Question

General Guidelines MATLAB Symbolic Computations are NOT allowed for this assignment. Generate ALL plots using MATLAB. Include your MATLAB codes as an appendix at the end of your document. All plots should be clearly labeled: provide a X-axes and y-axes labels, specify units as needed, add titles and legend as needed, add grids as needed, use different line stiles and/or symbols as needed. 6) Consider the CT signal X(t) = u(t)- r(t) +r(t-1), where r(t) = t.u(t). a) Derive the DT signal x[n] = x(nt), where Ts = 0.2 s; plot it for an appropriate interval using the MATLAB function stem. (2 pts) b) Derive the even and odd components of x[n];plot them and verify graphically that they add to x[n]. (2 pts) c) Derive the DT signal z[n] = x[n] + x[-n]; plot it for an appropriate interval. (2 pts) d) Calculate the energy of z[n], and compare it to the sum of the energies of its even and odd components. (3 pts)

Answer 1

© H = uki-vtey arte-1) d - -TCH ult-ret SE uch-r1H +r11+1) rt-1) - - CH । के 75 = 0.2 x]] = Arms) .20... @1234 -- < <1 11-2 o elswhere

Z(NT) { I-n's o oanisaa elsewhere =k(1) 20) = 31-0.22 ounas z elbesture eren Gomponent Brenn] = 27) +2E+] 2 : 270] '1008 AL 01234 270) n -4-32-10 943 110.4 xịn)+ [H] xe0 = 2 اع 23 throm any 2 - 60 = 2Th]-x[-2] -54-3-2-1. 1o12345 -0.1" -0.2 03 -04 © 2n = x[n]+x[-2] am 108 402 0 1 2 3 4 5

(4-]> Po 9.000 to -5-4-3-2- 10 xm) +x [n] = 21 0812018 06 oy o2 -5-4-3-2-1 01239sn Energy of Zin is I20212 02-00 - 6+2 [697607693703] = 4+250.6470.36 +0.16+0-04] = ut axl.q = 4+24 = 6:4 ]

Energy of thn) = 647 - Energies of El hem of h y eren and odd Components is 0.85 +0.6 = 1.45] Energies are not equal

E Energy of rein] luen? = 632 76.) +63)+62) 7601] = 1+2 70.16 +0.09+0.04 +0.012 - lt X013 --G] Energy of doin] E2 = 1 kolo)? = a (@49+09) +0.214698] = 280.3 = 0:6 3 Energy og 2 n) 's 6.4] Sum of energies of rein) and Koth is (16 and 0.6 = 2-2] energies are not equal

Matlab Script:

n = 0:5;
xn = 1-(0.2*n);
stem(n,xn);
xlabel('time n');
ylabel('x[n]');
title('n vs x[n]');

n = -5:5;
xen = [0 0.1 0.2 0.3 0.4 1 0.4 0.3 0.2 0.1 0];
xon = [0 -0.1 -0.2 -0.3 -0.4 0 0.4 0.3 0.2 0.1 0];
xn_new = xen+xon;
figure(2);
subplot(311);
stem(n,xen);

xlabel('time n');
ylabel('x_e[n]');
title('n vs x[n]');

subplot(312);
stem(n,xon);

xlabel('time n');
ylabel('x_o[n]');
title('n vs x[n]');

subplot(313);
stem(n,xn_new);

xlabel('time n');
ylabel('x[n] newly generated');
title('n vs x[n]');

figure(3);

zn = [0 0.2 0.4 0.6 0.8 2 0.8 0.6 0.4 0.2 0];
stem(n,zn);

xlabel('time n');
ylabel('z[n]');
title('n vs z[n]');

n vs x[n] 0.5 1 1.5 2 2.5 time n 3 3.5 4 4.5

n vs x[n] - -5 -4 -3 -2 -1 1 2 3 4 0 timen nvs x[n] F 0 G 아어 | DO 0 time n nvs x[n] F x[n] newly generated 20.5 ŽŽì 0 1 2 3 4 timen

nys z[n] z[n] -5 -4 -3 -2 -1 1 2 3 4 0 timen