Key focus: Rectangular pulse shaping with abrupt transitions eliminates intersymbol interference, but it has infinitely extending frequency response. Simulation discussed.
Rectangular pulse
A rectangular pulse with abrupt transitions is a natural choice for eliminating ISI. If an information sequence is shaped as rectangular pulses, at the symbol sampling instants, the interference due to other symbols are always zero. Easier to implement in hardware or software, a rectangular pulse 
of duration 
can be generated by the following function
The complete set of Matlab codes to generate a rectangular pulse and to plot the time-domain view and the frequency response is available in the book Wireless Communication Systems in Matlab.
This article is part of the book Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).
Program 1: rectFunction.m: Function for generating a rectangular pulse
function [p,t,filtDelay]=rectFunction(L,Nsym)
%Function for generating rectangular pulse for the given inputs
%L - oversampling factor (number of samples per symbol)
%Nsym - filter span in symbol durations
%Returns the output pulse p(t) that spans the discrete-time base
%-Nsym:1/L:Nsym. Also returns the filter delay.
Tsym=1;
t=-(Nsym/2):1/L:(Nsym/2); %unit symbol duration time-base
p=(t > -Tsym/2) .* (t <= Tsym/2);%rectangular pulse
%FIR filter delay = (N-1)/2, N=length of the filter
filtDelay = (length(p)-1)/2; %FIR filter delay endProgram 2: test rectPulse.m: Rectangular pulse and its manifestation in frequency domain
Matlab code for Program 2 is available is available in the book Wireless Communication Systems in Matlab.
As shown in Figure 1, the rectangular pulse in the time-domain manifests as a sinc function that extends infinitely on either side of the frequency spectrum (though only a portion of the frequency response is plotted in the figure) and thus its spectrum is not band-limited. When the infinitely extending frequency response is stuffed inside a band-limited channel, the truncation of the spectrum leads to energy spills in the time-domain. If we were to use sharp rectangular pulses, it needs a huge bandwidth that could violate practical design specs.
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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 |