Question

Problem 1 Let's consider an LTI system with intput and output relatex through the equation y(t) - --- (T 2) dr a) Find the impulse response h(t) for the given system (1). b) Is this system cansal or not? c) Determine the output of the system when the input x(t) is as shown below. Problem 2 Evaluate the following convolution where (t) and y(t) are plotted helow z(t) = z(t) * y(t) Hint. Expr the signals as a linear combination of time-lelayed wit step function and apply the lo .. (t + a) *wt+b) = (t = a + b)(t+a+b) Problem 3 Derive the following convolution (t) * (t) o(t) where x(t) = u(t+1)-u(t-1) is a rectangular pulse signal. Hins. Use the following lemma (1 + a) u (l+a) *u (1 + b) = +a+b) (+a+b)

Answer 1

O Civen that yeo= t ė (t- to be LT-2) de @ we know that yet) = 20 (2) *h(8) ... so y(t) = f g (Z). h (2-7) dt by compairing [n(t)= è tuets (4) causal. so hc2=0&tco, so it is given hits o = ct hit-I) Recunda - t recta) he IdZ both are same.

So ylt= hca)- [-1(2-2)] d I J- co hit rec-I) - 2 For teol For aletco =o E No common area- rhet) s nct-t) -att 1+t + t so ylt f et dz= (é z 1tt yitha l-e (ht) for alstrol

het Forostaa, dect-I) att Itt tt so ylts = f et dz= ine Cht) so y(t) = ln é (tt) for octea for to? ATT - rect-t) •2+t itt Itt so ylti e I dia Jt-2 (e-I) THE - (+-2) - è litt) for to $o y + lze -tao te(77) Outx2 e (2-2) eltt), t> L

a us-UCC -5411-21) Bolt) = uitstsuche26ce-2-un را ا) - ژر) را رالا 20 we know that- ultrasuct+b)= (t taoth ust+o+b) = (toen tults = rets so gets = luces + Sucki) 24-2) huts درد را را ) ylti= r(t)-lt-29454tagastus-29 +27ct-u-uret-3) + 16mct-5 11t (ts-3 (1-2) Asarlto-art-3) - 427 ct-wa Golts) given sits = ulti-ucta let yets nits autors Mtaalt = luctto-ulti Tuctti-uctes - = (4+2)~241tsar (2-2) we know that - (ttag uctta) *uctth) = 1 (to ttath on mitta) mult+16) = (+0+b) ucztQtb2 so it for (2+2)-2retary (+-2)][uctures] g(t) = 1 (t+3) Pultt3) 1 (& tighetti) = 2 ettig Faczty -24-13*u(+1) + Ct-zajuce-1-4 (Esfütt-3)

g(t)= {(+39%uC++3)- 3 (t-1 ?ucz-1) - 1 (tult) - 1 lt.gulta) Please like my solution.