Linear Optimal FIR Estimation of Discrete Time-Invariant State-Space Models

This paper addresses a general p-shift linear optimal finite impulse response (FIR) estimator intended for solving universally the problems of filtering (p = 0), smoothing (p < 0), and prediction (p > 0) of discrete time-invariant models in state space. An optimal solution is found in the batch form with the initial mean square state function self-determined by solving the discrete algebraic Riccati equation. An unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state. The mean square errors in both the optimal and unbiased estimates are found via the noise power gain (NPG) and a recursive algorithm for fast computation of the NPG is supplied. Applications are given for FIR filtering with fixed, receding, and full averaging horizons.

[1]  A. Jazwinski.  Limited memory optimal filtering , 1968 .

[2]  Stephen P. Boyd,et al.  Receding Horizon Control , 2011, IEEE Control Systems.

[3]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[4]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[5]  Chi-Tsong Chen,et al.  One-Dimensional Digital Signal Processing , 1979 .

[6]  George Epstein.  On finite-memory recursive filters (Corresp.) , 1970, IEEE Trans. Inf. Theory.

[7]  Wook Hyun Kwon,et al.  Minimum Variance FIR Smoothers for Discrete-Time State Space Models , 2007, IEEE Signal Processing Letters.

[8]  Henry Stark,et al.  Probability, Random Processes, and Estimation Theory for Engineers , 1995 .

[9]  Yuriy S. Shmaliy,et al.  Unbiased FIR Filtering of Discrete-Time Polynomial State-Space Models , 2009, IEEE Transactions on Signal Processing.

[10]  Yuriy S. Shmaliy,et al.  Optimal FIR filtering of the clock time errors , 2008 .

[11]  Li Danyang,et al.  Optimal state estimation without the requirement of a priori statistics information of the initial state , 1994 .

[12]  Wook Hyun Kwon,et al.  A receding horizon unbiased FIR filter for discrete-time state space models , 2002, Autom..

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

[14]  Yuriy S. Shmaliy,et al.  An unbiased p-step predictive FIR filter for a class of noise-free discrete-time models with independently observed states , 2009, Signal Image Video Process..

[15]  Yuriy S. Shmaliy,et al.  Optimal Gains of FIR Estimators for a Class of Discrete-Time State-Space Models , 2008, IEEE Signal Processing Letters.

[16]  W. Kwon,et al.  Receding Horizon Control: Model Predictive Control for State Models , 2005 .

[17]  P. J. Buxbaum.  Fixed-memory recursive filters (Corresp.) , 1974, IEEE Trans. Inf. Theory.

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

[19]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

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

[21]  Wook Hyun Kwon,et al.  A receding horizon Kalman FIR filter for discrete time-invariant systems , 1999, IEEE Trans. Autom. Control..

[22]  G. J. Bierman.  Fixed memory least squares filtering , 1975 .

[23]  Ji-Woong Choi,et al.  An FIR Channel Estimation Filter with Robustness to Channel Mismatch Condition , 2008, IEEE Transactions on Broadcasting.

[24]  Yuriy S. Shmaliy,et al.  Optimal Synchronization of Local Clocks by GPS 1PPS Signals Using Predictive FIR Filters , 2009, IEEE Transactions on Instrumentation and Measurement.

[25]  Yuriy S. Shmaliy,et al.  Optimal horizons for a one-parameter family of unbiased FIR filters , 2008, Digit. Signal Process..

[26]  Kent R. Johnson,et al.  Optimum, linear, discrete filtering of signals containing a nonrandom component , 1956, IRE Trans. Inf. Theory.

[27]  N. Wiener.  The Wiener RMS (Root Mean Square) Error Criterion in Filter Design and Prediction , 1949 .

[28]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[29]  Y.S. Shmaliy,et al.  An unbiased FIR filter for TIE model of a local clock in applications to GPS-based timekeeping , 2006, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[30]  M. N. Shanmukha Swamy,et al.  A nonlinear adaptive filter for narrowband interference mitigation in spread spectrum systems , 2005, Signal Process..

[31]  Jay H. Lee,et al.  Receding Horizon Recursive State Estimation , 1993, 1993 American Control Conference.

[32]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

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

[34]  Visa Koivunen,et al.  Detection and Tracking of MIMO Propagation Path Parameters Using State-Space Approach , 2009, IEEE Transactions on Signal Processing.

[35]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[36]  N. F. Toda,et al.  Divergence in the Kalman Filter , 1967 .

[37]  Wook Hyun Kwon,et al.  Optimal FIR filters for time-varying state-space models , 1990 .

[38]  Yuriy S. Shmaliy,et al.  A thinning algorithm for GPS-based unbiased FIR estimation of a clock TIE model , 2008 .

[39]  W. Kwon,et al.  Equivalence of finite memory filters , 1994 .

[40]  Jinhong Yuan,et al.  Joint Channel Tracking and Decoding for BICM–OFDM Systems Using Consistency Tests and Adaptive Detection Selection , 2009, IEEE Transactions on Vehicular Technology.

[41]  Alfred M. Bruckstein,et al.  Recursive limited memory filtering and scattering theory , 1985, IEEE Trans. Inf. Theory.

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

[43]  Frédéric Lehmann,et al.  Blind turbo-detection in the presence of phase noise , 2009, IET Commun..

[44]  Stephen Yurkovich,et al.  Fuzzy Control , 1997 .

[45]  Yuriy S. Shmaliy,et al.  FIR Smoothing of Discrete-Time Polynomial Signals in State Space , 2010, IEEE Transactions on Signal Processing.

[46]  Wook Hyun Kwon,et al.  FIR filters and recursive forms for discrete-time state-space models , 1987, Autom..

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

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

[49]  Gerald J. Bierman.  Fixed memory least squares filtering (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[50]  Y. Shmaliy,et al.  Efficient predictive estimator for holdover in GPS-based clock synchronization , 2008, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[51]  C. Brezinski.  Interpolation and Extrapolation , 2001 .