Digital Signal Processing Reference
In-Depth Information
the frequency response attenuates the input where the noise is powerful
while the signal is not, and it leaves the input almost unchanged otherwise,
hence the data-dependence of its expression.
The optimal filter above was derived for an arbitrary infinite two-sided im-
pulse response. Wiener's contribution was mainly concerned with the de-
sign of a
causal
response; the derivation is a little complicated and we will
not detail it here. A third, interesting design choice is imposing that the im-
pulse response be an
N
-tap FIR. In this case (8.27) becomes
l
g
r
,
y
i
d
.
,
©
,
L
s
N
−
1
h
[
k
]
r
X
[
n
−
k
]=
r
S
[
n
]
k
=
0
and by picking
N
successive values for
n
we can build the system of equa-
tions:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
r
X
[
0
]
r
X
[
1
]
...
r
X
[
N
−
1
]
h
[
0
]
r
S
[
0
]
r
X
[
1
]
r
X
[
0
]
...
r
X
[
N
−
2
]
h
[
1
]
r
S
[
1
]
r
X
[
2
]
r
X
[
1
]
...
r
X
[
N
−
3
]
h
[
2
]
r
S
[
2
]
=
.
.
.
.
.
.
.
.
[
]
[
]
[
]
[
]
r
X
N
−
1
r
X
N
−
2
...
r
X
0
h
[
N
−
1
]
r
S
N
−
1
where the Toeplitz nature of the matrix comes from the fact that
r
X
[
−
n
]=
r
X
[
. This is a classical Yule-Walker system of equations and it is a funda-
mental staple of adaptive signal processing.
n
]
Further Reading
A good introductory reference on the subject is E. Parzen's classic
Stochas-
tic Processes
(Society for Industrial Mathematics, 1999). For adaptive sig-
nal processing, see P. M. Clarkson's
Optimal and Adaptive Signal Process-
ing
(CRC Press, 1993). For an introduction to probability, see the textbook:
D. P. Bertsekas and J. N. Tsitsiklis,
Introduction to Probability
(Athena Sci-
entific, 2002). The classic topic by A. Papoulis,
Probability, Random Vari-
ables, and Stochastic Processes
(McGraw Hill, 2002) still serves as a good,
engineering-oriented introduction. A more contemporary treatment of
stochastic processes can be found in P. Thiran's excellent class notes for the
course “Processus Stochastiques pour les Communications”, given at the
Swiss Federal Institute of Technology (EPFL) in Lausanne.
