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A Tutorial on Non-Parametric Bilinear Time-Frequency Signal Representations

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Time and Frequency Representation of Signals and Systems

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 309))

Abstract

Nonstationary signals have a time-dependent spectral content. This is in contrast to stationary signals whose energy spectrum characterizes their spectral content and that is independent of time. Therefore, nonstationary signals require joint time—frequency representations.

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References

  • Wigner, E.P., On the quantum correction for thermo-dynamic equilibrium, Phys. Rev. 40, 749–759. 1932

    Article  Google Scholar 

  • Gabor, D., Theory of communication, J. IEE (London) 93 (III) (November), 429–457. Koenig, R., H.K. Dunn and L.Y. Lacy, The sound spectrograph, J. Acoust. Soc. Amer. 18, 19–49. 1946

    Google Scholar 

  • Potter, R.K., G.A. Kopp and H.C. Green, Visible Speech ( Van Nostrand, New York). Republished by Dover Publications, 1966. 1947

    Google Scholar 

  • Ville, J., Théorie et applications de la notion de signal analytique, Câbles et Transmission 2A(1), 61–74 [Engl. Transl.: I. Selin, Rand Corp. Report T-92, August 1, 1958. 1948

    Google Scholar 

  • Moyal, J.E., Quantum mechanics as a statistical theory. Proc. Camb. Phil. Soc. 45 (January), 99–124. 1949

    Article  MathSciNet  MATH  Google Scholar 

  • Fano, R.M., Short-time autocorrelation functions and power spectra, J. Acoust. Soc. Amer. 22(5) (September), 546–550. 1950

    Google Scholar 

  • Page, C.H., Instantaneous power spectra, J. Appl. Phys. 23. 103–106. 1952

    MATH  Google Scholar 

  • Woodward, P.M., Probability and information theory with application to radar ( Pergamon Press, London ). 1953

    Google Scholar 

  • Blanc-Lapierre, A., and B. Picinbono, Remarques sur la notion du spectre instantané de puissance, Publ. Sci. Univ. d’Alger, Série B1, 2–32. 1955

    Google Scholar 

  • More, H., I. Oppenheim and J. Ross, Some topics in quantum statistics: the Wigner function and transport theory, in: Studies in Statistical Mechanics, ed. G.E. Uhlenbeck, ( North-Holland Publishing, Amsterdam ) pp. 213–298. 1962

    Google Scholar 

  • Schroeder, M.R., and B.S. Atal, Generalized short-time power spectra and auto-correlation function, J. Acoust. Soc. Amer. 34(11), (November), 1679–1683. 1964

    Google Scholar 

  • Levin, M.J., Instantaneous power spectra and ambiguity function. IEEE Trans. Inf. Theory IT-10, 95–97. 1966

    Google Scholar 

  • Cohen, L., Generalized phase-space distribution functions, J. Math. Phys. 7, 781–786. 1966

    Article  Google Scholar 

  • Helström, C.W., An expansion of a signal into Gaussian elementary signals. IEE Trans. Inf. Theory IT-12, 81–82. 1966

    Google Scholar 

  • DeBruijn, N.G., Uncertainty principles in Fourier analysis, in: Inequalities, ed. O. Shisha ( Academic Press, New York ) pp. 57–71. 1967

    Google Scholar 

  • Levin, M.J., Instantaneous spectra and ambiguity functions, IEEE Trans. Inf. Theory IT-13, 95–97. 1967

    Google Scholar 

  • Margenau, H., and L. Cohen. Probabilities in quantum mechanics, in: Quantum theory and reality, ed. M. Bunge (Springer, Berlin) ch. 4, pp. 71–89. 1967

    Chapter  Google Scholar 

  • Montgomery, L.K., and I.S. Reed, A generalization of the Gabor-Helström transform. IEEE Trans. Inf. Theory IT-13, 344–345. 1968

    Google Scholar 

  • Bonnet, G., Considérations sur la représentation et l’analyse harmonique des signaux déterministes ou aléatoires, Ann. Télécom. 23 (3–4), 62–86. 1968

    MathSciNet  MATH  Google Scholar 

  • Rihaczek, A.W., Signal energy distribution in time and frequency, IEEE Trans. Inf. Theory IT-14, 369–374. 1968

    Google Scholar 

  • Ackroyd, M.H., Short-time spectra and time—frequency energy distributions, J. Acoust. Soc. Amer. 50 (5), 1229–1231. 1970

    Google Scholar 

  • Ackroyd, M.H., Instantaneous and time-varying spectra — An introduction, Radio Electron. Eng. 39, 145–152. 1970

    Google Scholar 

  • Ackroyd, M.H., Instantaneous spectra and instantaneous frequency, Proc. IEEE 58 (1). p. 141. 1970

    Google Scholar 

  • Mark, W.D., Spectral analysis of the convolution and filtering of non-stationary stochastic processes, J. Sound Vib. 11(1), 19–63. 1970

    Google Scholar 

  • Oppenheim, A.V., Speech spectrograms using the fast Fourier transform, IEEE Spectrum 7 (August), 57–62. 1970

    Article  Google Scholar 

  • Ruggeri, G.J., On phase-space description of quantum mechanics, Prog. Theor. Phvs. 46 (6), 1703–1712. 1971

    Article  MathSciNet  MATH  Google Scholar 

  • Escudié, B., Représentation temps—fréquence dans l’analyse et la synthèse des signaux. in: 6è Congrès International de Cybernétique, pp. 277–299. 1972

    Google Scholar 

  • Levshin, A., V.F. Pisarenko and G.A. Pogrebinsky, On a frequency—time analysis of oscillations, Ann. Geophys. 28, 211–218. 1972

    Google Scholar 

  • Nelson, G.A., Signal analysis in time and frequency using Gaussian wavefunctions. in: NATO Advanced Studies Institute on Network and Signal Theory, ed. J.O. Scalan (September 1972) pp. 454–460. 1972

    Google Scholar 

  • DeBruijn, N.G., A theory of generalized functions with applications to Wigner distribution and Weyl correspondence, Nieuw Arch. Wiskunde 21. 205–280. 1973

    MathSciNet  Google Scholar 

  • Lacoume, J.L., and W. Kofman. Description des processus non-stationnaires par la représentation temps—fréquence. 56 Coll. Traitement du Signal et applications GRETSI, 14/1–14/7 (Nice). 1975

    Google Scholar 

  • Escudié, B., and J. Gréa, Sur une formulation générale de la représentation en temps et fréquence dans l’analyse de signaux d’énergie finie, C. R. Acad. Sci. Paris, série A 283, 1049–1051. 1976

    Google Scholar 

  • Kodera, K., C. DeVilledary and R. Gendrin. A new method for the numerical analysis of non-stationary signals, Phys. Earth Planet. Inter. 12, 142–150. 1976

    Google Scholar 

  • Krüger, J.G., and A. Poffyn, Quantum mechanics in phase space. Part I — Unicity of the Wigner distribution function. Physica 85A. 84–100. 1976

    Article  MathSciNet  Google Scholar 

  • Tjostheim, D., Spectral generating operators for non-stationary processes. Adv. Appl. Prob. 8. 831–846. 1976

    Article  MathSciNet  MATH  Google Scholar 

  • Allen, J.B., and L.R. Rabiner, A unified approach to short-time Fourier analysis and synthesis, Proc. IEEE 65, 1558–1564. 1977

    Article  Google Scholar 

  • Berry, M.V., Semi-classical mechanics in phase space: a study of Wigner’s function, Phil. Trans. Roy. Soc. A 287, 237. 1977

    Article  MATH  Google Scholar 

  • Escudié, B., and J. Gréa, Représentation Hilbertienne et représentation conjointe en temps et fréquence des signaux d’énergie finie, Coll. GRETSI, Nice, France (April 1977). 1977

    Google Scholar 

  • Royer, A., Wigner distribution as the expectation value of a parity operator, Phys. Rev. A15(2), 449, 450. 1977

    Google Scholar 

  • Bastiaans, M.J., The Wigner distribution function applied to optical signal and systems, Opt. Commun. 25, 26–30. 1978

    Article  Google Scholar 

  • Kodera, K., R. Gendrin and C. DeVilledary, Analysis of time-varying signals with small BT values, IEEE Trans. Acoust., Speech Signal Process. ASSP-26(1), 64–76. 1978

    Google Scholar 

  • Melard, G., Propriétés du spectre évolutif d’un processus non-stationnaire, Ann. Inst. Henri Poincaré B XIV (4), 411–424. 1978

    MathSciNet  Google Scholar 

  • Bastiaans, M.J., Wigner distribution function and its application to first order optics, J. Opt. Soc. Amer. 69, 1710–1716. 1979

    Google Scholar 

  • Bastiaans, M.J., The Wigner distribution function and Hamiltonian’s characteristics of a geometrical-optical system, Opt. Commun. 30 (3), 321–326. 1979

    Article  Google Scholar 

  • Bastiaans, M.J., Transport equations for the Wigner distribution function. Opt. Acta 26, 1256–1272. 1979

    MathSciNet  Google Scholar 

  • Bastiaans, M.J., Transport equations for the Wigner distribution function in inhomogeneous and dispersive media, Opt. Acta 26, 1334–1344. 1979

    MathSciNet  Google Scholar 

  • Bouachache, B., B. Escudié, P. Flandrin and J. Gréa, Sur une condition nécessaire et suffisante de positivité de la représentation conjointe en temps et fréquence des signaux d’énergie finie, C. R. Acad. Sci. Paris, Série A 288, 307–309. 1979

    Google Scholar 

  • Bouachache, B., B. Escudié and J.M. Komatitsch, Sur la possibilité d’utiliser la représentation conjointe en temps et fréquence dans l’analyse des signaux modulés en fréquence émis en vibrosismique, 7ème Coll. GRETSI sur le Traitement du Signal et Applications. Nice, France (1979) pp. 121/1–121/6. 1979

    Google Scholar 

  • Bremmer, H., The Wigner distribution and transport equations in radiation problems, J. Appl. Soc. Eng. A 3, 251–260. 1979

    Google Scholar 

  • Escudié, B., “Représentation en temps et fréquence des signaux d’énergie finie: analyse et observation des signaux. Ann. Télécommun. 34 (3–4), 101–111. 1979

    Google Scholar 

  • Escudié, B., P. Flandrin and J. Gréa. Positivité des représentations en temps et fréquence des signaux d’énergie finie, représentation Hilbertienne et conditions d’observation des signaux, 7ème Coll. GRETSI sur le Traitement du Signal et Applications, Nice, France (1979) pp. 28/5–2/6/79. 1979

    Google Scholar 

  • Fargetton, H., Fréquences instantanées de signaux multicomposantes, Thèse (Grenoble). 1979

    Google Scholar 

  • Gendrin, R., and C. DeVilledary, Unambiguous determination of fine structures in multicomponent time-varying signals, Ann. Télécommun. 35 (3–4), 122–130. 1979

    Google Scholar 

  • Janssen, A.J.E.M., Application of Wigner distributions to harmonic analysis of generalized stochastic processes, MC-tract 114 (Mathematisch Centrum, Ansterdam). 1979

    Google Scholar 

  • Rabiner, L.R., and J.B. Allen, Short-time Fourier analysis techniques for FIR system identification and power spectrum estimation, IEEE Trans. Acoust., Speech Signal Process. ASSP-27. 1979

    Google Scholar 

  • Wigner, D.P.; Quantum-mechanical distribution functions revisited, in: Perspectives in quantum theory, eds. W. Yourgran and A. van de Merwe ( Dover, New York ). 1979

    Google Scholar 

  • Altes, R.A., Detection, estimation and classification with spectrograms. J. Acoust. Soc. Am. 67 (4), 1232–1246. 1980

    Google Scholar 

  • Bastiaans, M.J., Gabor’s expansion of the signal into Gaussian elementary signals, Proc. IEEE 68 (4) p. 538. 1980

    Article  Google Scholar 

  • Bastiaans, M.J., Wigner distribution function display: a supplement to ambiguity display using a single 1-D Input, Appl. Opt. 19 (2), 192. 1980

    Google Scholar 

  • Bartelt, H.O., K-H. Brenner and A.W. Lohmann, The Wigner distribution function and its optical production, Opt. Commun. 32, 32–38. 1980

    Article  Google Scholar 

  • Claasen; T.A.C.M.. and W.F.G. Mecklenbräuker, The Wigner distribution — a tool for time—frequency signal analysis — Part I: Continuous-time signals, Philips J. Res. 35 (3), 217–250. 1980

    MathSciNet  Google Scholar 

  • Claasen, T.A.C.M., and W.F.G. Mecklenbräuker, The Wigner distribution — a tool for time—frequency signal analysis — Part II: Discrete-time signals, Philips J. Res. 35 (4/5), 276–300. 1980

    MATH  Google Scholar 

  • Claasen, T.A.C.M., and W.F.G. Mecklenbräuker, The Wigner distribution — a tool for time—frequency signal analysis — Part III: Relations with other time—frequency signal transformations, Philips J. Res. 35 (6), 372–389. 1980

    MathSciNet  MATH  Google Scholar 

  • Escudié, B., and P. Flandrin, Sur quelques propriétés de la représentation conjointe et de la fonction d’ambiguité des signaux d’énergie finie, C. R. Acad. Sci. Paris A 291, 171–174. 1980

    MATH  Google Scholar 

  • Flandrin, P., Application de la théorie des catastrophes a l’étude du comportement asymptotique de la représentation conjointe de Wigner—Ville, DEA, CEPHAG, Grenoble, 1980. 1980

    Google Scholar 

  • Flandrin, P., and B. Escudié, Time and frequency representation of finite energy signals: a physical property as a result of a Hilbertian condition, Signal Process. 2. 93–100. 1980

    Article  MathSciNet  Google Scholar 

  • Marcuvitz, N., Quasiparticle view of wave propagation, Proc. IEEE, 68, pp. 1380–1395. 1980

    Article  Google Scholar 

  • Portnoff, M.R., Time—Frequency Representation of Digital Signals and Systems based on Short-time Fourier Analysis, IEEE Trans. Acoust., Speech Signal Process. ASSP-28(1), pp. 55–69. 1980

    Google Scholar 

  • Bastiaans, M.J., The Wigner distribution function and its applications to optics, in: Optics in Four Dimensions, ed. L.M. Narducci ( American Institute of Physics, New York ) pp. 292–312. 1981

    Google Scholar 

  • Bastiaans, M.J., A sampling theorem for the complex spectrogram and Gabor expansion of a signal in Gaussian elementary signals, Opt. Eng. 20(4), 594–598; in: Proc. Int. Opt. Computing Conf., Washington, D.C., 8–11 April, 1980. 1981

    Google Scholar 

  • Bertrand, P., and C. Fugier-Garrel, Formulation de la théorie de la communication dans le plan temps—frequence — aspects practiques,“ 8ème Coll. GRETSI, Nice (1981), pp. 829–834. 1981

    Google Scholar 

  • Claasen, T.A.C.M., and W.F.G. Mecklenbräuker, Time-frequency signal analysis by means of the Wigner distribution, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, Atlanta, CA (1981), pp. 69–72. 1981

    Google Scholar 

  • Fargetton, H., R. Gendrin and J.L. Lacoume, Adaptive methods for spectral analysis of time-varying signals, in: Signal Processing: Theories and Applications — II, (Proc. EUSIPCO-80 Conf., Lausanne, September 1980) eds. M. Kunt and F. De Coulon (North-Holland, Amsterdam) pp. 787–792. 1981

    Google Scholar 

  • Flandrin, P., and B. Escudié, “Géometrie des fonctions d’ambiguité et des représentations conjointes de Ville: l’approche de la théorie des catastrophes, 8ème Coll. GRETSI, pp. 69–74, Nice (1981). 1981

    Google Scholar 

  • Flandrin, P., B. Escudié, J. Gréa and Y. Biraud, Joint representation and Hilbertian analysis: “elementary” and “asymptotic” finite energy signals, Proc. EUSIPCO-80 Conf., Lausanne (1980), eds. M. Kunt and F. de Coulon ( North-Holland, Amsterdam ) pp. 25–26. 1981

    Google Scholar 

  • Grace, O.D., Instantaneous power spectra, J. Acoust. Soc. Amer. 69 (1), 191–198. 1981

    Google Scholar 

  • Janssen, A.J.E.M., Positivity of weighted Wigner distributions, SIAM J. Math. Anal. 12 (5). 752–758. 1981

    MathSciNet  MATH  Google Scholar 

  • Priestly, M.B., Spectral analysis and time series, Vols. I and II (Academic Press, New York). 1981

    Google Scholar 

  • Szu, H.H., and J.A. Blodgett, Wigner distribution and ambiguity function. in: Optics in Four Dimensions, ed. L.M. Narducci ( American Institute of Physics. New York ) pp. 355–381. 1981

    Google Scholar 

  • Bouachache. B., Représentation temps—fréquence — Application à la mesure de l’absorption du sous-sol, Thèse D. I. (INPG, Grenoble). 1982

    Google Scholar 

  • Bouachache, B., and P. Flandrin, Wigner—Ville analysis of time-varying signals. Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP82, Paris. France (May 1982) pp. 1329–1333. 1982

    Google Scholar 

  • Chan. D.S.K., A non-aliased discrete-time Wigner distribution for time—frequency signal analysis. Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing. ICASSP82. Paris, France (May 1982) pp. 1333–1336. 1982

    Google Scholar 

  • Chester. D.B., The Wigner distribution and its application to speech recognition and analysis. Ph.D. Dissertation (University of Cincinnati, OH). 1982

    Google Scholar 

  • Flandrin. P., Représentation des signaux dans le plan temps—fréquence. Thèse D. I. (Grenoble). 1982

    Google Scholar 

  • Flandrin, P., and B. Escudié, Sur la localisation des représentations conjointes dans le plan temps—fréquence, C. R. Acad. Sci. Paris, série I 295(7), 475–478, presented by A. Blanc-Lapierre. 1982

    Google Scholar 

  • Flandrin, P., B. Escudié and J. Gréa, Applications of operators and deformation techniques to time—frequency problems, IEEE ISIT, Les Arcs. 1982.

    Google Scholar 

  • Jacobson, L., and H. Wechsler, A paradigm for invariant object recognition of brightness. optical flow and binocular disparity images, Pattern Recognition Lett. 1. 61–68. 1982

    Article  Google Scholar 

  • Jacobson, L., and H. Wechsler, The Wigner distribution and its usefulness for 2-D image processing, Sixth Int. Joint Conference on Pattern Recognition, Munich, FRG (October 19–22, 1982 ).

    Google Scholar 

  • Janssen, A.J.E.M., On the locus and spread of pseudo-density functions in the time-frequency plane, Philips J. Res. 37, 79–110.

    Google Scholar 

  • Martin, W., Time-frequency analysis of non-stationary processes, IEEE-ISIT, Les Arcs, 1982.

    Google Scholar 

  • Martin, W.. Time-frequency analysis of random signals, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP82, Paris, France (May 1982) pp. 1325–1328.

    Google Scholar 

  • Nawab, S.H., T.F. Quatieri and J.S. Lim, Signal reconstruction from short-time Fourier transform magnitude, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP82, Paris, France (May 1982) pp. 1046–1048.

    Google Scholar 

  • Szu, H.H. Two-dimensional optical processing of one-dimensional acoustic data, Opt. Eng. 21. 804–813. 1982

    Google Scholar 

  • Bastiaans, M.J., Signal description by means of a local frequency spectrum, doctoral thesis (Technical University, Eindhoven, The Netherlands ). 1983

    Google Scholar 

  • Bouachache, B., Wigner-Ville analysis of time-varying signals: an application in seismic prospecting, EUSIPCO-83: Second European Association for Signal Processing Conference, Erlangen, FRG (September 12–16, 1983 ).

    Google Scholar 

  • Boudreaux-Bartels, G.F., Time-frequency signal processing algorithms: analysis and synthesis using Wigner distributions. Ph.D. Dissertation (Rice University, Houston. TX). 1983

    Google Scholar 

  • Boudreaux-Bartels. G.F., and T.W. Parks. Reducing aliasing in the Wigner distribution using implicit spline interpolation, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP83, Boston, MA (April 1983) pp. 1438–1441.

    Google Scholar 

  • Brenner, K.H., A discrete version of the Wigner distribution function, EUSIPCO-83. Second European Association for Signal Processing Conference, Erlangen, FRG (September 12–16, 1983 ).

    Google Scholar 

  • Chester, D.. F. Taylor and M. Doyle, On the Wigner distribution, Proc. IEEE Int. Conf. on Acoustics. Speech and Signal Processing ICASSP83, Boston. MA (April 1983) Vol. 2. pp. 491–494.

    Google Scholar 

  • Chester, D., F.J. Taylor and M. Doyle, Application of the Wigner distribution to speech processing, IEEE Acoustics. Speech and Signal Processing, Spectrum Estimation Workshop II. Tampa. Florida (November 10–11. 1983 ) pp. 98–102.

    Google Scholar 

  • Claasen. T.A.C.M.. and W.F.G. Mecklenbräuker, The aliasing problem in discrete-time Wigner distributions. IEEE Trans. Acoust., Speech Signal Process. ASSP-31. 1067–1072. 1983

    Google Scholar 

  • Flandrin. P., and W. Martin, Sur les conditions physiques assurant l’unicité de la représentation de Wigner-Ville comme représentation temps-fréquence, Neuvième Colloque sur là Traitement du Signal et ses Applications, pp. 43–49 (May 1983).

    Google Scholar 

  • Flandrin. P.. and W. Martin, Pseudo-Wigner estimators, IEEE ASSP Spectrum Estimation Workshop II, Tampa, Florida (November 10–11, 1983 ) pp. 181–185.

    Google Scholar 

  • Grenier. Y., Time-dependent ARMA modeling of non-stationary signals, IEEE Trans. Acoust. Speech Signal Process. ASSP-31, pp. 899–911. 1983

    Google Scholar 

  • Grenier, Y., and D. Aboutajdine, Comparison des représentations temps—fréquence de signaux présentant des discontinuités spectrales, Ann. Telecom. 38(11–12), 429–442.

    Google Scholar 

  • Griffin, D.W. and J.S., Lim, Signal estimation from modified short-time Fourier transform, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP83, Boston, MA (April 1983) pp. 804–807.

    Google Scholar 

  • Jacobson, L., and H. Wechsler, The composite pseudo Wigner distribution (CPWD): A computable and versatile approximation to the Wigner distribution (WD), Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP83, Boston, MA (April 1983) pp. 254–256.

    Google Scholar 

  • Janse, C.P. and A.J.M. Kaizer, Time—frequency distributions of loudspeakers: the application of the Wigner distribution, J. Audio Eng. Soc. 31(4), 198–223.

    Google Scholar 

  • Kraus, G., Eine Systematik der linearen, bilinearen und quadratischen Analyse deterministischer Signale, Arch. Elektron. Uebertragungstech. 37, Heft 5/6, 160–164.

    Google Scholar 

  • Martin, W., and P. Flandrin, Analysis of non-stationary processes: short-time periodo- grams versus a pseudo-Wigner estimator, EUSIPCO’83, ( North-Holland, Amsterdam ). 1983

    Google Scholar 

  • Nawab, S.H., T.F. Quatieri and J.S. Lim, Algorithms for signal reconstruction from short-time Fourier transform magnitude, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP83, Boston, MA (April 1983) pp. 800–803.

    Google Scholar 

  • O’Connell, R.F., The Wigner distribution function — 50th birthday, Found. Phys. 13 (1), 83–92. 1983

    Google Scholar 

  • Taylor, F., M. Doyle and D. Chester, On the Wigner distribution, Proc. IEEE Im. Conf. on Acoustics, Speech and Signal Processing ICASSP83, Boston, MA (April 1983) pp. 491–494.

    Google Scholar 

  • Auslander. L., and R. Rolimieri, Characterizing the radar ambiguity functions, IEEE Trans. Inf. Theory IT-30(6), 832–836. 1984

    Google Scholar 

  • Bouachache. B., and F. Rodriguez. Recognition of time-varying signals in the time—frequency domain by means of the Wigner distribution, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP84, San Diego, CA (March 1984) pp. 22.5.1–22. 5. 4. 1984

    Google Scholar 

  • Boudreaux-Bartels, G.F., and T.W. Parks, Signal estimation using modified Wigner distributions, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP84, San Diego, CA (March 1984).

    Google Scholar 

  • Brud, B.R., and T.E. Posch, A range and azimuth estimator based on forming the spatial Wigner distribution, Proc. IEEE Im. Conf. on Acoustics, Speech and Signal Processing ICASSP84, San Diego, CA (March 1984) pp. 41.B.9.1–41.B. 9. 2.

    Google Scholar 

  • Chester, D., F.J. Taylor and M. Doyle, The Wigner distribution in speech processing applications, J. Franklin Inst. 318(6), 415–430.

    Google Scholar 

  • Claasen, T.A.C.M., and W.F.G. Mecklenbräuker, On the time—frequency discrimination of energy distributions: can they look sharper than Heisenberg?, Proc. IEEE Im. Conf. on Acoustics, Speech and Signal Processing ICASSP84, San Diego, CA (March 1984) pp. 41.B.7.1–41.B. 7. 4.

    Google Scholar 

  • Cohen, L., Distributions in signal theory, Proc. IEEE Int. Conf. on Acoustics. Speech and Signal Processing ICASSP84, San Diego, CA (March 1984) pp. 41.B.1.1–41.B. 1. 4.

    Google Scholar 

  • Escudié, B., and J. Gréa, Joint representation (JR) in signal theory (ST) and Hilbertian analysis: a powerful tool for signal analysis, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP84, San Diego, CA (March 1984) pp. 41.B.6.1–41.B. 6. 4.

    Google Scholar 

  • Flandrin, P., Some features of time-frequency representations of multicomponent signals, IEEE ICASSP84, San Diego (1984) pp. 41.B.4.1–41.B. 4. 4.

    Google Scholar 

  • Flandrin, P., and B. Escudié, An interpretation of the pseudo-Wigner-Ville distribution, Signal Process 6(1), 27–36.

    Google Scholar 

  • Flandrin, P., and W. Martin, A general class of estimators for the Wigner-Ville spectrum of non-stationary processes, in: Lectures. Notes in Control and Information Sciences, Vol. 62, Analysis and Optimization of Systems (Springer-Verlag, Heidelberg) pp. 15–23. 1984

    Google Scholar 

  • Flandrin, P., W. Martin and M. Zakharia, On a hardware implementation of the Wigner-Ville transform, in: Digital Signal Processing-84, eds. V. Cappellini and A.G. Constantinides (North-Holland, Amsterdam) pp. 262–266. 1984

    Google Scholar 

  • Grenier, Y., Modélisation de signaux non-stationnaires, Thèse (Université de Paris-Sud). 1984

    Google Scholar 

  • Hillery, M., R.F. O’Connell, M.O. Scully and E.P. Wigner, Distribution functions in physics: fundamentals, Phys. Rep. 106 (3), 123–167. 1984

    MathSciNet  Google Scholar 

  • Hlawatsch, F., PseudoWignerverteilung modulierter Schwingungen, Fünfter Aachener Kolloquium, ( Aachen, 1984 ) S. 344–347.

    Google Scholar 

  • Hlawatsch, F., Interference terms in the Wigner distribution, in: Digital Signal Processing-84, eds. V. Cappellini and A.G. Constantinides (North-Holland, Amsterdam) pp. 363–367. 1984

    Google Scholar 

  • Janssen, A.J.E.M., Positivity properties of phase-space distribution function, J. Math. Phys. 25 (7), 2240–2252. 1984

    Article  MathSciNet  Google Scholar 

  • Janssen, A.J.E.M., A note on Hudson’s theorem about functions with nonnegative Wigner distributions. SIAM J. Math. Anal. 15 (1), 170–176. 1984

    MathSciNet  MATH  Google Scholar 

  • Janssen, A.J.E.M.. Gabor representations and Wigner distribution of signals, Proc. IEEE Int. Conf. on Acoustics. Speech and Signal Processing ICASSP84, San Diego, CA (1984) pp. 41.B.2.1–41.B. 2. 4. 1984

    Google Scholar 

  • Kumar, B.V.K.V., and C.W. Carroll, Performance of Wigner distribution function based detection methods, Opt. Eng. 23 (6), 732–737. 1984

    Google Scholar 

  • Marinovic, N.M.. and W.A. Smith, Use of the Wigner distribution to analyse the time-frequency response of ultrasonic transducers, IEEE Ultrasonics Symposium. Dallas, TX (1984).

    Google Scholar 

  • Martin. N., Développement de méthodes d’analyse spectrale autoregressive, Applications a des signaux réels non-stationnaires ou a N dimensions, Thèse D. I. (INPG, Grenoble) 1984

    Google Scholar 

  • Martin, W., Measuring the degree of non-stationarity by using the Wigner-Ville spectrum, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP84. San Diego, CA (1984) pp. 41.B.3.1–41.B. 3. 4.

    Google Scholar 

  • Martin, W., Spectral analysis of nonstationary processes. Sixth Int. Conf. on Analysis and Optimisation of Systems (special session on nonstationary processes), Nice. France (June 1984).

    Google Scholar 

  • Martin. W., Wigner-Ville-Spektralanalyse nichtstationärer Prozesse. Theorie und Anwendung in biologischen Fragestellungen, Habilitationsschrift (Universität Bonn).

    Google Scholar 

  • Riley, M.D., Detecting time-varying spectral energy concentrations, IEEE Digital Signal Processing Workshop, Chatham (1984) pp. 5.6.1–5. 6. 2.

    Google Scholar 

  • Schempp, W., Radar ambiguity functions, nilpotent harmonic analysis, and holomorphic theta series. Special Functions: Group Theoretical Aspects and Applications. eds. R.A. Askey et al., pp. 217–260. 1984

    Google Scholar 

  • Subotic, N., and B.E.A. Salek, Generation of the Wigner distribution function of twodimensional signals by a parallel optical processor, Opt. Lett. 9 (10), 471–473. 1984

    Article  Google Scholar 

  • Szu, H.H., and H.J. Caulfield, The mutual time-frequency content of two signals. Proc. IEEE 72 (7), 902–908. 1984

    Article  Google Scholar 

  • Boashash, B., On the anti-aliasing and computational properties of the Wigner—Ville distribution, Int. Symp. on Applied Signal Processing and Digital Filtering IASTED85, Paris, (June 1985) ed. M.H. Hamza ( Acta Press, Anaheim ). 1985

    Google Scholar 

  • Chester, D., and J. Wilbur, Time and spatial varying CAM and AI signal analysis using the Wigner distribution, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP85, Tampa. FL (1985) pp. 1045–1048. 1985

    Article  Google Scholar 

  • Cohen, L., Properties of the positive time—frequency distribution functions, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP85, Tampa, FL (1985) pp. 548–551. 1985

    Article  Google Scholar 

  • Cohen, L., Positive and negative joint quantum distributions, in: Non-equilibrium Quantum Statistical Physics, eds. J. Moore and M.O. Scully, 1985.

    Google Scholar 

  • Cohen, L., and T. Posch, Positive time—frequency distribution functions, IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 31–38. 1985

    Google Scholar 

  • Flandrin, P., Séparation de fréquences modulées proches par analyse de Wigner—Ville autoregressive, 106 Coll. Traitement du Signal GRETSI, Nice, France (May 1985). 1985

    Google Scholar 

  • Flandrin, P., and B. Escudié, Principe et mise en oeuvre de l’analyse temps—fréquence par transformation de Wigner—Ville, Traitement du Signal, GRETSI, Nice, France. 1985

    Google Scholar 

  • Flandrin, P.. B. Escudié and W. Martin, Représentations temps—fréquence et causalité. Coll. GRETSI-85, Nice, France (May 1985).

    Google Scholar 

  • Friedman, D.H., Instantaneous-frequency distribution vs. time: an interpretation of the phase structure of speech, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP85, Tampa. FL (1985) pp. 1121–1124. 1985

    Article  Google Scholar 

  • Garudadri, H., M.P. Beddoes, J.H.V. Gilbert and A.-P. Benguered, Identification of invariant acoustic cues in stop consonants using the Wigner distribution, Int. Symp. on Applied Signal Processing and Digital Filtering IASTED85, Paris (June 1985) ed. M.H. Hamza ( Acta Press, Anaheim ) pp. 196–200. 1985

    Google Scholar 

  • Hammond. J., and R. Harrison, Wigner—Ville and evolutionary spectra for covariant equivalent nonstationary random processes. Proc. IEEE Int. Conf. on Acoustics. Speech and Signal Processing ICASSP85, Tampa. FL (1985) pp. 1025–1028.

    Google Scholar 

  • Hlawatsch, F., Transformation, inversion and conversion of bilinear signal representations. Proc. IEEE Int. Conf. on Acoustics. Speech and Signal Processing ICASSP85. Tampa. FL (1985) pp. 1029–1032.

    Google Scholar 

  • Hlawatsch. F., Duality of time—frequency signal representations: energy density domain and correlation domain. Int. Symp. on Applied Signal Processing and Digital Filtering. IASTED85. Paris (June 1985) ed. M.H. Hamza (Acta Press, Anaheim).

    Google Scholar 

  • Hlawatsch. F.. Notes on bilinear time—frequency signal representations. Institutsbericht ( Institut für Nachrichten-technik. Technische Universität Wien ). 1985

    Google Scholar 

  • Kay, S., and G. Boudreaux-Bartels. On the optimality of the Wigner distribution for detection. Proc. IEEE Int. Conf. on Acoustics. Speech and Signal Processing ICASSP85. Tampa. FL (1985) pp. 1017–1020.

    Google Scholar 

  • Marinovic. N.M., and G. Eichmann. An expansion of Wigner distribution and its applications. Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP85. Tampa. FL (1985) pp. 1021–1024.

    Google Scholar 

  • Martin, W.. and P. Flandrin. Detection of changes of signal structure by using the Wigner—Ville spectrum, Signal Process. 8, 215–233.

    Google Scholar 

  • Yu, K., and S. Cheng, Signal synthesis from Wigner distribution, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing ICASSP85, Tampa, FL (1985) pp. 1037–1040.

    Google Scholar 

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Mecklenbräuker, W. (1989). A Tutorial on Non-Parametric Bilinear Time-Frequency Signal Representations. In: Longo, G., Picinbono, B. (eds) Time and Frequency Representation of Signals and Systems. International Centre for Mechanical Sciences, vol 309. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2620-2_2

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