Digital Signal Processing Reference
In-Depth Information
add the first
L
columns of the matrix of
c
(
n
)'s to the last
L
columns and write
the preceding equation more compactly as follows:
⎡
⎤
⎛
⎞
⎡
⎤
y
0
(
n
)
c
(0)
c
(3)
c
(2)
c
(1)
s
0
(
n
)
s
1
(
n
)
s
2
(
n
)
s
3
(
n
)
y
1
(
n
)
c
(1)
c
(0)
c
(3)
c
(2)
⎣
⎦
⎝
⎠
⎣
⎦
=
.
y
2
(
n
)
c
(2)
c
(1)
c
(0)
c
(3)
y
3
(
n
)
c
(3)
c
(2)
c
(1)
c
(0)
circulant
C
This explains in yet another way why the channel appears to be a circulant when
we use a cyclic prefix of appropriate length.
Summary (cyclic-prefix systems)
1. If we divide the transmitted symbol stream
s
(
n
)intoblocksoflength
M
and insert a cyclic prefix of length
L
at the beginning of each block, where
L
channel order and
M>L
, the effect of the channel appears to be a
circular convolution, and interblock interference is eliminated.
≥
2. Equivalently the vector
y
(
n
)isthevector
s
(
n
) multiplied with a circulant
channel matrix
C
.
3. Since circulants can be diagonalized by the DFT matrix, as in Eq. (7.24a)
(or equivalently since circular convolution implies pointwise multiplication
of DFT coe
cients), the receiver structure can be implemented as in Fig.
7.13 with the help of a DFT, an IDFT, and frequency-domain equalizers.
Some variations of the cyclic-prefix system will be presented in the next section.
Improved receivers which take into account the effect of channel noise will be
presented in later chapters.
7.5 Variations of the cyclic-prefix system
We now derive some variations of the cyclic-prefix system. It will be convenient
to use a diagram where blocked versions of signals are used (the reader should
review Sec. 3.5 on blocking and unblocking conventions).
Figure 7.14 shows the cyclic-prefix scheme in block diagram form. In the
figure,
p
(
n
) represents the cyclic-prefix part, which is merely a repetition of the
last
L
entries of
s
(
n
)
.
The channel noise is not shown, but will be analyzed in later
chapters. The matrix
Λ
e
on the receiver side is a diagonal matrix representing
the zero-forcing equalizer coe
cients in the DFT domain, that is,
⎡
⎤
1
/C
[0]
0
...
0
0
1
/C
[1]
...
0
⎣
⎦
Λ
e
=
.
.
.
.
.
.
.
0
0
...
1
/C
[
M
−
1]
We now show how to obtain other equivalent forms of the cyclic-prefix equalizer.
In Fig. 7.15(a) we have shown a portion of the system that is relevant for further
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