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ESTIMATION ALGORITHM FROM RANDOMLY DELAYED
OBSER
VATIONS WITH WHITE PLUS COLOURED NOISES
S. Nakamori
Department of Technology. Faculty of Education.
Kagoshima University 1-20-6, Kohrimoto, Kagoshima 890-0065, Japan.
A. Hermoso-Carazo, J. Linares-P
´
erez and M. I. S
´
anchez-Rodr
´
ıguez
Departamento de Estad
´
ıstica e I. O. Universidad de Granada.
Campus Fuentenueva s/n, 18071 Granada.
Keywords:
Signal estimation, randomly delayed observations, coloured noise, covariance information.
Abstract:
A recursive algorithm for the least-squares linear one-stage prediction and filtering problems of discrete-time
signals using randomly delayed measurements perturbed by additive white plus coloured noises are presented.
It is assumed that the autocovariance function of the signal and the coloured noise are expressed in a semi-
degenerate kernel form and the delay is modelled by a sequence of independent Bernoulli random variables,
which indicate if the measurements arrive in time or are delayed by one sampling time. The estimators are
obtained by an innovation approach and do not use the state-space model of the signal, but only the covariance
information about the signal and the observation noises and the delay probabilities.
1 INTRODUCTION
There are many situations, such as the ones relative
to telecommunication scope, in which it is possible
that the measurements available to estimate the state
of a system may not arrive in time, but delayed by
a any sampling time. Although sometimes these
delays have been treated as measurement errors or as
deterministic functions of the time, these assumptions
are not always accurate and, in these cases, the
best way to model the delay is to interpret it as a
stochastic process, including its statistical properties
in the system model.
Many recent works have used stochastic time-delay
models to treat estimation problems. For example, the
state estimation in a model with randomly varying
sensor delays has been described as a estimation
problem in systems with stochastic parameters (Yaz
and Ray, 1998). Also, the state estimation has been
treated in the case where a finite-state Markov chain is
applied to model the random delay in the observations
(Evans and Krishnamurthy, 1999).
The above studies consider that the state-space
generating the signal is known but, in many situations,
it is not available and estimation algorithms using
another kind of information, such as covariance one,
must be used. In (Nakamori et. al, 2004b), the least-
squares linear filtering and fixed-point smoothing
problems from measurements with stochastic delays,
perturbed by white noise, is treated by using
covariance information.
In this paper, we treat the least-squares linear
prediction and filtering problems of signals using
randomly delayed measurements which are perturbed
by additive white plus coloured noises. The delay is
modelled by a binary white noise, whose values, zero
or one, indicate if the measurements arrive in time or
are delayed by one sampling period.
This study also generalizes the work (Nakamori
et. al, 2004a), which consider uncertain observations
affected by additive white plus coloured noises
without delay in time.
The estimators are obtained without requiring the
state-space model generating the signal, but just using
the covariance functions of the signal and the noises,
assuming a semi-degenerate kernel form for the signal
and coloured noise autocovariance functions, and
the delay probabilities. Finally, the effectiveness
of the proposed algorithms is shown in a computer
simulation example.
2 PROBLEM FORMULATION
We consider the estimation problem of a n × 1 signal
z
k
from delayed observations described by
ey
k
= z
k
+ v
k
+ w
k
, k ≥ 0,
y
k
= (1 − γ
k
)ey
k
+ γ
k
ey
k−1
, k ≥ 1.
19
Nakamori S., Hermoso-Carazo A., Linares-Pérez J. and Sánchez-Rodríguez M. (2004).
ESTIMATION ALGORITHM FROM RANDOMLY DELAYED OBSERVATIONS WITH WHITE PLUS COLOURED NOISES.
In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 19-23
DOI: 10.5220/0001127000190023
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SciTePress