Description of spherically invariant random processes by means of g-functions

  • H. Brehm
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 969)


Gaussian Process Speech Signal Gaussian Case High Transcendental Function Concentric Ellipse 
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]
    Lord, R. D.: The Use of the Hankel Transform in Statistics. Biometrika 41 (1954), pp. 44–55.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Kingman, J.F.C.: Random Walks with Spherical Symmetry. Acta Math. (Stockholm) 109 (1963), pp. 11–53.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Vershik, A.M.: Some Characteristic Properties of Gaussian Stochastic Processes. Theory of Probability and its Applications, IX (1964), pp. 353–356.CrossRefzbMATHGoogle Scholar
  4. [4]
    Blake, I.F. and Thomas, J.B: On a Class of Processes Arising in Linear Estimation Theory. IEEE Trans. Information Theory, IT-14 (1968), pp. 12–16CrossRefzbMATHGoogle Scholar
  5. [5]
    McGraw, D.K. and Wagner, J.F: Elliptically Symmetric Distributions. IEEE Trans. Information Theory, IT 14 (1968), pp. 110–120.CrossRefGoogle Scholar
  6. [6]
    Picinbono, B: Spherically Invariant and Compound Gaussian Processes. IEEE Trans. Information Theory, IT-16 (1970), pp. 77–79.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Kingman, J.F.C.: On Random Sequences with Spherical Symmetry. Biometrika 59 (1972), pp. 492–494.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Yao, K.: A Representation Theorem and its Applications to Spherically-Invariant Random Processes. IEEE Trans. Information Theory, IT-19 (1973), pp. 600–607.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Bochner, S.: Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math. Annalen 108 (1933), pp. 399–408.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Bochner, S.: Completely Monotone Functions of the Laplace Operator for Torus and Sphere. Duke Math. J. 3 (1937), pp. 488–502.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Bochner, S.: Stable Laws of Probability and Completely Monotone Functions. Duke Math. J. 3 (1937), pp. 726–728.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Bochner, S.: Lectures on Fourier Integrals. Princeton University Press (1959)Google Scholar
  13. [13]
    Schoenberg, I.J.: On Certain Metric Spaces Arising from Euclidian Spaces by a Change of Metric and their Imbedding in Hilbert Space. Annals of Mathematics 38 (1937), pp. 787–793.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Schoenberg, I.J.: Metric Spaces and Completely Monotone Functions. Annals of Mathematics 39 (1938), pp. 811–841.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Schoenberg, I.J.: Metric Spaces and Positive Definite Functions. Trans. Am. Math. Soc. 44 (1938), pp. 522–536.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Widder, D.V.: Necessary and Sufficient Conditions for the Representation of a Function as a Laplace Integral. Trans. Am. Math. Soc. 33 (1931), pp. 851–892.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Picinbono, B. and Vezzosi, G.: Détection d'un Signal Certain dans un Bruit non Stationnaire et non Gaussien. Ann. Télécommunic. 25 (1970) pp. 433–439.zbMATHGoogle Scholar
  18. [18]
    Vezzosi, G. and Picinbono, B.: Détection d'un Signal Certain dans un Bruit Sphériquement Invariant, Structure et Characteristiques des Récepteurs. Ann. Télécommunic. 27 (1972), pp. 95–110.Google Scholar
  19. [19]
    Goldmann, J.: Statistical Properties of a Sum of Sinusoids and Gaussian Noise and its Generalization to Higher Dimensions. Bell Sys. Tech. J. 53 (1974), pp. 557–580.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Goldmann, J.: Detection in the Presence of Spherically Symmetric Random Vectors. IEEE Trans. Information Theory, IT-22 (1976), pp. 52–59.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Leung, H.M. and Cambanis, S.: On the Rate Distortion Functions of Spherically Invariant Vectors and Sequences. IEEE Trans. Information Theory, IT-24 (1978), pp. 367–373.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Brehm, H.: Sphärisch Invariante Stochastische Prozesse. Habilitationsschrift, Fachbereich Physik, Universität Frankfurt am Main (1978).Google Scholar
  23. [23]
    Wolf, D. and Brehm, H.: Experimental Studies on one-and two-dimensional Amplitude Probability Densities of Speech Signals. Proc. 1973 Int. Symp. on Inf. Theory, Ashkelon (Israel), IEEE-Cat. 73 CH 0753-4 IT, B4-6.Google Scholar
  24. [24]
    Brehm, H. and Wolf D.: Nth-Order Joint Probability Densities of Non-Gaussian Stochastic Processes. Proc. 1976 Int. Symp. on Inf. Theory, Ronneby (Schweden), IEEE-Cat. 76, CH 1095-9 IT, pp. 121–122.Google Scholar
  25. [25]
    Wolf, D. and Brehm H.: Mathematical Treatment of Speech Signals. Proc. 1977 Int. Symp. on Inf. Theory, Ithaca (USA), IEEE-Cat. 77, CH 1277-3 IT, E3, p. 105.Google Scholar
  26. [26]
    Wolf, D.: Analytische Beschreibung von Sprachsignalen. AEÜ 31 (1977), pp. 392–398.Google Scholar
  27. [27]
    Meijer, C.S.: Nieuw. Arch. Wisk. 18 (1936), pp. 10–39, Meijer, C.S.: Nederl. Akad. Wetensch. Proc. Ser. A 49 (1946), pp. 227–237, 344–356, 457–469, 632–641, 765–772, 936–943, 1063–1072 and 1165–1175.MathSciNetGoogle Scholar
  28. [28]
    Erdelyi, A. (ed.): Higher Transcendental Functions. Vol. I, II, III. McGraw-Hill Book Company, New York, Toronto, London, 1953.zbMATHGoogle Scholar
  29. [29]
    Luke, Y.: The Special Functions and their Approximations. Vol. I, II. Academic Press New York, San Francisco, London, 1969.zbMATHGoogle Scholar
  30. [30]
    Luke, Y.: Mathematical Functions and their Approximations. Academic Press, New York, San Francisco, London 1975.zbMATHGoogle Scholar
  31. [31]
    Papoulis, A.: Signal Analysis. McGraw-Hill Book Company, New York, etc., 1977.zbMATHGoogle Scholar
  32. [32]
    Oberhettinger, F. and Badii, L.: Tables of Laplace Transforms. Springer-Verlag, Berlin, Heidelberg, New York, 1973.CrossRefzbMATHGoogle Scholar
  33. [33]
    Abramowitz, M. and Stegun I. (ed.): Handbook of Mathematical Functions. Dover Publications, New York, 1965.Google Scholar
  34. [34]
    Oberhettinger, F.: Tables of Bessel Transforms. Springer-Verlag, Berlin, Heidelberg, New York, 1972.CrossRefzbMATHGoogle Scholar
  35. [35]
    Abut, H., Gray, R.M. and Rebolledo, G.: Vector Quantization of Speech and Speech-Like Waveforms. IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, (1982), pp. 423–435.CrossRefzbMATHGoogle Scholar
  36. [36]
    Berger, T.: Rate Distortion Theory. Prentice Hall, Englewood Cliffs, New Jersey, 1971.Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • H. Brehm
    • 1
  1. 1.Lehrstuhl für NachrichtentechnikUniversität Erlangen — NürnbergErlangen

Personalised recommendations