a la physique de l'information - Lisa - Université d'Angers

IMPROVED BAYESIAN ESTIMATION OF WEAK SIGNALS
IN NON-GAUSSIAN NOISE BY OPTIMAL QUANTIZATION
P. R. BHAT 1∗ , D. ROUSSEAU 2∗ , G. V. ANAND 3
1 Department of Electrical Engineering
Indian Institute of Technology MADRAS, CHENNAI, India.
2 Laboratoire d’Ingénierie des SystÁemes Automatisés (LISA, CNRS, 2656)
Université d’Angers, 62 avenue Notre Dame du Lac, 49000 ANGERS, France.
3 Department of Electrical Communication Engineering
Indian Institute of Science, BANGALORE 560012, India.
e-mail: anandgv@ece.iisc.ernet.in
ABSTRACT
In this paper, we present a new improved method for
signal shape estimation in non-Gaussian noise with low signal
to noise ratio. We combine a nonlinear preprocessing
with Wiener £ltering. In the proposed method, the data received
is £rst quantized by a symmetric 3–level quantizer
before processing by the Wiener £lter. A complete theoretical
analysis of this quantizer–estimator is worked out under
low signal to noise ratio conditions. In this framework,
we show that if the noise is suf£ciently non-Gaussian and
the quantizer thresholds are optimally chosen, the quantization,
although limited to 3–levels, leads to an enhancement
of the estimation performed by the Wiener £lter. Numerical
results comparing the quantizer–estimator with the
Wiener £lter applied alone are presented to con£rm the theory.
Non-Gaussian noise distributions speci£cally relevant
for an underwater acoustic environment are chosen for illustration.
1. INTRODUCTION
Shape estimation of a signal buried in noise is a problem of
great interest for many signal processing applications [1]. In
presence of Gaussian noise, the optimal Bayesian estimator
minimizing the estimation mean square error is the Wiener
£lter that has the advantage of being linear. In non-Gaussian
noisy environment, the optimal estimators are usually nonlinear
and hence more dif£cult to implement and, sometimes,
even to determine explicitly. Non-Gaussian noise
can be found in many practical situations like, for example,
in the £eld of ocean acoustics [2]. Therefore, there is a
* First and second authors performed the work while at 3
This work was supported by the Defense Research and Development
Organisation under the DRDO-IISc Joint Programme on Advanced Research
in Mathematical Engineering.
need to develop simple nonlinear estimators capable of outperforming
the Wiener estimator with non-Gaussian noise.
Such estimators would be especially useful when the signal
to be estimated is very weak compared to the noise; indeed
in these conditions the estimation performance of the optimal
linear estimator is very poor and really needs to be
enhanced. In this paper, we describe, analyze and evaluate
a new simple nonlinear estimator capable of outperforming
the Wiener £lter under conditions of weak signal and non-
Gaussian noise.
2. AN ESTIMATION PROBLEM
We consider the problem of estimating the shape of a weak
signal s(t) buried in an additive noise η(t). The observable
signal x(t) = s(t) + η(t) is uniformaly sampled to provide
N data points xi = x(i × T ) with i = 0, 1, . . . , N − 1 and
1/T the sampling rate. From the data set x = (x1, . . . xN )
the N-dimensional vector s = (s1, . . . sN ) is to be estimated.
The noise samples ηi are assumed independent and
identically distributed with cumulative distribution Fη(u),
and probability density function (pdf) fη(u) = dFη/du,
with zero mean and unit variance.
E [ηi] = 0; E [ηiηj] = δij, (1)
where δij = 1 if j = i and 0 otherwise. The signal s is
assumed to be a random wide-sense stationary process with
exponentially decaying autocorrelation. The signal samples
si are assumed identically distributed with cumulative distribution
Fs(u), pdf fs(u) = dFs/du, with zero mean and
variance σ 2 s. Therefore we have
with
E [sj] = 0; E [sisj] = σ 2 sρij , (2)
ρij = exp(−ζ|i − j|). (3)
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