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Sec. 3–6 Intersymbol Interference 191
Example 3–14 RAISED COSINE-ROLLOFF FILTERING
Plot the magnitude transfer function and the impulse response for the raised cosine-rolloff filter.
Compare with Fig. 3–26. See Example3_14.m for the solution.
Furthermore, assume that a PCM signal as described in Example 3–4 is a polar NRZ signal,
and it passes through a communication system with a raised cosine-rolloff filtering characteristic.
Let the rolloff factor be 0.25. The bit rate of the digital signal is 64 kbitss. Determine the absolute
bandwidth of the filtered digital signal.
From Eq. (3–74), the absolute bandwidth is B = 40 kHz. This is less than the unfiltered
digital signal null bandwidth of 64 kHz.
Let us now develop a formula which gives the baud rate that the raised cosine-rolloff system
can support without ISI. From Fig. 3–26b, the zeros in the system impulse response occur at
t = n2f 0 , where n Z 0. Therefore, data pulses may be inserted at each of these zero points without
causing ISI. That is, referring to Eq. (3–66) with t = 0, we see that the raised cosine-rolloff
filter satisfies Nyquist’s first criterion for the absence of ISI if we select the symbol clock period
to be T s = 1(2f 0 ). The corresponding baud rate is D = 1T s = 2f 0 symbolss. That is, the 6-dB
bandwidth of the raised cosine-rolloff filter, f 0 , is designed to be half the symbol (baud) rate.
Using Eqs. (3–70) and (3–72), we see that the baud rate which can be supported by the system is
2B
D =
(3–74)
1 + r
where B is the absolute bandwidth of the system and r is the system rolloff factor.
Example 3–15 ISI AND RAISED COSINE-ROLLOFF FILTERING
Plot the output waveform when a channel filters a unipolar NRZ signal. Assume that the equivalent
filter (i.e., for the combined input data pulse shape, transmitter, channel, and the receiver) is a raised
cosine-rolloff filter, where r = 0.5. Assume that the unipolar NRZ input signal has a bit rate of
R b = 1 Hz and that the data on the unipolar NRZ signal is [1 0 0 10 11010]. Calculate the absolute
bandwidth of the channel and plot the waveform at the receiver output. The solution is evaluated and
plotted by Example3_15.m. Observe that there is no ISI at the sampling times of nT b , where n is an
integer. Compare this result with that for Example 3–13.
The raised cosine filter is also called a Nyquist filter. It is only one of a more general
class of filters that satisfy Nyquist’s first criterion. This general class is described by the following
theorem:
THEOREM.
A filter is said to be a Nyquist filter if the effective transfer function is
w a f b + Y(f),
2f 0
H e (f) = c
|f| 6 2f 0
0, f elsewhere
(3–75)