Digital Signal Processing Reference
In-Depth Information
]
T
where each vector component is
k
th tap of each subchannel. By stacking
N
received vector samples into an
the channel impulse response in vector form as
h
(
k
)
=
[
h
0
(
k
)
,
...
,h
P
−
1
(
k
)
[
y
T
,
y
T
(
NP
×
1
)
-vector
y
N
(
k
)
=
(
k
)
,
...
(
k
−
+
)
]
T
and doing the same for the noise vectors
w
N
(
)
=
[
w
T
(
)
...
,
w
T
(
−
N
1
k
k
,
k
+
)
]
T
,wecan write a matrix equation (see Reference 3):
N
1
y
N
(
k
)
= H
N
s
N
(
k
)
+
w
N
(
k
).
(8.10)
The
(
L
+
N
−
1
)
×
1 input signal vector (where
L
is the length of the channel)
]
T
and the channel
coefficients are collected into a Sylvester resultant matrix with dimension
NP
is defined as
s
N
(
k
)
=
[
s
(
k
)
,s
(
k
−
1
)
,
...
,s
(
k
−
L
−
N
+
2
)
×
(
L
+
N
−
1
)
.
h
(
0
)
h
(
1
)...
h
(
L
−
1
)
0
...
0
0
h
(
0
)...
h
(
L
−
2
)
h
(
L
−
1
)
...
0
H
=
(8.11)
.
.
.
.
.
.
.
.
.
.
.
N
0
0
...
h
(
0
)
h
(
1
)
...
h
(
L
−
1
)
There are quite a few blind equalizers employing the above matrix model;
see, for example, References 3, and 34 to 36. Next, two commonly used SOCS-
based blind equalization methods are briefly outlined.
8.3.4.1 FS-CMA Algorithm
The FS-CMA algorithm is probably the simplest and very reliable fractionally
spaced equalization method.
3
,
4
,
32
In case we have oversampling factor
P
, the
equalizer taps
d
are updated using the stochastic gradient method as follows:
y
∗
(
2
2
,
d
(
k
+
1
)
=
d
(
k
)
+
µ
k
)
x
(
k
)(
|
x
(
k
)
|
−
γ)
(8.12)
where
γ
is the dispersion factor for the modulation scheme employed,
x
(
k
)
is the equalizer output, and
[
y
k
,
,y
k
−
(
N
−
1
)
,y
k
,
,y
k
−
(
N
−
1
)
,y
k
,y
k
−
(
N
−
1
)
]
T
,
y
(
k
)
=
...
···
,
···
,
···
(8.13)
where superscript denotes the subchannel, i.e., fractionally sampled data are
organized on a subchannel basis.
8.3.4.2 Prediction Error Filtering
Slock proposed using linear prediction for blind equalization.
36
The value
of the current received symbol is predicted on the basis of a set of previ-
ously received symbols. The desired response is thus the eventually received
symbol. The filter coefficients are found by minimizing the prediction error.
The transmitted sequence is assumed to be white. A prediction-error filter
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