google Scholar
  23. [23]
    M.I. Pupin, “Wave transmission over non-uniform cables and long distance air lines,” Trans. AIEE, vol. 17, pp. 445–507, 1900.Google Scholar
  24. [24]
    R. Schaumann, M.S. Ghausi and K.R. Laker, Design of Analog Filters(Appendix), Prentice-Hall, 1981.Google Scholar
  25. [25]
    A.S. Sedra and P.O. Bracket,Filter Theory and Design: Active and Passive,ISBS Inc, Forest Grove, OR, 1978.Google Scholar
  26. [26]
    C.E. Shannon, “Communication in the Presence of Noise,” Proc. IRE, pp. 10–12, January 1949.Google Scholar
  27. [27]
    A.K. Shaw, “Optimal identification of discrete-time systems from impulse response,” IEEE Trans. on Signal Processing, vol. 42, no. 1, pp. 113–120, January 1994.CrossRefGoogle Scholar
  28. [28]
    A.K. Shaw, “Optimal design of digital IIR filters by model-fitting frequency response data,” IEEE Trans. on Circuits and Systems, vol. 42, pp. 702–710, November 1995.CrossRefGoogle Scholar
  29. [29]
    G.C. Temes and J.W. LaPatra, Introduction to Circuit Synthesis and Design, McGraw-Hill, 1977.Google Scholar
  30. [30]
    G.C. Temes and D.Y.F. Zai, “Least pth approximation,” IEEE Trans. on Circuit Theory, CT-16, pp. 235–237, May 1969.CrossRefGoogle Scholar
  31. [31]
    W. Thomson, “On the theory of the electric telegraph,” Phil. Mag. vol. 11, pp. 146–160, 1856.Google Scholar
  32. [32]
    P.P. Vaidyanathan, “Optimal design of linear phase FIR digital filters with very flat passband and equiripple stopbands,” IEEE Trans. on Circuits and Systems, vol. CAS-32, pp. 904–907, September 1985.CrossRefGoogle Scholar
  33. [33]
    A.I. Zverev, Handbook of Filter Synthesis, John Wiley, NY, 1967.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Belle A. Shenoi
    • 1
  1. 1.Electrical Engineering DepartmentWright State UniversityDaytonUSA

Personalised recommendations