Question

1. Use combinations of STEP FUNCTIONS to describe each continuous-time signal shown below. f(t) 0 2 4 6 0 1 2 3 0 1 2 3 4 2. Sketch the following signals: (a) x (t)=1 [u(t+2)-u(t-1)] (c) X(t)=\fety (b) X(t)=t.e (d) x (t) = u(t) u(t-1).ult-2).u(t-3) 3. Determine whether the systems below are linear and time invariant. Justify your answer! (a) y(t) = x(31) (b) y(t)= 2x(1-t) y(t)=cos(x(t)] 4. Simplify the expressions: (a) y(t)=1.8(t+2)+(t +1) 8(1-1)+(t+3). 8(t) (b) y(t) = (1-1) 8(t-1)+(-2)(+2)' + cos(-27) 8(t+27)

Answer 1

f (t) can be writen as 15. 04 f (t) = 58 (t) - 2.5 8(+-2) - 58 (+-2) +2.5 r(+-6) fH) = 5861)-7.5764-2) +2.5 t-26 where 8 A) is a ramp signal oct) 2 - - 990 3 X g (t) = u(t) of 4 (4-1) + ult-2)-3 ut-3) th(t) 4 h(t) = 28 (4)-28A4) - 28A-3)+28tt - 4) a (t) = u(1) -24 (4-1) +2u (4-2) - u(43)!

*4)=+. (44+2) 42-)] u(++2) = t3,-2 " + u A-1)= uct+2) - (t-1) 9 + [u (t +2) – u(t-1)] (t) 2061)= telt! - 1 tl

*.4) - rid = -tet to - tet +3,0 Izol t sol 2 ti 30+ .ut-). 44-2). U(1-3 d) 264) A744) - 4(4-2) 443)

@ g (t) = x (34) Linear day, 4) + by El) = an, (4) + bx, (t) | 44) = T[X] y (t) = x(34) is a linear system Tim Invarient YET) {t=t-T= Y/1G) = x(4-7) y = x(3+) 964, 0))+-+-t = x(36-1) x(3t-T) |x (t)=uC7-T) = © + Hence y (t) = x(3t is TIME varient y (t) = 2a(1-1) system is Linear system ay,+) = a 24, (-+) by (H) = 62% (1-4) Let / (t) = T[23(+) 026) = am (+)+ b^2() 434) = T (ax, 6) + b x31+)] = a Thai(t)] + BT (*2))

Y(t) = a 2x, (1-t) + b2% Clot) - aytt byt) 193 (t) = ay, (t) + by 4) | Hence Linear system TIME invarient 4Ct)- 23(-+) y (17) /tatut = ax (1-4-1)] = 2011-t+T) 499 a cs= x 4-1) = 21–+-+) Hence Time varient system, © 4H) = cos(act)] By observation given signal cos (u(t)} [instead of ? system cost is non-linear Hence 4A) = cos(x(+)) is Non-Linear TIME Invarient y =) = cos fue +)] GAIT) = cos fu(t-T)] yt/ = cos(x )) TIME invarient A-T)

@ )+(43) 6 4) 7 (4) = + G+2) + (1+() 8 According to the propesties of impulse - - signal. (H) 8+) = xco&A) 4, (t) = 62) & (+2) + 28 (t-1) +38(t) * 4.64) = -1)? 864-1) + 84-2).C++233 + cos 4-27) detan y(t) = o. &(t-1) + 2+2) 3. 84-2) + cos (27-27). 84+27) - of 64 8A-2) + 8 (t +27) 42(t) = 648 (+-2) + 8(++27)