Digital Signal Processing Reference
In-Depth Information
E
|
2
, and, from (2.118), it comes that
Notice that
R
Y
(
0
)
=
Y
(
t
)
|
E
2
∞
H
)
2
S
X
(
|
Y
(
t
)
|
=
(
f
f
)
df
(2.119)
−∞
A similar reasoning lead us to the discrete-time counterpart of this result.
The autocorrelation of the output of an LTI system with a discrete-time
random process as an input is given by
h
∗
(
R
Y
(
k
)
=
R
X
(
k
)
∗
h
(
−
n
)
∗
n
)
(2.120)
where, in this case,
∗
stands for the discrete-time convolution, and
H
2
S
X
S
Y
(
exp
(
j
2π
f
))
=
(
exp
(
j
2π
f
))
(
exp
(
j
2π
f
))
(2.121)
2.5 Estimation Theory
Estimation theory is the field of statistical signal processing that deals with
the determination of one or more parameters of interest, based on a set of
available measured or empirical data. This problem is rather general, and
a number of scientific domains derive great benefit from the application of
estimation techniques.
From the perspective of this topic, it is particularly relevant to the case in
which the parameters of interest are associated with a system to be designed
or analyzed. This corresponds to applying statistical methods to the optimal
filtering problem, subject of Chapter 3. Before that, however, it is useful to
present the foundations of estimation theory in general terms.
Different methods can be built according to the hypotheses we assume
concerning the parameters to be estimated [70]. If they are considered to
be deterministic parameters, we may derive the so-called classical estima-
tion methods, as that of
maximum likelihood
(ML) estimation. Dealing with
the parameters to be estimated as r.v.'s gives rise to the Bayesian estima-
tion methods, like the
minimum mean-squared error
(MMSE) and the
maximum
a posteriori
(MAP) methods. Finally, an estimation method may be derived
regardless of the nature of the unknown parameters, as the
least-squares
(LS)
estimation method that may be applied to either random or deterministic
parameters.
As a well-established discipline, estimation theory has been treated in
important topics, in which existing methods are studied in detail [165,
181, 265, 283]. In this section, we provide a brief exposition of the main
