Digital Signal Processing Reference
In-Depth Information
as before. Figure 7.9 shows the cyclic-prefixing system in the form of a block
diagram. If we wish the original data rate to be preserved in spite of introduction
of the redundancy then the samples of
x
(
n
) have to be spaced closer than those
of
s
(
n
)byafactor
γ.
We will show that if this new symbol stream
x
(
n
)is
transmitted through the channel
C
(
z
), then, from its output
y
(
n
)
,
we can recover
s
(
n
) perfectly (ignoring noise, of course), with
no IIR filtering
. Thus cyclic
prefixing serves the same purpose as zero padding, but channel equalization now
uses frequency-domain computations, as we shall show.
7.3.1 Working principle of the cyclic-prefix system
The signal
x
(
n
) containing the cyclic prefix is convolved with the channel impulse
response
c
(
n
)
,
which has length
L
+ 1. Figure 7.10(b) shows how the samples of
c
(
n
) enter the computation of the output sample
y
(
L
)
.
The samples involved in
the computation of
y
(
M
+
L −
1) are also shown. Note that causality ensures
that none of the output blocks is affected by future input blocks. Moreover, the
prefix of length
L
ensures that the last
M
samples in the
m
th block depend
only on the input samples in that block but not on the previous block. So, once
again,
interblock interference is eliminated
as in zero padding. Since the first
L
input samples are identical to those at the end of the block, we see that the
last
M
samples of the output are computed as if we are performing a
circular
convolution
or cyclic convolution of
x
(
n
)with
c
(
n
) [Oppenheim and Schafer,
1999]. That is, if we denote the last
M
samples of the
m
th block temporarily as
a
(
n
)=
x
(
Pm
+
L
+
n
)
,b
(
n
)=
y
(
Pm
+
L
+
n
)
,
for 0
≤
n
≤
M
−
1
,
(7
.
12)
then
M−
1
b
(
n
)=
c
(
)
a
((
n
−
)) (cyclic convolution)
,
(7
.
13)
=0
where
a
((
i
)) denotes the periodic extension of
a
(
i
) with period
M
. Defining the
M
-point DFTs
M−
1
M−
1
a
(
n
)
W
kn
,
b
(
n
)
W
kn
,
A
[
k
]=
B
[
k
]=
(7
.
14)
n
=0
n
=0
where
W
=
e
−j
2
π/M
,
(7
.
15)
we then have, from the circular convolution theorem [Oppenheim and Schafer,
1999]
B
[
k
]=
C
[
k
]
A
[
k
]
,
(7
.
16)
where
C
[
k
]isthe
M
-point DFT of the channel, that is,
L
c
(
n
)
W
kn
C
[
k
]=
(7
.
17)
n
=0
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