Advertisement

Polynomial Filters

  • I. Pitas
  • A. N. Venetsanopoulos
Chapter
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 84)

Abstract

In many problems of digital signal processing it is necessary to introduce nonlinear systems. For example, it is well known that in detection and estimation problems, nonlinear filters arise in the case where the Gaussian assumption is not valid or the noise is not signal independent and/or additive. In the search for optimum signal processing systems, the task is to obtain general characterization procedures for nonlinear systems that retain at least a part of the simplicity that the impulse response method has for linear filters.

Keywords

Gaussian Random Variable Kernel Matrix Volterra Series Nonlinear Filter Volterra Kernel 
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.

References

  1. [1]
    M. Schetzen, The Volterra and Wiener theories of nonlinear systems, Wiley, 1980.Google Scholar
  2. [2]
    E. Biglieri, “Theory of Volterra processors and some applications”, in Proc. ICASSP-82, Paris, pp. 294–297, 1982.Google Scholar
  3. [3]
    S. Benedetto, E. Biglieri, “Nonlinear equalization of digital satellite chan- nels”, presented at the 9th AIAA Conf. Comm. Satellite Syst., San Diego, CA, March 1982.Google Scholar
  4. [4]
    D.D. Falconer, “Adaptive equalization of channel nonlinearities in QAM data transmission systems”, Bell Sys. Tech. J., vol. 57, pp. 2589–2611, Sept. 1978.zbMATHGoogle Scholar
  5. [5]
    G.L. Sicuranza, A. Bucconi, P. Mitri, “Adaptive echo cancellation with nonlinear digital filters”, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, San Diego, CA, pp. 3.10.1–3.10. 4, 1984.Google Scholar
  6. [6]
    O. Agazzi, D.G. Messerschmitt, D.A. Hodges, “Nonlinear echo cancellation of data signals”, IEEE Trans. Commun., vol. COM-30, pp. 24212433, Nov. 1982.Google Scholar
  7. [7]
    G.L. Sicuranza, G. Ramponi, “Distributed arithmetic implementation of nonlinear echo cancellers”, Proc. IEEE Int. Conf. on Acoust., Speech and Signal Processing, ICASSP-85, pp. 42.5.1–42.5.4, Tampa FL, 1985.Google Scholar
  8. [8]
    B.E.A. Saleh, “Optical bilinear transformation: General properties”, Optica Acta, vol. 26, no. 6, pp. 777–799, 1979.Google Scholar
  9. [9]
    G. Ramponi, “Quadratic filters for image enhancement”, Proc. Fourth European Signal Processing Conf., EUSIPCO-88, Grenoble, France, pp. 239–242, 1988.Google Scholar
  10. [10]
    G. Ramponi, “Enhancement of low-contrast images by nonlinear operators”, Alta Frequenza, vol. LVII, no. 7, pp. 451–455, Sept. 1988.Google Scholar
  11. [11]
    G. Ramponi, G. Sicuranza, “Quadratic digital filters for image processing”, IEEE Trans. on Acoust., Speech, Signal Processing, vol. ASSP-36, no. 6, pp. 937–939, June 1988.CrossRefGoogle Scholar
  12. [12]
    G. Ramponi, “Edge extraction by a class of second-order nonlinear filters”, Electronics Letters, vol. 22, no. 9, April 24, 1986.Google Scholar
  13. [13]
    R. Glavina, G. Ramponi, S. Cucchi, G. Sicuranza, “Interframe image coding with nonlinear prediction”, Tecnica Italiana, no. 2, 1988.Google Scholar
  14. [14]
    G.L. Sicuranza, G. Ramponi, “Adaptive nonlinear prediction of TV image sequences”, Electronics Letters, vol. 25, no. 8, pp. 526–527, April 13, 1989.Google Scholar
  15. [15]
    R. Glavina, S. Cucchi, G. Sicuranza, “Nonlinear interpolation of TV image sequences”, Electronics Letters, vol. 23, no. 15, pp. 778–780, July 16, 1987.Google Scholar
  16. [16]
    M.J. Hinich, D.M. Patterson, “Evidence of nonlinearity in daily stock returns”, Journal of Business and Economic Statistics, 3, pp. 69–77, 1985.Google Scholar
  17. [17]
    B. Picinbono, “Quadratic filters”, Proc. of IEEE ICASP, Paris, France, pp. 298–301, 1982.Google Scholar
  18. [18]
    H.H. Chiang, C.L. Nikias, A.N. Venetsanopoulos, “Efficient implementations of quadratic filters”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. ASSP-34, no. 6, pp. 1511–1528, Dec. 1986.CrossRefGoogle Scholar
  19. [19]
    G.L. Sicuranza, “Theory and realization of nonlinear digital filters”, in Proc. IEEE ISCAS-84, Montreal, Canada, May 1984.Google Scholar
  20. [20]
    G. Ramponi, G. Sicuranza, W. Ukovich, “An optimization approach to the design of nonlinear Volterra filters”, Proc. of EUSIPCO-86, The Hague, The Netherlands, 1986.Google Scholar
  21. [21]
    G. Ramponi, G. Sicuranza, W. Ukovich, “A computational method for the design of 2-D nonlinear Volterra filters”, IEEE Trans. on Circuits and Systems, vol. CAS-35, no. 9, Sept. 1988.Google Scholar
  22. [22]
    J. Katzenelson, L.A. Gould, “The design of nonlinear filters and control systems”, Inform. Contr., vol. 5, pp. 108–143, 1962.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    J.F. Barret, “The use of functionals in the analysis of nonlinear system”, J. Electron. Contr., vol. 15, no. 6, pp. 567–615, 1963.CrossRefGoogle Scholar
  24. [24]
    P. Eykhoff, “Some fundamental aspects of process-parameter estimation”, IEEE Trans. Automat. Contr., vol. AC-8, pp. 347–357, Oct. 1963.Google Scholar
  25. [25]
    A.V. Barakrishnan, “A general theory of nonlinear estimation problems in control systems”, J. Math. Anal. Appl., vol. 8, pp. 4–30, Feb. 1964.MathSciNetCrossRefGoogle Scholar
  26. [26]
    M. Schetzen, “Nonlinear system modeling based on the Wiener theory”, Proc. IEEE, vol. 69, pp. 1557–1573, Dec. 1981.CrossRefGoogle Scholar
  27. [27]
    A.S. French, E.G. Butz, “Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm”, Int. J. Contr., vol. 17, no. 3, pp. 529–539, 1973.zbMATHCrossRefGoogle Scholar
  28. Y.W. Lee, M. Schetzen, “Measurement of the Wiener kernels of a nonlinear system by cross-correlation”, Int. J. Contr.,vol. 2, no. 3, pp. 237254, 1965.Google Scholar
  29. [29]
    A.V. Barakrishnan, “On a class of nonlinear estimation problems”, IEEE Trans. Inform. Theory, vol. IT-10, pp. 314–320, Oct. 1964.Google Scholar
  30. [30]
    D.R. Brillinger, “The identification of polynomial systems by means of higher order spectra”, J. Sound Vib., vol. 12, no. 3, pp. 301–313, 1970.zbMATHCrossRefGoogle Scholar
  31. [31]
    M.J. Coker, D.N. Simkins, “A nonlinear adaptive noise canceller”, in Proc. 1980 IEEE Int. Conf. Acoust., Speech, Signal Processing, pp. 470473, 1980.Google Scholar
  32. [32]
    D.M. Mansour, A.H. Gray Jr., “Frequency domain non-linear filter”, in Proc. 1981 IEEE Int. Conf. Acoust., Speech, Signal Processing, pp. 550553, 1981.Google Scholar
  33. [33]
    S.R. Parker, F.A. Perry, “A discrete ARMA model for nonlinear system identification”, IEEE Trans. Circuits Syst., vol. CAS-28, pp. 224–233, March 1981.Google Scholar
  34. [34]
    T. Koh, E.J. Powers, “An adaptive nonlinear digital filter with lattice orthogonalization”, in Proc. 1983 IEEE Int. Conf. Acoust., Speech, Signal Processing, pp. 37–40, 1983.Google Scholar
  35. T. Koh, E.J. Powers, R.W. Miksad, F.J. Fischer, “Application of nonlinear digital filters to modeling low-frequency drift oscillations of moored vessels in random seas”, in Proc. 1984 Offshore Technol. Conf.,pp. 309314,1984.Google Scholar
  36. [36]
    S.R. Parker, “An autoregressive moving average (ARMA) discrete nonlinear model”, in Proc. IEEE Int. Symp. Circuits Syst., pp. 918–920, 1980.Google Scholar
  37. [37]
    S.Y. Fakhouri, “Identification of the Volterra kernels of nonlinear systems”, Proc. IEE, vol. 127, pt. D, pp. 296–304, Nov. 1980.Google Scholar
  38. [38]
    C.M. Nikias, M.R. Raghuveer, “Bispectrum estimation: A digital signal processing framework”, Proc. IEEE, vol. 75, pp. 869–891, July 1987.CrossRefGoogle Scholar
  39. [39]
    S.A. Diant, M.R. Raghuveer, “Estimation of the parameters of a second-order nonlinear system”, Int. Contr. on ACC, Baton Rouge, Louisiana, Oct. 1988.Google Scholar
  40. [40]
    V.Z. Marmarelis, D. Sheby, “Bispectrum analysis of weakly nonlinear quadratic systems”, in Proc. ASSP Spectrum Estimation and Modeling Workshop III, Boston, MA, pp. 14–16, Nov. 1986.Google Scholar
  41. [41]
    W.B. Davenport, W.L. Root, “An introduction to the theory of random signals and noise”, McGraw-Hill, New York, 1958.zbMATHGoogle Scholar
  42. [42]
    W.J. Lawless, M. Schwartz, “Binary signaling over channels containing quadratic nonlinearities”, IEEE Trans., vol. COM-22, pp. 288–298, 1974.Google Scholar
  43. [43]
    S. Benedetto, E. Biglieri, R. Daffara, “Performance of multilevel baseband digital systems in a nonlinear environment”, IEEE Trans. Commun., pp. 1166–1175, Oct. 1976.Google Scholar
  44. [44]
    S.A. Alshebeili, Linear and nonlinear modeling of non-Gaussian processes, Ph.D. Thesis, Department of Electrical Engineering, University of Toronto, in preparation.Google Scholar
  45. [45]
    G. Sicuranza, G. Ramponi, “Theory and realization of M-D nonlinear digital filters”, Proc. of IEEE Int. Conf. on Acoust., Speech, Signal Processing, Tokyo, Japan, April 7–11, 1986.Google Scholar
  46. [46]
    A.N. Venetsanopoulos, K.M. Ty, A.C.P. Loui, “High speed architectures for digital image processing”, IEEE Trans. on Circuits and Systems, vol. CAS-34, no. 8, pp. 887–896, Aug. 1987.CrossRefGoogle Scholar
  47. [47]
    G. Sicuranza, G. Ramponi, “Adaptive nonlinear digital filters using distributed arithmetic”, IEEE Trans. on Acoust., Speech, Signal Processing, vol. ASSP-34, no. 3, pp. 518–526, June 1986.CrossRefGoogle Scholar
  48. [48]
    G. Ramponi, G. Sicuranza, “Decision-directed nonlinear filter for image processing”, Electronics Letters, vol. 23, no. 23, pp. 1218–1219, Nov. 5, 1987.Google Scholar
  49. [49]
    V.G. Mertzios, G.L. Sicuranza, A.N. Venetsanopoulos, “Efficient structures for two-dimensional quadratic filters”, Photogrammetria, vol. 43, pp. 157–166, 1989.CrossRefGoogle Scholar
  50. [50]
    E.V.D. Eijnd, J. Scfionkens, J. Renneboog, “Parameter estimation in rational Volterra models”, IEEE International Symposium of Circuits and Systems, Philadelphia, PA, pp. 110–114, May 1987.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • I. Pitas
    • 1
  • A. N. Venetsanopoulos
    • 2
  1. 1.Aristotelian University of ThessalonikiGreece
  2. 2.University of TorontoCanada

Personalised recommendations