Digital Signal Processing Reference
In-Depth Information
If we consider an efficient model, in this case an FIR filter with three
coefficients, the corresponding input vector is
)
=
s
)
T
s
(
(
n
)
s
(
n
−
1
)
s
(
n
−
2
(3.34)
n
and the correlation matrix is given by
⎡
⎣
⎤
⎦
.
100
010
001
R
=
(3.35)
To obtain the cross-correlation vector, it is necessary to determine the desired
signal. From the above discussion, the error signal must express a comparison
between the system and the model outputs. This leads to the natural choice,
d
(
n
)
=
x
(
n
)
(3.36)
Since the signal
s
(
n
)
and the noise are mutually independent, it follows that
p
(
0
)
=
E
[
x
(
n
)
s
(
n
)
]
E
h
0
s
)
s
)
=
(
)
+
(
−
1
)
+
(
−
2
)
+
ν
(
(
n
h
1
s
n
h
2
s
n
n
n
h
0
E
s
2
=
(
n
)
=
(3.37)
h
0
p
(
1
)
=
E
[
x
(
n
)
s
(
n
−
1
)
]
E
h
0
s
)
s
)
=
(
n
)
+
h
1
s
(
n
−
1
)
+
h
2
s
(
n
−
2
)
+
ν
(
n
(
n
−
1
h
1
E
s
2
=
(
n
−
1
)
=
h
1
(3.38)
and
p
(
2
)
=
E
[
x
(
n
)
s
(
n
−
2
)
]
E
h
0
s
)
s
)
=
(
n
)
+
h
1
s
(
n
−
1
)
+
h
2
s
(
n
−
2
)
+
ν
(
n
(
n
−
2
s
2
=
h
2
E
[
(
n
−
2
)
]
=
(3.39)
h
2
so that
⎡
⎣
⎤
⎦
h
0
h
1
h
2
p
=
(3.40)
and finally
⎡
⎣
⎤
⎦
h
0
h
1
h
2
R
−1
p
w
w
=
=
(3.41)
