From Kalman Filtering to Innovations, Martingales, Scattering and Other Nice Things

This paper is an account of the development of some of several researches inspired by Kalman’s seminal work on linear least-squares estimation for processes with known state-space models.

[1]  Robert E. Kalaba,et al.  Multiple scattering processes: Inverse and direct , 1975 .

[2]  R. Bucy Nonlinear filtering theory , 1965 .

[3]  P. Brémaud Point processes and queues, martingale dynamics , 1983 .

[4]  Mark H. A. Davis,et al.  Exact and approximate filtering in signal detection: An example (Corresp.) , 1977, IEEE Trans. Inf. Theory.

[5]  Thomas Kailath,et al.  Likelihood ratios for Gaussian processes , 1970, IEEE Trans. Inf. Theory.

[6]  L. D. Collins Realizable whitening filters and state-variable realizations , 1968 .

[7]  T. Kailath,et al.  An inverse scattering framework for several problems in signal processing , 1987, IEEE ASSP Magazine.

[8]  Numerical experiments in linear control theory using generalized X - Y equations , 1976 .

[9]  L. Ljung,et al.  Generalized Krein-Levinson Equations for Efficient Calculation of Fredholm Resolvents of Non-Displacement Kernels , 1978 .

[10]  J. Chun,et al.  Fast array algorithms for structured matrices , 1990 .

[11]  H. Kunita,et al.  Stochastic differential equations for the non linear filtering problem , 1972 .

[12]  T. Kailath,et al.  Complementary models and smoothing , 1989 .

[13]  D. Fraser,et al.  The optimum linear smoother as a combination of two optimum linear filters , 1969 .

[14]  Fred C. Schweppe,et al.  Evaluation of likelihood functions for Gaussian signals , 1965, IEEE Trans. Inf. Theory.

[15]  Thomas Kailath,et al.  Generalized Bezoutians and families of efficient zero-location procedures , 1991 .

[16]  I. V. Girsanov On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures , 1960 .

[17]  H. Kunita,et al.  On Square Integrable Martingales , 1967, Nagoya Mathematical Journal.

[18]  Alfred M. Bruckstein,et al.  Inverse scattering for discrete transmission—line models , 1987 .

[19]  Thomas Kailath,et al.  A further note on a general likelihood formula for random signals in Gaussian noise , 1970, IEEE Trans. Inf. Theory.

[20]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

[21]  R. E. Kalman,et al.  Contributions to the Theory of Optimal Control , 1960 .

[22]  Thomas Kailath,et al.  A coding scheme for additive noise channels with feedback-I: No bandwidth constraint , 1966, IEEE Trans. Inf. Theory.

[23]  Thomas Kailath,et al.  An RKHS approach to detection and estimation problems- III: Generalized innovations representations and a likelihood-ratio formula , 1972, IEEE Trans. Inf. Theory.

[24]  Thomas Kailath,et al.  Displacement structure for Hankel, Vandermonde, and related (derived) matrices , 1991 .

[25]  M. Morf,et al.  Inverses of Toeplitz operators, innovations, and orthogonal polynomials , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[26]  M. Hitsuda Representation of Gaussian processes equivalent to Wiener process , 1968 .

[27]  T. Kailath Some extensions of the innovations theorem , 1971 .

[28]  R. Kálmán Algebraic characterization of polynomials whose zeros lie in certain algebraic domains. , 1969, Proceedings of the National Academy of Sciences of the United States of America.

[29]  T. Kailath,et al.  Orthogonal functionals of independent-increment processes , 1976, IEEE Trans. Inf. Theory.

[30]  Lennart Ljung,et al.  A unified approach to smoothing formulas , 1976, Autom..

[31]  John L. Casti,et al.  A new initial-value method for on-line filtering and estimation (Corresp.) , 1972, IEEE Trans. Inf. Theory.

[32]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[33]  T. Kailath,et al.  On a generalized Szegö- Levinson realization algorithm for optimal linear predictors based on a network synthesis approach , 1978 .

[34]  R. Redheffer On the Relation of Transmission-Line Theory to Scattering and Transfer† , 1962 .

[35]  D. Pal Fast algorithms for structured matrices with arbitrary rank profile , 1990 .

[36]  Thomas Kailath,et al.  Some new algorithms for recursive estimation in constant linear systems , 1973, IEEE Trans. Inf. Theory.

[37]  Demetrios G. Lainiotis,et al.  Partitioned estimation algorithms, I: Nonlinear estimation , 1974, Inf. Sci..

[38]  J. Willems Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[39]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[40]  Thomas Kailath,et al.  The modeling of randomly modulated jump processes , 1975, IEEE Trans. Inf. Theory.

[41]  H. W. Sorenson,et al.  Kalman filtering : theory and application , 1985 .

[42]  Donald L. Snyder,et al.  Random point processes , 1975 .

[43]  M. Morf Fast Algorithms for Multivariable Systems , 1974 .

[44]  Albert Wilansky,et al.  Topics in Functional Analysis , 1967 .

[45]  E.D. Denman,et al.  An introduction to invariant imbedding , 1977, Proceedings of the IEEE.

[46]  Thomas Kailath,et al.  An inverse scattering approach to the partial realization problem , 1984, The 23rd IEEE Conference on Decision and Control.

[47]  R. Kalaba,et al.  Exact Solution of a Family of Matrix Integral Equations for Multiply Scattered Partially Polarized Radiation. II , 1970 .

[48]  Alfred M. Bruckstein,et al.  Fast matrix factorizations via discrete transmission lines , 1986 .

[49]  V. Sobolev,et al.  A treatise on radiative transfer. , 1963 .

[50]  P. Brémaud Point Processes and Queues , 1981 .

[51]  T. Kailath,et al.  An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise , 1968 .

[52]  T. Kailath,et al.  Scattering theory and linear least-squares estimation, part III: The estimates , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[53]  Thomas Kailath,et al.  Divide-and-conquer solutions of least-squares problems for matrices with displacement structure , 1991 .

[54]  Michiel Hazewinkel,et al.  Preface : Stochastic systems : the mathematics of filtering and identification and applications , 1981 .

[55]  T. Kailath A Note on Least Squares Estimation by the Innovations Method , 1972 .

[56]  George C. Verghese,et al.  A scattering framework for decentralized estimation problems , 1983, Autom..

[57]  J. Schalkwijk,et al.  Center-of-gravity information feedback , 1968, IEEE Trans. Inf. Theory.

[58]  Thomas Kailath,et al.  A general likelihood-ratio formula for random signals in Gaussian noise , 1969, IEEE Trans. Inf. Theory.

[59]  Thomas Kailath,et al.  Generalized Gohberg-Semencul Formulas for Matrix Inversion , 1989 .

[60]  R. E. Kalman,et al.  On the Hermite-Fujiwara theorem in stability theory , 1965 .

[61]  Thomas Kailath,et al.  Orthogonal transformation (square-root) implementations of the generalized Chandrasekhar and generalized Levinson algorithms , 1979 .

[62]  David Q. Mayne,et al.  A solution of the smoothing problem for linear dynamic systems , 1966, Autom..

[63]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[64]  L. Ljung,et al.  The asymptotic behavior of constant-coefficient Riccati differential equations , 1976 .

[65]  R. Kálmán THE THEORY OF OPTIMAL CONTROL AND THE CALCULUS OF VARIATIONS , 1960 .

[66]  T. Kailath,et al.  An innovations approach to least squares estimation--Part IV: Recursive estimation given lumped covariance functions , 1971 .

[67]  David Chapman,et al.  Random signals and noise , 1992 .

[68]  L. Ljung,et al.  Scattering theory and linear least squares estimation—Part I: Continuous-time problems , 1976, Proceedings of the IEEE.

[69]  L. Zadeh,et al.  An Extension of Wiener's Theory of Prediction , 1950 .

[70]  V. E. Bene On kailath's innovations conjecture hold , 1976, The Bell System Technical Journal.

[71]  R. Mehra On the identification of variances and adaptive Kalman filtering , 1970 .

[72]  P. Meyer Sur un probleme de filtration , 1973 .

[73]  T. Kailath The innovations approach to detection and estimation theory , 1970 .

[74]  M. Morf,et al.  Some new algorithms for recursive estimation in constant, linear, discrete-time systems , 1974 .

[75]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[76]  M. A. Kaashoek,et al.  The Gohberg Anniversary Collection , 1989 .

[77]  H. W. Bode,et al.  A Simplified Derivation of Linear Least Square Smoothing and Prediction Theory , 1950, Proceedings of the IRE.

[78]  T. Kailath The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures , 1971 .

[79]  Thomas Kailath,et al.  A further note on backwards Markovian models (Corresp.) , 1979, IEEE Trans. Inf. Theory.

[80]  M. Morf,et al.  Square-root algorithms for least-squares estimation , 1975 .

[81]  S. Mitter,et al.  New results on the innovations problem for non-linear filtering , 1981 .

[82]  Lennart Ljung,et al.  Backwards Markovian models for second-order stochastic processes (Corresp.) , 1976, IEEE Trans. Inf. Theory.