Model Based Processing of Signals: A State Space Approach

This paper is a tutorial on linear, state space, model-based methods for certain nonlinear estimation problems commonly encountered in signal and data analysis. A protoypica1 problem that is studied is that of estimating the frequencies of multiple, superimposed sinusoids from a short record of noise-corrupted data. The approach expounded however, is applicable to a vast range of nonlinear signal analysis problems, and applications in direction finding and damped sinusoid retrieval are dealt with in some detail. The benefits that result from using a state space description of the signal are highlighted in this paper. It is shown that state space models provide an elegant tool for exposing the structure present in the problem. The approach also allows for robust parameterization of the model with respect to finite precision errors. The robustness of the parameter set is complemented by the availability of numerically robust tools to estimate the parameters. The resulting algorithms are compatible with multiprocessor implementations.

[1]  Bhaskar D. Rao,et al.  Performance analysis of Root-Music , 1989, IEEE Trans. Acoust. Speech Signal Process..

[2]  Masanao Aoki,et al.  State Space Modeling of Time Series , 1987 .

[3]  H. T. Kung.  Why systolic architectures? , 1982, Computer.

[4]  R. Vaccaro.  Deterministic balancing and stochastic model reduction , 1985 .

[5]  Bjorn Ottersten,et al.  Analysis of algorithms for sensor arrays with invariance structure , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[6]  A. Bax,et al.  Two-dimensional nuclear magnetic resonance spectroscopy. , 1986, Science.

[7]  Thomas Kailath,et al.  ESPIRT-estimation of signal parameters via rotational invariance techniques , 1989 .

[8]  Gene H. Golub,et al.  Matrix computations , 1983 .

[9]  J.-J. Fuchs,et al.  State-space modeling and estimation of time differences of arrival , 1986, IEEE Trans. Acoust. Speech Signal Process..

[10]  G. Stewart.  Introduction to matrix computations , 1973 .

[11]  Bjorn Ottersten,et al.  Asymptotic robustness of sensor array processing methods , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[12]  Thomas Kailath,et al.  Linear Systems , 1980 .

[13]  Petre Stoica,et al.  Performance comparison of subspace rotation and MUSIC methods for direction estimation , 1990, Fifth ASSP Workshop on Spectrum Estimation and Modeling.

[14]  Tapan K. Sarkar,et al.  Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise , 1990, IEEE Trans. Acoust. Speech Signal Process..

[15]  S. S. Rao,et al.  A criterion for identifying dominant singular values in the SVD based method of harmonic retrieval , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[16]  J. Cadzow,et al.  Signal processing via least squares error modeling , 1990, IEEE ASSP Magazine.

[17]  Mats Viberg,et al.  Subspace fitting concepts in sensor array processing , 1990 .

[18]  S.M. Kay,et al.  Spectrum analysis—A modern perspective , 1981, Proceedings of the IEEE.

[19]  James E. Evans,et al.  Application of Advanced Signal Processing Techniques to Angle of Arrival Estimation in ATC Navigation and Surveillance Systems , 1982 .

[20]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[21]  Zhi-Dong Bai,et al.  Remarks on certain criteria for detection of number of signals , 1987, IEEE Trans. Acoust. Speech Signal Process..

[22]  Thomas Kailath,et al.  Detection of signals by information theoretic criteria , 1985, IEEE Trans. Acoust. Speech Signal Process..

[23]  K. Arun,et al.  Balanced approximation of stochastic systems , 1990 .

[24]  Petre Stoica,et al.  MUSIC, maximum likelihood, and Cramer-Rao bound , 1989, IEEE Transactions on Acoustics, Speech, and Signal Processing.

[25]  Ramdas Kumaresan,et al.  ESTIMATING THE PARAMETERS OF EXPONENTIALLY DAMPED OR UNDAMPED SINUSOIDAL SIGNALS IN NOISE , 1982 .

[26]  J. Cadzow,et al.  Spectral estimation: An overdetermined rational model equation approach , 1982, Proceedings of the IEEE.

[27]  G. Stewart.  On the Perturbation of Pseudo-Inverses, Projections and Linear Least Squares Problems , 1977 .

[28]  U. Desai,et al.  A realization approach to stochastic model reduction and balanced stochastic realizations , 1982, 1982 21st IEEE Conference on Decision and Control.

[29]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .

[30]  R. R. Boorstyn,et al.  Multiple tone parameter estimation from discrete-time observations , 1976, The Bell System Technical Journal.

[31]  A. Benveniste,et al.  Single sample modal identification of a nonstationary stochastic process , 1985, IEEE Transactions on Automatic Control.

[32]  J. Helton,et al.  Power spectrum reduction by optimal Hankel norm approximation of the phase of the outer spectral factor , 1985 .

[33]  V. Pisarenko.  The Retrieval of Harmonics from a Covariance Function , 1973 .

[34]  S. Kung,et al.  VLSI Array processors , 1985, IEEE ASSP Magazine.

[35]  Sylvie Mayrargue.  ESPRIT and TAM (Toeplitz approximation method) are theoretically equivalent , 1988, ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing.

[36]  R. Kumaresan,et al.  Estimation of frequencies of multiple sinusoids: Making linear prediction perform like maximum likelihood , 1982, Proceedings of the IEEE.

[37]  S. Hwang.  Minimum uncorrelated unit noise in state-space digital filtering , 1977 .

[38]  Jean-Jacques Fuchs,et al.  Estimating the number of sinusoids in additive white noise , 1988, IEEE Trans. Acoust. Speech Signal Process..

[39]  Bhaskar D. Rao.  Sensitivity analysis of state space methods in spectrum estimation , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[40]  A. Ouamri,et al.  Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds , 1989, IEEE Trans. Acoust. Speech Signal Process..

[41]  K. S. Arun,et al.  A Unitarily Constrained Total Least Squares Problem in Signal Processing , 1992, SIAM J. Matrix Anal. Appl..

[42]  R. Gerchberg.  Super-resolution through Error Energy Reduction , 1974 .

[43]  S. Mayrargue,et al.  A new application of SVD to harmonic retrieval , 1989 .

[44]  W. M. Carey,et al.  Digital spectral analysis: with applications , 1986 .

[45]  P. Faurre.  Stochastic Realization Algorithms , 1976 .

[46]  Kai-Bor Yu,et al.  Total least squares approach for frequency estimation using linear prediction , 1987, IEEE Trans. Acoust. Speech Signal Process..

[47]  Lothar Thiele,et al.  On the sensitivity of linear state-space systems , 1986 .

[48]  Bhaskar D. Rao.  PERFORMANCE ANALYSIS OF SUBSPACE BASED METHODS , 1988 .

[49]  Sun-Yuan Kung,et al.  A Toeplitz approximation approach to coherent source direction finding , 1986, ICASSP '86. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[50]  Joos Vandewalle,et al.  A variety of applications of singular value decomposition in identification and signal processing , 1989 .

[51]  Bhaskar D. Rao.  Perturbation analysis of an SVD-based linear prediction method for estimating the frequencies of multiple sinusoids , 1988, IEEE Trans. Acoust. Speech Signal Process..

[52]  Sun-Yuan Kung,et al.  A new identification and model reduction algorithm via singular value decomposition , 1978 .

[53]  S. Deans.  The Radon Transform and Some of Its Applications , 1983 .

[54]  D. V. Bhaskar Rao.  Analysis of coefficient quantization errors in state-space digital filters , 1986, IEEE Trans. Acoust. Speech Signal Process..

[55]  Sabine Van Huffel.  Analysis of the Total Least Squares Problem and its Use in Parameter Estimation , 1987 .

[56]  R. Kumaresan,et al.  Estimating the Angles of Arrival of Multiple Plane Waves , 1983, IEEE Transactions on Aerospace and Electronic Systems.

[57]  K. S. Arun,et al.  Multiply Replicated Arrays For Direction Of Arrival Estimation , 1988, Twenty-Second Asilomar Conference on Signals, Systems and Computers.

[58]  J. Vandewalle,et al.  Analysis and solution of the nongeneric total least squares problem , 1988 .

[59]  A. Laub,et al.  The singular value decomposition: Its computation and some applications , 1980 .

[60]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[61]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[62]  A. Papoulis.  A new algorithm in spectral analysis and band-limited extrapolation. , 1975 .

[63]  D. J. De Rosier,et al.  Reconstruction of Three Dimensional Structures from Electron Micrographs , 1968, Nature.

[64]  Thomas Kailath,et al.  ESPRIT-A subspace rotation approach to estimation of parameters of cisoids in noise , 1986, IEEE Trans. Acoust. Speech Signal Process..

[65]  K S Arun,et al.  Bandlimited rational models for signal reconstruction , 1990 .