Digital Signal Processing Reference
In-Depth Information
q
(
n
)
s
(
n
)
s
(
n
)
W
M
Λ
−
1/2
W
M
Λ
−
1/2
α
−1
H
α
circulant channel
diagonal
normalized IDFT
normalized DFT
diagonal
Figure 17.4
.
Structure of the zero-forcing MMSE cyclic prefix transceiver.
The
diagonal matric
Λ
c
has the channel DFT coe
cients
C
[
k
] on the diagonal.
We now prove:
Theorem17.1.
ZF-MMSE cyclic-prefix system.
The cyclic-prefix transceiver
optimized to have minimum MSE under the zero-forcing constraint has the form
shown in Fig. 17.4. Here
W
is the
M
♠
×
M
DFT matrix, and
Λ
c
is the diagonal
matrix given by
⎡
⎤
C
[0]
0
...
0
0
C
[1]
...
0
⎣
⎦
Λ
c
=
,
.
.
.
.
.
.
0
0
...
C
[
M
−
1]
with
C
[
k
] representing the DFT coe
cients of the scalar channel as shown in
Eq. (17.5). The constant
α
is chosen to satisfy the power constraint at the
transmitter. The minimized mean square error is given by
2
,
M−
1
E
mmse
=
σ
s
σ
q
p
0
1
(17
.
19)
|
C
[
k
]
|
k
=0
where
p
0
is the channel input power.
♦
Before proceeding to the proof, some remarks are in order.
1. The optimal system splits the equalizer
Λ
−
1
c
equally between the trans-
mitter and the receiver.
2. Since
s
(
n
) is the blocked version of a scalar symbol stream
s
(
n
)
,
the
average
error per sample
of
s
(
n
) is obtained by dividing Eq. (17.19) by
M.
3. Even though the expression for the mean square error (17.19) depends on
σ
q
and the power
p
0
, the optimal transceiver matrices do not depend on
these quantities (except for the constant
α
which is chosen to satisfy the
power constraint).
Proof of Theorem 17.1.
The diagonal matrix
Λ
g
in the optimal structure
of Fig. 17.3 is given by
Σ
g
Λ
−
1
θ
Λ
g
=
(using Eq. (17.16))
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