Digital Signal Processing Reference
In-Depth Information
Since the correlation matrix presents a Toeplitz structure, it can be
partitioned into
r
0
σ
e
f
0
P
r
I
A
H
·
0
P
···
−
=
(5.88)
R
(K
−
1
)
r
H
(
n
)
x
Finally, we get to the equations that allow us to obtain the prediction coef-
ficients and the prediction-error variance based on second-order statistics of
the received signal:
⎧
⎨
r
R
(K
−
1
)
x
−
1
r
H
σ
e
f
=
r
0
−
(
n
)
(5.89)
R
(K
−
1
)
−
1
⎩
r
H
A
L
−
1
=
(
n
)
x
From this, it is possible to obtain an expression for the ZF equalizer with
zero equalization delay. The development here assumes an ideal noiseless
case. Let us rewrite Equation 5.43 for the sake of convenience:
x
(
n
)
=
Hs
(
n
)
(5.90)
The estimation of the prediction error in terms of x
(
n
−
1
)
is now done in
terms of
s
(
n
−
1
)
:
)
x
(n
−
1
)
=
)
s
(n
−
1
)
e
f
(
n
e
f
(
n
s
(n
−
1
)
=
(
n
)
−ˆ
(
n
)
x
x
s
(n
−
1
)
(5.91)
L
−
1
L
−
1
=
h
(
i
)
s
(
n
−
i
)
−
h
(
i
)
ˆ
s
(
n
−
i
)
i
=
0
i
=
0
)
s
(n
−
1
)
=
h
(
0
)
˜
s
(
n
with
s
˜
(
n
)
representing the prediction error that arises when
s
(
n
)
is estimated
with samples of the vector
s
.
From Equation 5.91, if the transmitted signal is assumed to be composed
of i.i.d. samples (or, at least, uncorrelated),
(
n
−
1
)
˜
s
(
n
)
will correspond to
s
(
n
)
, i.e.,
the prediction-error filter represents a ZF equalizer for a null delay.
Such a result consolidates the possibility of using second-order statis-
tics in SIMO channels blind equalization, as linear prediction is essentially
