A signal x (t) is periodic and one period of the signal is described by
Find the samples of this signal over one period sampled at a rate of 1 samples/sec (beginning at time, t=0). Then graph, on the same scale, two periods of the original signal and two periods of a periodic signal which is bandlimited to 0.5 Hz or less that would have these same samples.
% Solution to Exercise 65 in Sampling and the DFT
close all ;
fs = 1 ; Ts = 1/fs ; T = 8 ; N = T/Ts ;
% Set up a vector of sampling times, ts.
ts = [0:Ts:(N-1)*Ts]’ ;
% Set up a vector of corresponding sample values, xs.
xs = 3*ramp(ts).*rect((ts-2.75)/5.5) ;
% Set up vectors of times and signal values much closer
% for plotting the continuous signal, x.
nPts = 256 ; dt = T/nPts ; t = [0:dt:(nPts-1)*dt]’ ;
x = 3*ramp(t).*rect((t-2.75)/5.5) ;
% Find the CTFS of the signal, x(t).
[Xs,k] = CTFS(xs,ts,k) ;
% Generate the bandlimited signal, xbl, which passes through
% all the samples, xs[n]. Sum all the complex frequency components
% from n = -N/2 to n = +N/2.
xbl = zeros(nPts,1) ; f0 = 1/T ;
for nn = -N/2:N/2-1, xbl = xbl + Xs(nn+N/2+1)*exp(j*2*nn*pi*f0*t) ; end
% Clean up any small imaginary parts left over due to
% round off error.
xbl=real(xbl) ;
% Form two periods from the one period computed so far for
% each time-domain function computed; s, x and sbl.
xbl2 = [xbl;xbl] ; t2 = [t;t+T] ; x2 = [x;x] ;
xs2 = [xs;xs] ; ts2 = [ts;ts+T] ;
% Plot the original signal, samples and bandlimited signal.
p = xyplot({t2,t2,ts2},{x2,xbl2,xs2},[0,16,0,20],’\itt’,...
‘x({\itt}),x_b_l({\itt}) and x[{\itn}]’,’Times’,18,’Times’,14,...
“,’Times’,24,{’n’,’n’,’n’},{’c’,’c’,’d’},{’k’,’k’,’k’}) ;
set(p{1},’LineWidth’,0.5) ;
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