Question

Please show all your works. Thanks.

2) Let r(t) be the continuous-time signal x(t) t(0.97)' [a(t)-u(t-100)| where u(t) is the unit step function. We sample r(t) with a sampling period of T 0.4 over ootoo to obtain r/n]. Assume that the sample for n 0 is taken at t 0. We then use sinc interpolation to generate Tr(t) from xml a) Plot r(t) using MATLAB over the smallest range that includes all nonzero values of r(t) b) Find r[n] c) Find x,(t) d) Plot x(t) - x,(t) using MATLAB over the same range that you used for part a e) What is the maximum value of |x(t)-xr(t)| over-oo < t < oo? f) Find the value of x(t) for t = 50 g) Find the value of xr(t) for 50 h) Find the value of r(t) for t-50.2 i) Find the value of xr(t) for t = 50.2 j) Using only your answers to f,g,h.i can you determine whether T Wmax < where wmax corresponds to the frequency domain representation for r(t)? Explain your answer

Answer 1

b)

$x[n]=x(nT)=(nT)(0.97^{nT})(u(nT)-u(nT-100))$

=>

$x[n]=(0.4n)(0.97^{0.4n})(u(n)-u(n-100/0.4))$

$x[n]=(0.4n)(0.99^{n})(u(n)-u(n-250))$

c)

with sinc interpolation

$x_r(t)=\sum_{n=-\infty}^{\infty}x[n]sinc((t-nT)/T)=\sum_{n=-\infty}^{\infty}(0.4n)(0.99^{n})(u(n)-u(n-250))sinc((t-nT)/T)$ $x_r(t)=\sum_{n=0}^{250}(0.4n)(0.99^{n})sinc((t-nT)/T)$

d)

t=0:0.01:100;
xt=t.*(0.97.^t);
plot(t,xt);
xlabel('t');
ylabel('x(t)');
T=0.4;
xr=zeros(size(t));
for k=1:length(t)
for n=0:250
xr(k)=xr(k)+(0.4.*n).*(0.99.^n).*sinc((t(k)-n.*T)./T);
end
end
figure
plot(t,abs(xr-xt));
xlabel('t');
ylabel('|x_r(t)-x(t)|');

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