Question

MATLAB Assignment #4 Due date: Monday May 7, 2018 Exercise (Sampling): Consider the CT signals (t) a) (2 Points) Derive an analytical expression for the Fourier transform X(jw). Note that since a(t) is real and even, so is X(jw). Plot X(jw) for we [-100,100. Is a(t) band limited? b) (4 Points) Applying an idea low-pass filter with cutoff frequency wM to(), the output can be computed as: Write a MATLAB program to numerically evaluate the above integral. In the same figure, plot x(t) and x2(t) for wM = 2,5, 10, 100 and t E 1-10, 10, what do you observe )(4 Points) The reconstructed signal using the DT samples (nT can be computed as: i(t) = ? r(nT)h(t-nT) sin Write a MATLAB program to where T is the sampling period, and h(t) = numerically evaluate the above summation. (You need to think about how to deal with the infinite sum.) In the same figure, plot r(t) ard î(t) for T-,,?, and te -10,10. What do you observe?

Answer 1

Itl 2 1t 지 ) もーう Room Raure it is seen ttrad both *)&XGa) are erev en nahure e aah stheC) incr and the banducth ralueao Sin xCt) 72 Sap

clear all t--10: . 01: 10; x-exp (-abs (t)) subplot (1,2,1); plot (t, x): title ('x(t) ) k-0: for w--100:1:100 X(k)trapz (t, exp (-i*wt)) end w=-100:1:100; subplot (1,2,2); plot (w, X); title ( "X (jw)" )

clear all t -10:.01:10: x-exp (-abs (t)) subplot (1,2,1); plot (t,x); title('x(t) ') k=0; for w=-100: 1 : 100 k=k+1; x (k) =trap2(t, exp (-rytt)) ; end wm=100;舌enter TT. value | k1-0: for t-10: .01:10 k1=k1 +1 ; xl (k1) = (1/ (2 *pi) ) *trapz (wil, exp (jtył *t) ) ; end t=-10:.01:10 subplot (1,2,2); plot (t, xL); title ('xL (t) for wm-100)

t -10:.01:10: T-input ('enter T' n-t./T x-exp (-abs (t) subplot (1,3,1); plot (t,x); title('x(t) ') h-sin (a) ./a; subplot (1,3,2); plot (t,h); title ('h(t) '; x1-conv (x, h) ; z-length (x) +length (h)-1: initial finalt--10-10 end_finalt-10+10: final t-linspace (initial finalt,end finalt, z); subplot (1,3,3) plot (final t, x1);

x(t) 20 0.9 0.8 15 0.7 0.6 10 0.5 0.4 바 0.3 0.2 0.1 10 6 4 0 2 46 8 100 -80 -60 40 20 020 40 60 80 100

xL(t) for wm 5 xL(t) for wm-2 1.6 1.4 1.2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.7 0.6 0.5 0.4 0.3 0.2 0.8 0.6 0.4 0.2 0.2 -0.1 0.4 10 10 xL(t) for wm=100 35 30 25 20 15 10 xL(t) for wm-10 3.5 2.5 1.5 0.5 0 0.5 10 5 10 10 0 10 10 0

x(t) h(t) x1(t) for pi/2 0.03 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.8 0.025 0.6 0.02 0.4 0.015 0.2 0.01 0.005 0.2 0 10-0 00 10 0 10 0 20 x(t) h(t) x1(t) for pi/5 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.05 0.8 0.1 0.6 0.15 0.4 -0.2 0.2 0.25 0.3 -0.2 0.35 0.4 10 0 10 10 0 10 0 0 20

x(t) h(t) x1(t) for pi/10 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.01 -0.02 0.03 0.04 0.05 0.06 0.07 -0.08 -0.09 10 20 0.8 0.6 0.4 0.2 -0.2 20