Digital Signal Processing Reference
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are proportional to the
M
uniformly spaced samples of the polyphase
components
R
0
(
e
jω
)and
R
1
(
e
jω
), i.e., the
M
-point DFTs of
r
0
(
n
)=
c
c
(
nT
)and
r
1
(
n
)=
c
c
(
nT
+0
.
5
T
)
.
Thus, instead of inverting the
M
DFT coe
cients of the channel, as is normally done in a cyclic-
prefixed ZF system, the ZF condition in the cyclic-prefixed FSE sys-
tem does something different, namely it combines the
M
DFT coe
-
cients of
r
0
(
n
)and
r
1
(
n
)toforce
C
0
[
k
]
E
0
[
k
]+
C
1
[
k
]
E
1
[
k
]=1
,
0
≤
k
≤
M
−
1
,
(P7
.
26
b
)
where
E
i
[
k
]isthe
k
th diagonal element of
Λ
e,i
.
Thus, unless
C
0
[
k
]and
C
1
[
k
] are both very small for a certain
k,
the
equalizer coe
cients
E
0
[
k
]and
E
1
[
k
] do not get unduly large. Furthermore,
for a given pair of numbers
C
0
[
k
]and
C
1
[
k
], the numbers
E
0
[
k
]and
E
1
[
k
]
satisfying Eq. (P7.26b) are not unique. For example if
E
0
[
k
]
,E
1
[
k
] satisfies
Eq. (P7.26b), then so does the pair
E
0
[
k
]+
α
(
k
)
C
1
[
k
]
,
E
1
[
k
]
α
(
k
)
C
0
[
k
]
,
−
(P7
.
26
c
)
for any set of
M
numbers
α
(
k
). This freedom offered by
α
[
k
] can be used
[Vaidyanathan and Vrcelj, 2002] to minimize reconstruction error due to
channel noise (which has been ignored in this problem).
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