CONTINUOUS-TIME SIGNAL FILTERING FROM
NON-INDEPENDENT UNCERTAIN OBSERVATIONS
S. Nakamori
Department of Technology. Faculty of Education, Kagoshima University
1-20-6, Kohrimoto, Kagoshima 890-0065, Japan
A. Hermoso-Carazo
Departamento de Estad
´
ıstica e Investigaci
´
on Operativa, Universidad de Granada
Campus Fuentenueva, s/n, 18071 Granada, Spain
J. Jim
´
enez-L
´
opez
Departamento de Estad
´
ıstica e Investigaci
´
on Operativa, Universidad de Ja
´
en
Paraje Las Lagunillas, s/n, 23071 Ja
´
en, Spain
J. Linares-P
´
erez
Departamento de Estad
´
ıstica e Investigaci
´
on Operativa, Universidad de Granada
Campus Fuentenueva, s/n, 18071 Granada, Spain
Keywords:
Uncertain observations, Chandrasekhar-type filter, covariance information
Abstract:
Filtering algorithms are presented as solution of the least mean-squared error linear estimation problem of
continuous-time wide-sense stationary scalar signals from uncertain observations perturbed by white and
coloured additive noises. These algorithms, one of them based on Chandrasekhar-type equations and the
other on Riccati-type ones, are derived assuming a specific type of dependence between the Bernoulli ran-
dom variables describing the uncertainty and do not require the whole knowledge of the state-space model.
By comparing both algorithms it is deduced that the Chandrasekhar-type one is more advantageous from a
computational viewpoint.
1 INTRODUCTION
In the mid-seventies, the replacement of the Riccati-
type equations by a set of Chandrasekhar-type ones
in the algorithms proposed as solution of the least
mean-squared error (LMSE) linear estimation prob-
lem led to more advantageous algorithms from a com-
putational point of view since the Chandrasekhar-type
algorithms contain, generally, less difference or dif-
ferential equations than the ones based on Riccati-
type equations. For continuous-time invariant sys-
tems, Kailath (1973) was the first author who pro-
posed an algorithm of this kind to solve the LMSE
linear estimation problem. This work was the starting
point for many posterior contributions. We shall men-
tion, among others, Sayed and Kailath (1994) who,
assuming a full knowledge of the state-space model,
obtained Chandrasekhar-type algorithms for a class of
time-variant models. Recently, Nakamori (2000) has
proposed a Chandrasekhar-type algorithm to estimate
a continuous-time wide-sense stationary signal from
observations perturbed by white and coloured addi-
tive noises but assuming, in contrast to the above pa-
pers, that the state-space model is not available and
using covariance information.
On the other hand, in the last decades, consider-
able attention has been given to systems with uncer-
tain observations, since they model many real situ-
ations. These systems are characterized by includ-
ing in the observation model, besides additive noise,
a multiplicative noise described by Bernoulli random
variables, which determine the presence or absence of
signal in the observations. These systems are then ap-
propriate to model situations in which there exist in-
termittent failures in the observation mechanism and,
hence, the observations may contain noise plus sig-
nal or only noise in a random manner; for example,
communication systems with random interruptions.
The LMSE linear estimation problem of discrete
signals from uncertain observations has been ap-
322
Nakamori S., Hermoso-Carazo A., Jiménez-López J. and Linares-Pérez J. (2004).
CONTINUOUS-TIME SIGNAL FILTERING FROM NON-INDEPENDENT UNCERTAIN OBSERVATIONS.
In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 324-327
DOI: 10.5220/0001127703240327
Copyright
c
 SciTePress

proached under different hypotheses. For example,
Hermoso and Linares (1994) proposed a Riccati-
type algorithm in discrete-time systems, considering
that the Bernoulli random variables are independent;
on the other hand, Hadidi and Schwartz (1979) ob-
tained an estimation algorithm based also on Ric-
cati equations, but considering a specific type of de-
pendence between the Bernoulli variables. Both pa-
pers are based on a full knowledge of the state-space
model; recently, Nakamori et al. (2004) have pro-
posed a Chandrasekhar-type filtering algorithm for
wide-sense stationary signals from uncertain observa-
tions without using the state-space model but covari-
ance information.
In this paper, we analyze the LMSE linear filter-
ing problem of continuous-time wide-sense stationary
scalar signals from uncertain observations perturbed
by white and coloured additive noises. Assuming
that the Bernoulli random variables present a type of
dependence analogous to that considered by Hadidi
and Schwartz (1979), we propose a Chandrasekhar
and a Riccati-type algorithm, derived by using covari-
ance information. The comparison between both al-
gorithms shows the computational advantages of the
Chandrasekhar-type one.
2 ESTIMATION PROBLEM
Let us consider a continuous-time scalar observation
equation described by
y(t) = u(t)z(t)+v(t)+v
0
(t), z(t) = Hx(t), t ≥ 0
(1)
where y(t) represents the observation of the signal
z(t), perturbed by a multiplicative noise, u(t), and
by white and coloured additive noises, v(t) and v
0
(t),
respectively; the signal is expressed as a linear com-
bination of the components of the n-dimensional state
vector x(t).
Denoting by Φ and Φ
0
the system matrices of the
state and the coloured noise, respectively, we have as-
sumed the following hypotheses on the processes ap-
pearing in equation (1):
(H.1) The signal process {z(t); t ≥ 0} is wide-sense
stationary with zero mean, being its autoco-
variance function K
z
(t, s) = E[z(t)z(s)] =
K
z
(t − s), for t, s ≥ 0. Moreover, the cross-
covariance function of the state x(t) and the
signal z(s), K
xz
(t, s), verifies the differential
equation
∂K
xz
(t, s)
∂t
= ΦK
xz
(t, s), s < t.
(H.2) The additive noise {v(t); t ≥ 0} is a zero-mean
white process whose autocovariance function is
given by E[v(t)v(s)] = Rδ
D
(t − s), for t, s ≥
0, being R 6= 0 and δ
D
the Dirac delta function.
(H.3) The coloured noise {v
0
(t); t ≥ 0} is
a zero-mean wide-sense stationary process
with autocovariance function K
v
0
(t, s) =
E[v
0
(t)v
0
(s)] = K
v
0
(t−s), for t, s ≥ 0, which
satisfies the differential equation
∂K
v
0
(t, s)
∂t
= Φ
0
K
v
0
(t, s), s < t.
(H.4) The multiplicative noise {u(t); t ≥ 0} de-
scribing the uncertainty in the observations is
modelled by identically distributed Bernoulli
random variables with initial probability vector
(1 − p, p)
T
and conditional probability matrix
P (t/s). We assume that the (2, 2)-element of
this matrix is independent of t and s, that is,
P (u(t) = 1/u(s) = 1) = p
22
for t 6= s. Under these considerations, it is clear
that
E [u(t)u(s)] =
½
p, if t = s
p p
22
, if t 6= s
(H.5) The processes {x(t); t ≥ 0}, {u(t); t ≥ 0},
{v(t); t ≥ 0} and {v
0
(t); t ≥ 0} are mutually
independent.
Under these considerations, our aim consists of de-
termining an algorithm to calculate the LMSE linear
estimator of the signal z(t) given the observations un-
til time t, that is {y(s); 0 ≤ s ≤ t}. It is clearly ob-
served that this estimator, denoted by bz(t), can be ex-
pressed as bz(t) = Hbx(t), where bx(t) is the LMSE
linear filter of the state. For this reason, we have fo-
cussed our interest on obtaining an algorithm for bx(t),
which can be expressed as
bx(t) =
Z
t
0
h(t, τ )y(τ )dτ (2)
where {h(t, τ), 0 ≤ τ ≤ t} denotes the impulse-
response function.
As a consequence of the Orthogonal Projection
Lemma (OPL) and the hypotheses on the model, bx(t)
satisfies the Wiener-Hopf equation, given by
pK
xz
(t, s) =
Z
t
0
h(t, τ )E [y(τ)y(s)] dτ, 0 ≤ s ≤ t
(3)
or, equivalently,
h(t, s)R = pK
xz
(t, s) −
Z
t
0
h(t, τ )
K(τ, s)dτ,
(4)
K(τ, s) = pp
22
HK
xz
(τ, s) + K
v
0
(τ, s).
In order to determine a differential equation for bx(t),
we differentiate (2) with respect to t and so, we obtain
dbx(t)
dt
=
Z
t
0
∂h(t, τ)
∂t
y(τ)dτ + h(t, t)y(t). (5)
CONTINUOUS-TIME SIGNAL FILTERING FROM NON-INDEPENDENT UNCERTAIN OBSERVATIONS
323

On the other hand, differentiating (3) with respect to t,
from (H.1) and the OPL, it is had that, for 0 < s < t,
Z
t
0
µ
Φh(t,τ)−
∂h(t,τ)
∂t
−h(t,t)[p
22
Hh(t,τ) + g(t,τ)]
´
E[y(τ )y(s)] dτ = 0
where {g(t, τ), 0 ≤ τ ≤ t} represents the impulse-
response function of the coloured noise filter, bv
0
(t).
Then, it is clear that this integral equation will be sat-
isfied if we consider a function h satisfying
∂h(t, τ)
∂t
= Φh(t, τ ) − h(t, t)[p
22
Hh(t, τ) + g(t, τ)]
(6)
for 0 ≤ τ ≤ t. Hence, by substituting (6) in (5), the
following differential equation for bx(t) is derived
dbx(t)
dt
= Φbx(t) + h(t, t) [y(t) − p
22
Hbx(t) − bv
0
(t)]
(7)
with initial condition bx(0) = 0.
By following an analogous reasoning, it is obtained
the Wiener-Hopf equation for bv
0
(t),
g(t, s)R = K
v
0
(t, s)−
Z
t
0
g(t, τ )
K(τ, s)dτ, 0 ≤ s ≤ t
(8)
and the following differential equation,
∂g(t, τ)
∂t
= Φ
0
g(t, τ )−g(t, t)[p
22
Hh(t, τ)+g(t, τ)].
(9)
Then, bv
0
(t) verifies the differential equation
d bv
0
(t)
dt
= Φ
0
bv
0
(t)+g(t, t) [y(t) − p
22
Hbx(t) − bv
0
(t)]
(10)
with initial condition bv
0
(0) = 0.
To complete the algorithm, in the following section
we show two different ways to calculate the filtering
gains, h(t, t) and g(t, t).
3 FILTERING ALGORITHMS
Next we derive two algorithms as a solution of the
LMSE filtering problem: in one of them, the filtering
gains are obtained from Chandrasekhar-type equa-
tions whereas, in the other, Riccati-type ones are used.
3.1 Chandrasekhar-type algorithm
Theorem 1. The filter of the signal, bz(t), is calcu-
lated from the relation bz(t) = H bx(t) where the state
filter, bx(t), satisfies the differential equation (7) and
the coloured noise filter, bv
0
(t), is given from (10).
The filtering gains are calculated as follows
dh(t, t)
dt
= −h(t, 0) [p
22
Hh(t, 0) + g(t, 0)] (11)
dg(t, t)
dt
= −g(t, 0) [p
22
Hh(t, 0) + g(t, 0)] (12)
where h(t, 0) and g(t, 0) satisfy the following differ-
ential equations
dh(t, 0)
dt
= Φh(t, 0) − h(t, t)[p
22
Hh(t, 0) + g(t, 0)]
(13)
dg(t, 0)
dt
= Φ
0
g(t, 0) − g(t, t)[p
22
Hh(t, 0) + g(t, 0)]
(14)
being the initial conditions
h(0, 0) = pR
−1
K
xz
(0). (15)
g(0, 0) = R
−1
K
v
0
(0). (16)
Proof. Differentiating (4) with respect to t and s, we
obtain the following expression, valid for 0 ≤ s ≤ t,
µ
∂h(t, s)
∂t
+
∂h(t, s)
∂s
¶
R
=−h(t, 0)K(0,s)−
Z
t
0
µ
∂h(t,τ)
∂t
+
∂h(t,τ)
∂τ
¶
K(τ,s)dτ
where we have used that, from the stationary property
of the signal process,
∂K
xz
(t, s)
∂t
+
∂K
xz
(t, s)
∂s
= 0.
Then, if we define a function J satisfying
J(t, s)R =
K(0, s)−
Z
t
0
J(t, τ)
K(τ, s)dτ, 0 ≤ s ≤ t
(17)
it is immediately obtained that
∂h(t, s)
∂t
+
∂h(t, s)
∂s
= −h(t, 0)J(t, s), 0 ≤ s ≤ t.
(18)
Next, if (4), replacing s by t − s, is multiplied on the
left by p
22
H and the resultant expression is added to
(8) replacing also s by t − s, we obtain
[p
22
Hh(t, t − s) + g(t, t − s)]R
=
K(s, 0)−
Z
t
0
[p
22
Hh(t, t−τ)+g(t, t−τ )]
K(s, τ )dτ
(19)
for 0 ≤ s ≤ t. Then, by comparing (17) and (19),
J(t, s) = p
22
Hh(t, t − s) + g(t, t − s), 0 ≤ s ≤ t
(20)
and, consequently, from (18) and (20), (11) is derived.
On the other hand, the differential equation (13)
and the initial condition (15) are respectively derived
by taking τ = 0 in (6) and s = t = 0 in (4).
By following an analogous reasoning, the differen-
tial equations (12) and (14) and the initial condition
(16) are deduced. ¤
ICINCO 2004 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL
324

3.2 Riccati-type algorithm
Theorem 2. The filter of the signal, bz(t), is calcu-
lated from the relation bz(t) = H bx(t) where the state
filter, bx(t), satisfies the differential equation (7) and
the coloured noise filter, bv
0
(t), is given from (10).
The filtering gains are given by
h(t, t) = R
−1
£
pK
xz
(0)−p
22
S(t)H
T
−T (t)
¤
(21)
g(t, t) = R
−1
£
K
v
0
(0)−p
22
T
T
(t)H
T
−U(t)
¤
(22)
where S(t) = E[bx(t)bx
T
(t)], T (t) = E[bx(t) bv
0
(t)]
and U (t) = E[ bv
0
2
(t)] satisfy the following differen-
tial equations
dS(t)
dt
= ΦS(t)+S(t)Φ
T
+ Rh(t, t) h
T
(t, t),
dT (t)
dt
= ΦT(t)+Φ
0
T (t)+ Rg(t, t)h(t, t),
dU(t)
dt
= 2Φ
0
U(t)+Rg
2
(t, t).
(23)
Proof. From the OPL and the hypotheses on the
model, we have that
S(t) = E[bx(t)x
T
(t)] = p
Z
t
0
h(t, τ )K
T
xz
(t, τ )dτ
T (t) = E[bx(t)v
0
(t)] =
Z
t
0
h(t, τ )K
v
0
(τ, t)dτ
U(t) = E[ bv
0
(t)v
0
(t)] =
Z
t
0
g(t, τ )K
v
0
(τ, t)dτ.
(24)
Then by putting s = t in (4) and (8) and using (24),
equations (21) and (22) are obtained. Again, from
(24), the differential equations given in (23) are de-
rived by using (6) and (9). ¤
By comparing the algorithms proposed in the above
theorems, it is observed that the Chandrasekhar-
type one contains less differential equations than the
Riccati-type algorithm; specifically, 3n + 3 are the
differential equations included in the Chandrasekhar-
type algorithm, and n
2
+ 2n + 2, in the Riccati-type
one. So, if n ≥ 2, there exists a reduction regarding
the number of equations in the Chandrasekhar-type
algorithm, which implies a decrease in the computa-
tion time. Hence, the Chandrasekhar-type algorithm
is more advantageous than the Riccati-type one in a
computational sense.
Finally, as a measure of the estimation accuracy, the
filtering error variance, which is defined by P (t) =
E
h
{z(t) − bz(t)}
2
i
, can be calculated as
P (t, t) = H
£
K
xz
(t, t) − S(t)H
T
¤
with S(t) given in Theorem 2.
4 CONCLUSION
In this paper, the LMSE linear filtering problem of
wide-sense stationary scalar signals in continuous-
time systems with uncertain observations perturbed
by white and coloured additive noises is analyzed.
Assuming uncertainty non-independent we derive two
algorithms without requiring the whole knowledge of
the state-space model but using covariance informa-
tion. Both algorithms differs in the way of calculat-
ing the filtering gains. From the comparison between
them it is deduced that the Chandrasekhar-type one
is computationally more appropriate than the Riccati-
type algorithm.
ACKNOWLEDGMENT
Supported by the ‘Ministerio de Ciencia y Tec-
nolog
´
ıa’. Contract BFM2002-00932.
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