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Volterra models are widely used in various application areas. Their usefulness is mainly due to their ability to approximate with an arbitrary precision any fading memory nonlinear system, and their property of linearity with respect to parameters, the kernels coefficients. The main drawback of these models is their parametric complexity needing to estimate a huge number of parameters. Considering Volterra kernels as symmetric tensors, we use their PARAFAC decomposition to derive the Volterra-Parafac models inducing a substantial parametric complexity reduction. We show that these models can be viewed as a set of Wiener models in parallel. Then, we apply the extended Kalman filter for recursively identifying such Volterra-Parafac models. Some simulation results illustrate the effectiveness of the proposed identification method, in the case of cubic Volterra systems.
RÉSUMÉ
Les modèles de Volterra sont très utilisés dans de nombreux domaines d’application du fait qu’ils permettent de représenter, avec une précision arbitraire, tout système non linéaire de mémoire finie. Ils possèdent de plus la propriété d’être linéaires vis-à-vis de leurs paramètres, les coefficients des noyaux. Le principal inconvénient de ces modèles est leur complexité paramétrique qui nécessite d’estimer un très grand nombre de paramètres. Cet article présente une nouvelle méthode permettant de réduire cette complexité paramétrique en considérant les noyaux de Volterra d’ordre supérieur à un comme des tenseurs symétriques et en les décomposant à l’aide de la décomposition PARAFAC. Les modèles de Volterra-Parafac ainsi obtenus peuvent être vus comme une série de modèles de Wiener mis en parallèle. En exploitant cette nouvelle formulation des modèles de Volterra, nous proposons un algorithme d’identification récursif basé sur le filtre de Kalman étendu. Des résultats de simulation illustrent le comportement de la méthode d’identification proposée, dans le cas de systèmes de Volterra cubiques.
Volterra models, nonlinear system modeling and identification, Volterra kernels, PARAFAC decomposition, extended Kalman filter, tensors
MOTS-CLÉS
modèles de Volterra, modélisation et identification de systèmes non linéaires, noyaux de Volterra, décomposition PARAFAC, filtre de Kalman étendu, tenseurs
Acar E., Aykut-Bingol C., Bingol H., Bro R., Yener B. (2007). « Multiway analysis of epilepsy tensors », Bioinformatics, vol. 23, p. 10-18.
Acar T., Yu Y., Petropulu A. P. (2006). « Blind MIMO system estimation based on PARAFAC decomposition of higher order output tensors », IEEE Tr. on Signal Processing, vol. 54, n° 11, p. 4156-4168, Nov..
Agazzi O., Messerschmitt D., Hodges D. (1982). « Nonlinear echo cancellation of data signals », IEEE Tr. Commun., vol. 30, n° 11, p. 2421-2433, Nov.
Albera L., Ferréol A., Comon P., Chevalier P. (2004). « Blind Identification of Over-complete MixturEs of sources (BIOME) », Linear Algebra Appl., vol. 391C, p. 3-30, Nov.
Benedetto S., Biglieri E. (1983). « Nonlinear equalization of digital satellite channels », IEEE J. Select. Areas Commun, vol. SAC-1, p. 57-62, January.
Benedetto S., Biglieri E., Daffara S. (1979). « Modeling and performance evaluation of nonlinear satellite links- A Volterra series approach », IEEE Tr. on Aerospace Electron. Systems, vol. AES - 15, n° 4, p. 494-507, July.
Boyd S., Chua L. O. (1985). « Fading memory and the problem of approximating nonlinear operators with Volterra series », IEEE Tr. on Circ. and Syst., vol. CAS-32, n° 11, p. 1150-1161, November.
Bro R. (1997). « PARAFAC. Tutorial and applications », Chemometrics and Intelligent Laboratory Systems, vol. 38, p. 149-171.
Campello R. J. G. B., Amaral W. C., Favier G. (2006). « A note on the optimal expansion of Volterra models using Laguerre functions », Automatica, vol. 42, n° 4, p. 689-693.
Campello R. J. G. B., Favier G., Amaral W. C. (2004). « Optimal expansions of discrete time Volterra models using Laguerre functions », Automatica, vol. 40, n° 5, p. 815-822.
Cardoso J. F. (1990). « Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem », Proc. Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Albuquerque, New Mexico, USA, p. 2655-2658, 3-6 April.
Cardoso J. F. (1991). « Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors », Proc. Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Toronto, Ontario, Canada„ p. 3109-3112, 14-17 May.
Carroll J., Chang J. J. (1970). « Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decompositions », Psychometrika, vol. 35, p. 283- 319.
Cheng C. H., Powers E. J. (2001). « Optimal Volterra kernel estimation algorithms for a nonlinear communication system for PSK and QAM inputs », IEEE Tr. on Signal Processing, vol. 49, n° 1, p. 147-163, January.
Chon K. H., Holstein-Rathlou N. H., Marsh D. J., Marmarelis V. Z. (1998). « Comparative nonlinear modeling of renal autoregulation in rats: Volterra approach versus artificial neural networks », IEEE Tr. Neural Networks, vol. 9, n° 3, p. 430-435, May.
Comon P., Golub G., Lim L.-H., Mourrain B. (2008). « Symmetric tensors and symmetric tensor rank », SIAM J. on Matrix Analysis and Appl., vol. 30, n° 3, p. 1254-1279.
da Rosa A., Campello R. J. G. B., Amaral W. C. (2007). « Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions », Automatica, vol. 43, n° 6, p. 1084-1091.
de Almeida A. L. F., Favier G., Mota J. C. M. (2007). « PARAFAC-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization », Signal Processing, vol. 87, n° 2, p. 337-351, Feb., 2007.
de Almeida A. L. F., Favier G., Mota J. C. M. (2008). « Constrained tensor modeling approach to blind multiple-antenna CDMA schemes », IEEE Tr. on Signal Proc., vol. 56, n° 6, p. 2417-2428, June.
de Almeida A. L. F., Favier G., Mota J. C. M. (2009a). Multiuser MIMO systems using STFMA PARAFAC tensor modeling, Springer, chapter 11 of "Optimizing wireless communication systems", Ed. Cavalcanti, F. R. P., Andersson, S., p. 421-457.
de Almeida A. L. F., Favier G., Mota J. C. M. (2009b). « Space-time spreading-multiplexing for MIMO wireless communication systems using the PARATUCK-2 tensor model », Signal Processing, vol. 89, p. 2103-2116.
de Lathauwer L., Castaing J., Cardoso J. F. (2007). « Fourth-order cumulant-based blind identification of underdetermined mixtures », IEEE Tr. on Signal Proc., vol. 55, n° 6, p. 2965-2973, June.
Dogançay K., Tanrikulu O. (2001). « Adaptive filtering algorithms with selective partial updates », IEEE Tr. on Circuits and Systems-II: Analog and Digital Signal Processing, vol. 48, n° 8, p. 762 - 769, Aug.
Douglas S. C. (1997). « Adaptive filters employing partial updates », IEEE Tr. on Circuits and Systems-II: Analog and Digital Signal Processing, vol. 44, n° 3, p. 209-216, March.
Doyle III F. J., Pearson R. K., Ogunnaike B. A. (2002). « Identification and control using Volterra models », Springer-Verlag.
Dumont G., Fu Y. (1993), « Non-linear adaptive control via Laguerre expansion of Volterra kernels », Int. J. of Adaptive Control and Signal Processing, vol. 7, p. 367-382.
Eun C., Powers E. J. (1997). « A new Volterra predistorter based on the indirect learning architecture », IEEE Tr. Signal Proc., vol. 45, n° 1, p. 223-227, Jan.
Favier G. (2009). « Nonlinear system modeling and identification using tensor approaches », 10th Int. Conf. on Sciences and Techniques of Automatic control and computer engineering (STA’2009), Hammamet, Tunisia, December.
Favier G., Bouilloc T. (2009a). « Identification de modèles de Volterra basée sur la décomposition PARAFAC », GRETSI Symp., Dijon, France, September.
Favier G., Bouilloc T. (2009b). « Parametric complexity reduction of Volterra models using tensor decompositions. », EUSIPCO, Glasgow, Scotland, August.
Favier G., Kibangou A. Y., Bouilloc T. (2010). « Nonlinear systems modeling and identification using Volterra-PARAFAC models », Submitted for publication.
Fernandes C. A. R., Favier G., Mota J. C. M. (2009a). « Blind identification of multiuser nonlinear channels using tensor decomposition and precoding », Signal Processing, vol. 89, n° 12, p. 2644-2656, December.
Fernandes C. E. R., Favier G., Mota J. C. M. (2008). « Blind channel identification algorithms based on the PARAFAC decomposition of cumulant tensors: The single and multiuser cases », Signal Processing, vol. 88, n° 6, p. 1382-1401, June.
Fernandes C. E. R., Favier G., Mota J. C. M. (2009b). « PARAFAC -based blind identification of convolutive MIMO linear systems », Proc. of the 15th IFAC Symp. on System Identification (SYSID), Saint-Malo, France, p. 1704-1709, July.
Ferréol A., Albera L., Chevalier P. (2005). « Fourth-order blind identification of underdetermined mixtures of sources (FOBIUM) », IEEE Tr. on Signal Proc., vol. 53, n° 5, p. 1640-1653, May.
Frank W. A. (1995). « An efficient approximation to the quadratic Volterra filter and its application in real-time loudspeaker linearization », Signal Processing, vol. 45, n° 1, p. 97-113.
Haber R., Keviczky L. (1999). « Nonlinear system identification. Input-ouput modeling approach. Vol. 1: Nonlinear system parameter identification », Kluwer Academic Publishers.
Hacioglu R.,Williamson G. (2001). « Reduced complexity Volterra models for nonlinear system identification », EURASIP J. on Applied Signal Processing, vol. 4, p. 257-265.
Harshman R. (1970). « Foundations of the PARAFAC procedure: models and conditions for an "explanatory" multimodal factor analysis », UCLA working papers in phonetics, vol. 16, p. 1-84.
Harshman R. (1972). « Determination and proof of minimum uniqueness conditions for PARAFAC 1 », UCLA working papers in phonetics, vol. 22, p. 111-117.
Heuberger P. S. C., Van den Hof P. M. J., Bosgra O. H. (1995). « A generalized orthonormal basis for linear dynamical systems », IEEE Tr. on Automatic Control, vol. 40, n° 3, p. 451-465.
Hitchcock F. (1927). « The expression of a tensor or a polyadic as a sum of products », J. Math. Phys. Camb., vol. 6, n° 3, p. 164-189.
Kajikawa Y. (2008). « Subband parallel cascade Volterra filter for linearization of loudspeaker systems », EUSIPCO, Lausanne, Switzerland, September.
Karfoul A., Albera L., De Lathauwer L. (2008). « Canonical decomposition of even higher order cumulant arrays for blind underdetermined mixture identification », 5th IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM’2008), Darmstadt, Germany, p. 501-505, July.
Khouaja A., Favier G. (2004). « Identification of PARAFAC-Volterra cubic models using an alternating recursive least squares algorithm », EUSIPCO, Vienna, Austria, p. 1903-1906, September.
Kibangou A. Y., Favier G. (2009a). « Blind equalization of nonlinear channels using tensor decompositions with code/space/time diversities », Signal Processing, vol. 89, n° 2, p. 133-143, February.
Kibangou A. Y., Favier G. (2009b). « Identification of Parallel-cascade Wiener systems using joint diagonalization of third-order Volterra kernel slices », IEEE Signal Proc. Letters, vol.16, n° 3, p. 188-191, March.
Kibangou A. Y., Favier G. (2009c). « Tensor-based identification of the structure of blockoriented nonlinear systems », Proc. of the 15th IFAC Symp. on System Identification (SYSID), Saint-Malo, France.
Kibangou A. Y., Favier G., Hassani M. M. (2005a). « Laguerre-Volterra filters optimization based on Laguerre spectra », EURASIP J. on Applied Signal Processing, vol. 17, p. 2874-2887.
Kibangou A. Y., Favier G., Hassani M. M. (2005b). « Selection of generalized orthonormal bases for second-order Volterra filters », Signal Processing, vol. 85, p. 2371-2385.
Kibangou A. Y., Favier G. (2010). « Tensor analysis-based model structure determination and parameter estimation for block-oriented nonlinear systems ». IEEE Journal of Selected Topics in Signal Processing, Special issue on Model Order Selection in Signal Processing Systems, vol. 4, p. 514-525.
Korenberg M. J. (1991). « Parallel cascade identification and kernel estimation for nonlinear systems », Annals of Biomedical Eng., vol. 19, p. 429-455, 1991.
Korenberg M. J., Hunter I. W. (1986). « The identification of nonlinear biological systems: LNL cascade models », Biological Cybernetics, vol. 55, p. 125-134.
Korenberg M. J., Hunter I. W. (1996). « The identification of nonlinear biological systems: Volterra kernel approaches », Annals of Biomedical Eng., vol. 24, p. 250-268.
Kroonenberg P. M. (2007). Applied multiway data analysis, John Wiley & Sons.
Kruskal J. B. (1977). « Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics », Linear Algebra Applicat., vol. 18(2), p. 95-138.
Kruskal J. B. (1989). « Rank, decomposition and uniqueness for 3-way and N-way arrays », in R. Coppi, S. Bolasco (eds), Multiway data analysis, Elsevier, Amsterdam, p. 7-18.
Lu H., Plataniotis K., Venetsanopoulos A. (2008). « MPCA: multilinear principal component analysis of tensor objects », IEEE Tr. on Neural Networks, vol. 19, p. 18-39.
Mäkilä P. M. (1990). « Approximation of stable systems by Laguerre filters », Automatica, vol. 26, n° 2, p. 333-345.
Maner B. R., Doyle III F. J., Ogunnaike B. A., Pearson R. K. (1996). « Nonlinear model predictive control of a simulated multivariable polymerization reactor using second-order Volterra models », Automatica, vol. 32, n° 9, p. 1285-1301.
Marmarelis V. Z. (1993). « Identification of nonlinear biological systems using Laguerre expansions of kernels », Annals of Biomedical Engineering, vol. 21, p. 573-589.
Marmarelis V. Z. (2004). « Nonlinear dynamic modeling of physiological systems », IEEE Press, John Wiley & Sons, 2004.
Mathews V. J., Sicuranza G. (2000). « Polynomial signal processing », John Wiley & Sons. Mileounis G., Koukoulas P., Kalouptsidis N. (2009). « Input-output identification of nonlinear channels using PSK, QAM and OFDM inputs », Signal Processing, vol. 89, p. 1359-1369.
Morup M., Hansen L., Arnfred S., Lim L.-H., Madsen K. (2008). « Shift-invariant multilinear decomposition of neuroimaging data », NeuroImage, vol. 42, p. 1439-1450.
Mumolo E., Francescato D. (1993). « Adaptive predictive coding of speech by means of Volterra predictors », Proc. of IEEE Winter Workshop on Nonlinear digital signal processing, Tampere, Finland, p. 2.1.4.1-2.1.4.4, January.
Ninness B., Gustafsson F. (1997). « A unifying construction of orthonormal bases for system identification », IEEE Tr. on Automatic Control, vol. 42, n° 4, p. 515-521.
Redfern A. J., Zhou G. (2001a). « Blind zero forcing equalization of multichannel nonlinear CDMA systems », IEEE Tr. on Signal Processing, vol. 49, n° 10, p. 2363-2371, October.
Redfern A. J., Zhou G. (2001b). « Multichannel and block based precoding methods for fixed point equalization of nonlinear communication channels », Signal Processing, vol. 81, p. 1041- 1052.
Salmi J., Richter A., Koivunen V. (2009). « Sequential unfolding SVD for tensors with applications in array signal processing », IEEE Tr. on Signal Process., vol. 57, n° 2, p. 4719-4733, Dec..
Sidiropoulos N. D., Bro R. (2000a). « On the uniqueness of multilinear decomposition of Nway arrays », Journal of Chemometrics, vol. 14, p. 229-239.
Sidiropoulos N. D., Bro R., Giannakis G. (2000b). « Parallel factor analysis in sensor array processing », IEEE Tr. on Signal Processing, vol. 48, n° 8, p. 2377-2388, Aug.
Sidiropoulos N. D., Giannakis G., Bro R. (2000c). « Blind PARAFAC receivers for DSCDMA systems », IEEE Tr. on Signal Processing, vol. 48, n° 3, p. 810-823, March.
Smilde A., Geladi P., Bro R. (2004). Multi-way analysis, Chichester, UL, Wiley.
Tan L., Jiang J. (2001). « Adaptive Volterra filters for active control of nonlinear noise processes », IEEE Tr. on Signal Processing, vol. 49, n° 8, p. 1667-1676, August.
Vasilescu M., Terzopoulos D. (2002). « Multilinear analysis for facial image recognition », Proc. Int. Conf. Pattern Recognition, Quebec, PQ, 2, vol. 3, p. 511-514.
Vasilescu M., Terzopoulos D. (2007). « Multilinear (tensor) image synthesis, analysis and recognition », IEEE Signal Proc. Magazine, vol. 24, p. 118-123, Nov.
Wahlberg B. (1991). « System identification using Laguerre models », IEEE Tr. on Automatic Control, vol. 36, n° 5, p. 551-562, May.
Wahlberg B. (1994). « System identification using Kautz models », IEEE Tr. on Automatic Control, vol. 39, n° 6, p. 1276-1282, June.
Wahlberg B., Mäkilä P. M. (1996). « On approximation of stable linear dynamical systems using Laguerre and Kautz functions », Automatica, vol. 32, n° 5, p. 693-708.
Wiener N. (1958). Nonlinear problems in random theory, Wiley, New-York.
Yu Y., Petropulu A. P. (2008). « PARAFAC-based blind estimation of possibly underdetermined convolutive MIMO systems », IEEE Tr. on Signal Process., vol. 56, n° 1, p. 111-124, Jan.
Zeller M., Kellerman W. (2007). « Fast adaptation of frequency-domain Volterra filters using inherent recursions of iterated coefficient updates », EUSIPCO, Poznan, Poland, p. 1605-1609, September.