Digital Signal Processing Reference
In-Depth Information
In practice, ISI is negligible in telephone channels for data rates less than 2400 bps.
For higher data rates, ISI is a problem that degrades the system performance.
One way to reduce the effect of ISI is to smooth the corners of the rectangular
pulses (that represent the symbols) to reduce the effective bandwidth of each
symbol. This is called pulse shaping, and this shaping is done by filtering the
original rectangular pulses with special-purpose filters such as the raised-cosine
filter. This filter has a transfer function which is given by:
8
<
9
=
0
j
f
j
1
a
2T
T
T
2
½
1
þ
cos
p
a
ðj
f
j
1
a
1
a
2T
j
f
j
1
þ
a
H
ð
f
Þ¼
2T
Þ
:
2T
;
j
f
j
[
1
þ
a
2T
0
where the constant 0 B a B 1 is called the roll-off factor and T is the symbol period.
This filter is useful in reducing ISI as long as the data rate R = 1/T \ 2W,W being
the channel bandwidth. The rate R = 2W is called the Nyquist rate.
The impulse response h(t) of the raised-cosine filter is given by:
cos
ð
pb
T
Þ
1
4b
2
t
2
T
2
Not only does the raised cosine filter reduce the time extent of the ISI, but it also
shapes the interference intelligently. In particular, the raised cosine filter transforms
the rectangular pulse into an oscillating pulse which happens to be zero at all
symbol detection points before and after the current data symbol as shown in
Fig.
3.27
. That is, whenever a data symbol needs to be detected, the interference
from all prior and subsequent symbols is zero at that point. This is so because the
impulse response h(t) is zero at all nT (where n is an integer), except at n = 0.
Therefore, if the transmitted waveform is correctly sampled at the receiver at
t = nT, the original symbol values can be recovered exactly in the absence of noise.
Note that a matched filer is necessary at the receiver to reduce the effect of noise.
t
T
h
ð
t
Þ¼
sinc
Fig. 3.27 Successive data
impulses (train of 1s) filtered
using a raised-cosine filter
Rx Pulses
t
−
T
T
2
T
3
T
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