Digital Signal Processing Reference
In-Depth Information
we have
F
=
W
†
/
√
M,
(17
.
56)
and the optimal equalizer is (Sec. 15.2)
⎧
⎨
Λ
−
c
W
√
M
ZF-MMSE case,
Λ
c
−
1
Λ
c
Λ
c
+
σ
q
G
=
(17
.
57)
W
√
M
σ
s
I
pure-MMSE case.
⎩
Λ
e
This example is shown in Fig. 17.13.
The reader will recall from Sec. 7.5 that this is nothing but the multicarrier
system or the OFDM system. We also use the term
MC-CP
(multicarrier cyclic-
prefix) to describe this. Note that the SC-CP system and the MC-CP system
can be schematically related as shown in Fig. 17.14. The mean square errors
of the SC-CP and MC-CP MMSE systems can be obtained from Eq. (15.34)
in Chap. 15, by substituting
σ
h,k
=
|
C
[
k
]
|
.
Thus, for the optimal cyclic-prefix
transceiver with unitary precoder,
⎧
⎨
⎩
M−
1
Mσ
q
σ
s
p
0
1
(ZF-MMSE)
|
C
[
k
]
|
2
k
=0
E
mmse
=
(17
.
58)
M−
1
Mσ
s
σ
q
p
0
1
(pure-MMSE).
Mσ
q
p
0
+
|
C
[
k
]
|
2
k
=0
The average MSE per symbol can be obtained by dividing these by
M.
We
summarize the results derived so far in the following lemma. Here “orthonormal
transceivers” simply means that the precoders are restricted to be unitary.
Theorem 17.4.
Optimality of cyclic-prefix systems.
The single-carrier (SC)
system and the multicarrier (MC) system are both optimal systems among the
class of orthonormal transceivers for cyclic-prefix systems. The optimality is in
the sense of minimizing the mean square error (with or without the zero-forcing
constraint). The minimized errors are given by Eq. (17.58), where
C
[
k
] denote
the channel DFT coe
cients.
♠
♦
17.6.2 Error covariances
The error covariance matrices in the MC-CP and SC-CP cases have some struc-
tural differences. These are important when we address the problem of minimiz-
ing the error probability. We first prove the following result:
Lemma 17.2.
Structure of error covariance
. For the MC-CP system the
error covariance is diagonal, whereas for the SC-CP system it is circulant. This
property holds whether there is zero forcing or not.
♠
♦
Search WWH ::
Custom Search