Generalized Chandrasekhar recursions from the generalized Schur algorithm

Presents a new approach to the Chandrasekhar recursions and some generalizations thereof. The derivation uses the generalized Schur recursions, which are O(N/sup 2/) recursions for the triangular factorization of N/spl times/N matrices having a certain Toeplitz-like displacement structure. It is shown that when the extra structure provided by an underlying state-space model is properly incorporated into the generalized Schur algorithm, it reduces to the Chandrasekhar recursions, which are O(Nn/sup 2/) recursions for estimating the n-dimensional state of a time-invariant (or constant-parameter) system from N measured outputs. It is further noted that the generalized Schur algorithm factors more general structured matrices, and this fact is readily used to extend the Chandrasekhar recursions to a class of time-variant state-space models, special cases of which often arise in adaptive filtering. >

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

[2]  T. Kailath,et al.  Triangular Factorization of Structured Hermitian Matrices , 1986 .

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

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

[5]  Tryphon T. Georgiou,et al.  On a Schur-algorithm based approach to spectral factorization: connection with the Riccati equation☆ , 1992 .

[6]  Thomas Kailath,et al.  State-space approach to factorization of lossless transfer functions and structured matrices☆ , 1992 .

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

[8]  Ali H. Sayed,et al.  Extended Chandrasekhar recursions , 1994, IEEE Trans. Autom. Control..

[9]  N. Levinson.  The Wiener (Root Mean Square) Error Criterion in Filter Design and Prediction , 1946 .

[10]  L. Ljung,et al.  Extended Levinson and Chandrasekhar equations for general discrete-time linear estimation problems , 1978 .

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

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

[13]  J. L. Hock,et al.  An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space , 1984 .

[14]  Thomas Kailath,et al.  Lattice filter parameterization and modeling of nonstationary processes , 1984, IEEE Trans. Inf. Theory.

[15]  T. Kailath,et al.  A state-space approach to adaptive RLS filtering , 1994, IEEE Signal Processing Magazine.

[16]  A. Lindquist.  A New Algorithm for Optimal Filtering of Discrete-Time Stationary Processes , 1974 .

[17]  M. Morf,et al.  Displacement ranks of matrices and linear equations , 1979 .

[18]  A. Lindquist.  Some reduced-order non-Riccati equations for linear least-squares estimation : the stationary, single-output case† , 1976 .