Digital Signal Processing Reference
In-Depth Information
approaches, having in mind the problems to be discussed in the subsequent
chapters.
2.5.1 The Estimation Problem
For the sake of clarity, the estimation problem can be divided into two
cases, depending on the number of parameters involved: single-parameter
estimation and multiple-parameter estimation.
2.5.1.1 Single-Parameter Estimation
Let us consider that a realization
x
of a discrete-time stochastic process
depends on an unknown parameter θ. The problem of parameter estimation
is then to estimate the parameter θ from a finite set of available observa-
tions
(
n
)
. Hence, we need to construct a function
that extracts the parameter from the measurements:
{
x
(
0
)
x
(
1
)
···
x
(
N
−
1
)
}
φ
x
)
θ
=
(
0
)
x
(
1
)
···
x
(
N
−
1
(2.122)
where φ
is a deterministic transformation to be determined. One important
aspect is that θ can be either deterministic or random, depending on the
problem at hand, whileθ is typically random, since it is a function of random
variables. The r.v. θ is called estimator of θ and a realization of such r.v. is
called estimate [165].
Let
x
[·]
]
T
.Ifθ is assumed to be deterministic,
=
[
x
(
0
)
x
(
1
)
···
x
(
N
−
1
)
we use the notation
p
X
(
to emphasize the dependence of the data on θ.
When θ is random,
x
and θ are related by means of the joint pdf denoted by
p
X
,θ
(
x
; θ
)
. In both cases, θ is a deterministic function of
x
and hence is also
statistically dependent on θ, but it is not a deterministic function on θ.This
fact tells us that
x
, θ
)
∂
θ
/∂
=
θ
0.
2.5.1.2 Multiple-Parameter Estimation
Let us now consider that the discrete stochastic process
x
(
n
)
depends on a
set of
L
unknown parameters θ
(
0
)
θ
(
1
)
···
θ
(
L
−
1
)
. Now the problem
consists in finding a transformation
φ
x
)
θ
=
(
0
)
x
(
1
)
···
x
(
N
−
1
(2.123)
in order to estimate the unknown parameters, being
φ
avectorof
L
functions to be determined. Similarly to the single-parameter case,
θ
can
be either deterministic or random while
θ
is typically random. Also,
x
(
·
)
=
]
T
is related to
θ
via its pdf
p
X
(
[
x
(
0
)
x
(
1
)
···
x
(
N
−
1
)
x
;
θ
)
when
θ
is
