FIR Smoothing of Discrete-Time Polynomial Signals in State Space

We address a smoothing finite impulse response (FIR) filtering solution for deterministic discrete-time signals represented in state space with finite-degree polynomials. The optimal smoothing FIR filter is derived in an exact matrix form requiring the initial state and the measurement noise covariance function. The relevant unbiased solution is represented both in the matrix and polynomial forms that do not involve any knowledge about measurement noise and initial state. The unique l-degree unbiased gain and the noise power gain are derived for a general case. The widely used low-degree gains are investigated in detail. As an example, the best linear fit is provided for a two-state clock error model.

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

[2]  Tamal Bose,et al.  Digital Signal and Image Processing , 2003 .

[3]  Choon Ki Ahn,et al.  Fixed-lag maximum likelihood FIR smoother for state-space models , 2008, IEICE Electron. Express.

[4]  John B. Moore,et al.  Discrete-time fixed-lag smoothing algorithms , 1973 .

[5]  Xin Wang.  NFIR nonlinear filter , 1991, IEEE Trans. Signal Process..

[6]  Steven W. Smith,et al.  The Scientist and Engineer's Guide to Digital Signal Processing , 1997 .

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

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

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

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

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

[12]  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.

[13]  B. Dumitrescu.  Positive Trigonometric Polynomials and Signal Processing Applications , 2007 .

[14]  Jenq-Tay Yuan,et al.  Order-recursive FIR smoothers , 1994, IEEE Trans. Signal Process..

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

[16]  T. G. Campbell,et al.  Predictive FIR filters with low computational complexity , 1991 .

[17]  Yuriy S Shmaliy.  Linear unbiased prediction of clock errors , 2009, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[18]  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.

[19]  Y.S. Shmaliy.  On Real-Time Optimal FIR Estimation of Linear TIE Models of Local Clocks , 2007, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[20]  Xiaobo Zhou,et al.  FIR-median hybrid filters with polynomial fitting , 2004, Digit. Signal Process..

[21]  Yrjö Neuvo,et al.  FIR-median hybrid filters with predictive FIR substructures , 1988, IEEE Trans. Acoust. Speech Signal Process..

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

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

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

[25]  Wook Hyun Kwon,et al.  A Note on Two-Filter Smoothing Formulas , 2008, IEEE Transactions on Automatic Control.

[26]  H. Rauch.  Solutions to the linear smoothing problem , 1963 .

[27]  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..

[28]  Wook Hyun Kwon,et al.  $L_{2}-E$ FIR Smoothers for Deterministic Discrete-Time State–Space Signal Models , 2007, IEEE Transactions on Automatic Control.

[29]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .