Key focus: Sinc pulse shaping of transmitted bits, offers minimum bandwidth and avoids intersymbol interference. Discuss its practical considerations & simulation.
Sinc pulse shaping
As suggested in the earlier post, the pulse shape that avoids ISI with the least amount of bandwidth is a sinc pulse of bandwidth 
. Here, 
is the baud rate of the system also called symbol rate. A sinc pulse described as time and frequency domain dual is given below
Following Matlab codes generate a sinc pulse with 
and plot the time-domain/frequency-domain response (Figure 1). From the time-domain plot, the value of the sinc pulse hits zero at integral multiple sampling instants 
seconds except at 
where it peaks to the maximum value. Thus the sinc pulse satisfies the Nyquist criterion for zero ISI.
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Program 1: sincFunction.m: Function for generating sinc pulse
Program 2: Sinc pulse and its manifestation in frequency domain
Tsym=1; %Symbol duration
L=16; %oversampling rate, each symbol contains L samples
Nsym = 80; %filter span in symbol duration
Fs=L/Tsym; %sampling frequency
[p,t]=sincFunction(L,Nsym); %Sinc Pulse
subplot(1,2,1); t=t*Tsym; plot(t,p); title('Sinc pulse');
[fftVals,freqVals]=freqDomainView(p,Fs,'double'); %See Chapter 1
subplot(1,2,2);
plot(freqVals,abs(fftVals)/abs(fftVals(length(fftVals)/2+1)));
The main drawback of the sinc pulse is that it decays too slowly at the rate of 
as 
. This implies that the samples that are far apart can cause intersymbol interference in the event of modest clock synchronization errors. A sinc pulse is of infinite duration and for practical implementations, it has to be truncated to finite length 
for some integer 
. This leads to problems in frequency domain as explained next.
Figure 2 shows the one-sided frequency response of the sinc pulse that is truncated to various lengths. It is evident that the truncation of sinc pulse in time domain to 
leads to sidelobes in the frequency domain and the sidelobes become wider for decreasing values of . This effect is closely related to Gibbs phenomenon – the ringing artifact due to approximation of discontinuities by spectral methods. As a result, no matter how large the value of is chosen, the first sidelobe is always only 
down from the main lobe. Also, the sinc pulse is very sensitive to the timing jitters at the receiver. These problems can be addressed when the transition band in the frequency domain is made less abrupt.
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Topics in this chapter
| Pulse Shaping, Matched Filtering and Partial Response Signaling ● Introduction ● Nyquist Criterion for zero ISI ● Discrete-time model for a system with pulse shaping and matched filtering □ Rectangular pulse shaping □ Sinc pulse shaping □ Raised-cosine pulse shaping □ Square-root raised-cosine pulse shaping ● Eye Diagram ● Implementing a Matched Filter system with SRRC filtering □ Plotting the eye diagram □ Performance simulation ● Partial Response Signaling Models □ Impulse response and frequency response of PR signaling schemes ● Precoding □ Implementing a modulo-M precoder □ Simulation and results |