Digital Signal Processing Reference
In-Depth Information
q
(
n
)
s
(
n
)
W
M
W
M
Λ
e
s
(
n
)
U
U
H
(a)
circulant
channel
equalizer
G
SC-CP
q
(
n
)
s
(
n
)
W
M
W
M
Λ
e
U
s
(
n
)
U
H
(b)
precoder
circulant
channel
equalizer
G
MC-CP
Figure 17.15
. (a), (b) The SC-CP system and MC-CP system, with unitary matrices
U
and
U
†
attached to them.
Discussions on bias removal.
Atthispointitisinsightfultorelatetheabove
discussions to Sec. 16.3, where we had a detailed discussion of bias removal in
MMSE receivers. If the cascade of the channel and precoder is a fixed diagonal
matrix, and if the equalizer is also restricted to be diagonal, then removing bias
from the MMSE equalizer simply yields the ZF equalizer (Sec. 16.5). The bias-
removal multipliers are defined by a diagonal matrix that immediately follows
Λ
e
.
For the MC-CP system, the diagonal equalizer
Λ
e
in Fig. 17.13(b) indeed
sees a diagonal precoder-channel cascade, and the MMSE equalizer (with bias
removal) is no better than a ZF equalizer. Compare this with an SC-CP system
(Fig. 17.12(c)). In this case, the equalizer
G
is not diagonal, and furthermore the
bias-removal multipliers are not inserted right after the diagonal part
Λ
e
;they
are inserted after the IDFT matrix. So, in this case, the bias-removed MMSE
system is different from the ZF equalizer. Thus, unlike in the MC-CP case, the
SC-CP MMSE system with bias removal does not reduce to a ZF system.
17.7 Cyclic-prefix optimization examples
We now present numerical examples which demonstrate the performance of op-
timal cyclic-prefix systems. We will compare MMSE systems with and without
zero-forcing equalizers. For each of these we will consider both systems with
orthonormal precoders and general precoders. The MMSE expressions for these
cases were summarized in Sec. 15.2 of Chap. 15 and are reproduced below for
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