Root Raised Cosine Filtering

We have now generated a source coded, 3.84 Mcps, I and Q streamed, HPSK formatted
signal. Although the bandwidth occupancy of the signal is directly a function of the
3.84 Mcps spreading code, the signal will contain higher frequency components
because of the digital composition of the signal. This may be verified by performing a
Fourier analysis of the composite signal. However, we only have a 5 MHz bandwidth
channel available to us, so the I and Q signals must be passed through filters to constrain
the bandwidth. Although high-frequency components are removed from the signal,
it is important that the consequent softening of the waveform has minimum
impact on the channel BER. This objective can be met by using a class of filters referred
to as Nyquist filters. A particular Nyquist filter is usually chosen, since it is easier to
implement than other configurations: the raised cosine filter.

If an ideal pulse (hence, infinite bandwidth) representing a 1 is examined, it is seen
that there is a large time window in which to test the amplitude of the pulse to check
for its presence, that is, the total flat top. If the pulse is passed through a filter, it is seen
that the optimum test time for the maximum amplitude is reduced to a very small time
window. It is therefore important that each pulse (or bit) is able to develop its correct
amplitude.
The frequency-limiting response of the filter has the effect of smearing or timestretching
the energy of the pulse. When a train of pulses (bit stream) is passed through
the filter, this ringing will cause an amount of energy from one pulse to still exist during
the next. This carrying forward of energy is the cause of Inter-Symbol Interference
(ISI),.
The Nyquist filter has a response such that the ringing energy from one pulse passes
through zero at the decision point of the next pulse and so has minimum effect on its
level at this critical time.

The Nyquist filter exhibits a symmetrical transition band, as shown in Figure 3.10.
The cosine filter exhibits this characteristic and is referred to as a raised cosine filter,
since its response is positioned above the base line.
It is the total communication channel that requires the Nyquist response (that is, the
transmitter/receiver combination), and so half of the filter is implemented in the transmitter
and the other half in the receiver. To create the correct overall response, a Root
Raised Cosine (RRC) filter is used in each location as: