Digital Signal Processing Reference
In-Depth Information
q
(
n
)
J
P
s
(
n
)
s
(
n
)
M
M
F
G
H
(a)
equalizer
precoder
channel, possibly
rectangular
q
(
n
)
s
(
n
)
J
s
(
n
)
M
M
P
M
Σ
g
Σ
f
U
h
V
h
H
P
−
M
J
−
M
diagonal
matrix
diagonal
matrix
unitary
matrix
unitary
matrix
channel
(b)
0
zero padding
ignore
precoder
equalizer
Figure 17.1
. (a) General form of the transceiver. (b) The structure of the MMSE
transceiver with or without the zero-forcing constraint. The optimal diagonal matrices
Σ
f
and
Σ
g
depend on whether the ZF constraint is imposed or not.
that the signal and noise covariance matrices are
1
R
ss
=
σ
s
I
M
R
qq
=
σ
q
I
J
.
and
(17
.
2)
The components
q
k
(
n
) of the channel noise
q
(
n
) are assumed to be Gaussian
with variance
σ
q
. More specifically, when the constellation is QAM, the channel
can be complex and the noise components
q
k
(
n
) are assumed to be complex
circular Gaussian (Sec. 6.6). For PAM constellations and real channels,
q
k
(
n
)is
real and Gaussian. Here
U
h
and
V
h
are the unitary matrices appearing in the
SVD of the channel:
V
h
H
=
U
h
J×J
Σ
h
J×P
.
(17
.
3)
P×P
Besides
U
h
and
V
h
,
the optimal transceiver contains only diagonal matrices,
namely
Σ
f
and
Σ
g
. These matrices depend on whether the ZF constraint is
imposed or not. Thus the MMSE transceivers with or without the ZF constraint
have identical form except for details of the diagonal matrices
Σ
f
and
Σ
g
.
1
When the channel noise
q
(
n
) is colored, as is the case in practical DSL systems [Starr
et
al.,
1999], the techniques described in Chap. 12 should be used to obtain an equivalent system
where the noise covariance has the form
σ
q
I
.
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