
Plenary
Lecture
DiscreteTime Optimal and Unbiased FIR Estimation of
State Space Models
Professor Yuriy S.
Shmaliy
Department of Electronics
FIMEE, Guanajuato University
Salamanca, 36885, Mexico
Email:
shmaliy@salamanca.ugto.mx
Abstract:
Optimal estimation of signal parameters and system
models is often required to formalize a posteriori
knowledge about undergoing processes in the presence of
noise. Therefore, filtering, smoothing, and prediction
have become key tools of statistical signal, image, and
speech processing and found applications in algorithms
of various electronic systems. Very often, estimation is
provided using methods of linear optimal filtering
employing either finite impulse response (FIR) or
infinite impulse response (IIR) structures. First
fundamental works on discretetime optimal linear
filtering of stationary random processes were published
in 19391941 by Kolmogorov as mathematicallyoriented.
Soon after, Wiener solved the problem for engineering
applications in continuoustime and Levinson used the
Wiener error criterion in filter design and prediction.
The solutions by Wiener were all in the frequency
domain, presuming IIR solutions. A FIR modification to
the Wiener filter was made by Zadeh and Ragazzini.
Thereafter, Johnson extended ZadehRagazzini's results
to discrete time. The roots of optimal FIR filtering can
be found namely in these basic works. Despite the
inherent bounded input/bounded output stability and
robustness against temporary model uncertainties and
roundoff errors, practical interest to FIR filtering
weakened after Kalman and Bucy presented in 19601961
complete results on the theory of linear filtering of
nonstationary Gaussian processes. In contrast to the FIR
solutions implying large computational burden and
memory, the recursive IIR KalmanBucy algorithm has
appeared to be simple, accurate, and fast. That has
generated an enormous number of papers devoted to the
investigation and application of this filter. It then
has been shown that the Kalman filter is a nice solution
if the model is distinct, there are no uncertainties,
and noise sources are all white sequences. Otherwise,
the algorithm may become unstable and its estimate may
diverge. An interest to FIR structures has grown in
recent decades owing to a dramatic development in
computational resources. In receding horizon predictive
control, significant results on optimal linear FIR
filtering of Gaussian processes have been achieved by
Jazwinski, Liu and Liu, Ling and Lim, and Kwon et al.
For image processing, predictive FIR filtering has been
proposed by Heinonen and Neuvo and thereafter developed
by many authors. For polynomial models, FIR structures
were used by Wang to design a nonlinear filter, by Zhou
and Wang in the FIRmedian hybrid filters, and a number
of publications keep growing.
In this presentation, we show that the general theory of
the pshift optimal linear FIR estimator follows
straightforwardly from the realtime state space model
(from n and n1 to n) used in signal processing, rather
than from the prediction model (from n to n+1) used in
control. This model allows for a universal estimator
intended for solving the problems of filtering (p = 0),
prediction (p > 0), and smoothing (p < 0) in
discretetime and state space on a horizon of N points.
In such an estimator, the initial state is
selfdetermined by solving the discrete algebraic
Riccati equation (DARE). The noise components are
allowed to have arbitrary distribution and covariance
functions with a particular case of white Gaussian
approximation. Depending on p, the estimator is readily
modified to solve several specific problems, such as the
receding horizon control one (p = 1), smoothing the
initial state (p = ?N+1), holdover in digital
communication networks (p > 0), etc. We show that the
optimal FIR estimator gain is a product of the unbiased
gain and the noisedependent function composed with the
covariance functions and the initial state function. An
important point is that the optimal and unbiased
estimates converge either when the convolution length is
large, N >> 1, or if the initial state error dominates
the noise components. The unbiased (near optimal) FIR
estimate associated with the best linear unbiased
estimator (BLUE) is considered in detail as having
strong engineering features. Along with the noise power
gain (NPG), this estimate can be represented in batch
and recursive Kalmanlike forms. A special attention is
paid to the polynomial state space models as being basic
for many applications. For this model, the unique
lowdegree polynomial gains are derived and investigated
in detail. Applications are given for polynomial state
space modeling, clock state estimation and
synchronization, and image processing. The tradeoff
with the Kalman algorithm is also discussed and
supported with experimental results.
Brief Biography of the Speaker:
Professor Yuriy S. Shmaliy is a Full Professor of
Electronics of the School of Mechanical, Electrical, and
Electronic Engineering (FIMEE) of the University of
Guanajuato, Mexico. He received the B.S., M.S., and
Ph.D. degrees in 1974, 1976 and 1982, respectively, from
the Kharkiv Aviation Institute, Ukraine, all in
Electrical Engineering. In 1992 he received the Doctor
of Technical Sc. degree from the Kharkiv Railroad
Institute. In March 1985, he joined the Kharkiv Military
University. He serves as Full Professor beginning in
1986. Since 1999 to 2009, he has been with the Kharkiv
National University of Radio Electronics.
Prof. Shmaliy has 250 Journal and Conference papers and
80 patents. His books ContinuousTime Signals (2006) and
ContinuousTime Systems (2007) were published by
Springer. His book GPSBased Optimal FIR Filtering of
Clock Models (2009) was published by Nova Science Publ.,
New York. He also contributed with several invited
Chapters to books. He was rewarded a title, Honorary
Radio Engineer of the USSR, in 1991. He was listed in
Marquis Who's Who in the World in 1998; Outstanding
People of the 20th Century, Cambridge, England in 1999;
and Contemporary Who's Who, American Bibliographical
Institute, in 2002. He is a Senior Member of IEEE. He
has Certificates of Recognition and Appreciation from
the IEEE, WSEAS, and IASTED. He serves as an Associate
Editor in Recent Patents on Space Technology. He is a
member of several Organizing and Program Committees of
Int. Symposia. He organized and chaired several
International Conferences on Precision Oscillations in
Electronics and Optics. He was multiply invited to give
tutorial, seminar, and plenary lectures. His current
interests include optimal estimation, statistical signal
processing, and stochastic system theory.

