Digital Signal Processing Reference
In-Depth Information
Equivalent diagonal form.
From Eq. (17.7) we have
√
M
H
W
†
√
M
=
Λ
c
=diag[
C
[0]
C
[1]
...
C
[
M −
1] ]
,
which shows that Fig. 17.3(a) can be redrawn in the equivalent diagonal form
shown in Fig. 17.3(b). Here
σ
f,k
are the diagonal elements of
Σ
f
,and
λ
g,k
are the
diagonal elements of
Λ
g
. This is nothing but a frequency-domain representation
of the channel because the channel multipliers
C
[
k
] are the DFT coecients of
c
(
n
)
.
In this diagonal form, each DFT coecient
C
[
k
] is independently equalized
by the precoder/equalizer pair
{
σ
f,k
,λ
g,k
}
.
17.3 Cyclic-prefix systems optimized for MSE: details
We now apply the results of the preceding section to derive the detailed structure
of the MMSE cyclic-prefix systems. We discuss the zero-forcing case and the non
zero-forcing case separately for clarity.
17.3.1 ZF-MMSE transceiver
For the MMSE transceiver with zero-forcing constraint (ZF-MMSE transceiver)
the diagonal matrices
Σ
f
and
Σ
g
were derived in Sec. 12.4.4. These are repro-
duced below:
⎡
⎣
⎤
⎦
σ
−
1
/
2
h,
0
0
...
0
σ
−
1
/
2
h,
1
0
...
0
=
α
Σ
−
1
/
2
h
Σ
f
=
α
(17
.
17)
.
.
.
.
.
.
σ
−
1
/
2
h,M−
1
0
0
...
and
⎡
⎤
σ
−
1
/
2
h,
0
0
...
0
⎣
⎦
σ
−
1
/
2
h,
1
1
α
0
...
0
1
α
Σ
−
1
/
2
Σ
g
=
=
.
(17
.
18)
.
.
.
.
.
.
h
σ
−
1
/
2
h,M−
1
0
0
...
Note that the channel singular values
σ
h,k
are assumed nonzero for all
k.
The
constant
α
ensures that the power constraint is satisfied. The
k
th set of multi-
pliers
σ
f,k
and
σ
g,k
depend only on the
k
th singular value
σ
h,k
.
None of these
multipliers is zero, and the ordering of the singular values
σ
h,k
does not matter.
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