Question

1) [10 pts. Let a system be defined as ta y(t) x(31 - 2a)dt 2a Is this system b) No b) No b) No vii) memoryless? a) Yes viii) Linear? a) Yes ix) Time invariant? a) Yes x) Causal? a) Yes xi) BIBO stable? a) Yes 2) [5 pts. What is the impulse response h(t)? 3) [10 pts.] Let a signal in s domain b) No b) No 2 Y(S) Sa What is the Roc so that the inverse exists? Find y(t). 4) (20 pts. Let the transfer function of an UTI system be 2a? H(s) = 5?30207 Find the unit step response of the system using the convolution integral analytically. 5) [20 pts. DT system is constructed by an LTI system with the unit pulse response h[n] = 0.1":[n] ah[n] a) Find the overall system transfer function Ha(z) b) Response to unit step input 8[n 1] - [n]

a = 3
signals and systems

Answer 1

tta 0 О g(t) a (31 - -2a) dT - za Given a=3 t+3 yet) S x(39-2(3)) at - 2(3) t+3 ylt 1 x (37-6) at -6 vil, memory less? ot3 x(3T-6) dT y (o) - 6 3 y (0) = x(370-6) dT 6 Here y (o) = present value depends on past limits ie - 6'. so it is memory System Answer (6

vili, Linear? 2) S t+3 y, lt ) = 2, (37-6) dt 0 - 6 (t+3) Yn it) - 6 y If yz(t) = 4,(t)+9,(t) 23(t) = 2, Ct) + x2lt) Yz1t) X₃ (37-6) d? t+3 -- t+3 (2,607-6) + 13(3-6)]47 6 ++3 t+3 s 4,137–6) dT + S x2(37-6) at -6 6 (t) 9(t) 9zlt) = 9,lt) + y(t) obeys super position fence it is Linear. Answer a yes.

present output depends on past values ix, Time invarient ? 3 Input shitt t+3 ylt) = 5 x(31-6-to) di L 0 If out put shitt – ie t= t-to t-to+3 y (t-to) = J X (3T-6) dT 6 L 2 Ⓡ So it is time varient. b 3 it is not time Invarient Answer x Causal ? as shown in Vio so it is non causal- So Answer (6

xi, BiBo stable ? 18 264) = ult) x(31-6) = u(31-6) = u(3 (7-2)) = u(1-2) t+3 9 (1) - U(T-2) dT 6 tt3 YIt ) = (01T-2)) 26 bounded olt) un un bou it is unstable. so Answer (b 2 Impulse Response :- output of the system when the input is Impulse input ie set) ylt) XH) hit 2 2ct) * ht ) = ylt)

xit) - Sit) 5 8(t)* bit) - ylt) ylt) = hit )