Question

3. Impulse Response and Step Response. (25 pts) Consider the following LTI systems: • T1: Has input-output relationship yı(t) = -X1(t – 5) 1<t<4 • T2: Has impulse response hz(t) = { therwise • T3: Has step response s3(t) = -4u(t + 3) • T4: Has step response s4(t) = -tu(t) (a) (5 pts) What is the impulse response hi(t) of system Tj? (b) (10 pts) What is the step response sz(t) of system T2? Write it in terms of unit step functions (c) (5 pts) What is the input-output relationship for system T3? (d) (5 pts) What is impulse response h4(t) of system T4? Simplify your answer as much as possible.

Answer 1

☺ 9,6+) = - 20, CD - 5) taking Leplace tocarboom on both side Dooperty - Yi (s) = - X,(s). e-35 helton as xes) a Y,(s2 = -6-5s 264) X(S) Get? t y(s) TiCs) = -e-IS (603= HILS) YO) 2 705) streaker fraction H (S) = -2-55 SH2 tua 12 ( Sy 55 to 6) -80-3) ht) = - SCA-5) Kearmed with Sambutea desfongezhlt) Lissan (o-As Cotas - soo

tarmober dem ction - H(S)= y(5) X15) when X(S) =1 le ult) = S(4) than 947-D gombulbe despange 8 y(s) = H65) ·X65) 705) = HCS) 1 o so ( when gorput is ampulse then" Jocel but gmbluse response. & it I is equal to hlt) le geral (444)=647) le GH)= hlfo embalse Terbarse so annycs) withcs) CamScanner

-) 24 24 -3) = (b ht۔ اد lo, elge [ (4- ا - 3 . 141 = (ht (اط ۸ س ا * 4 | نام (eg تا 10 سال) ).. لا .کسے ہے - یہ و کھولنے سے (به مه کے= *ے حه (4) cs Scanner CamScanner

14)= 3 [4,6++i) – (164-47] = 3^42 - 3464-9) H(S) = 3.fere - Bus] - (es-e-43) 1469 = 3(es.e *)] Now for step sertanse gorput = Step signal a 2ilt) le zilt) = Cilf) Cs) - $ HOD= y() =D YCS) = XCs). HCS) XCS) Y69 = .5 (23-4-1) = 2405-e*) - Bessons - 1 104-40 €7) e n 064-73 cm Sergisi die P 0 1 2 3 4 5 6 18(8-9) 3 - 1° 1 254 To i 2 3 4 5 h 7 it $24) = 44) = 3 [o14-7(4-43] rs Scanned aerhonge due to step goupied CamScanner Step perbense

golf+) 6 ar(44) gt)= 3 [&L&H)-84-45] [0-0 =0 3 Co-o] =3 3 [2-13 = 6 3 [3-0] = 3.3=9 3 [4-07 23.4 =12 5 3 [5-0] = 3-5 = 15 6 lī 3 [6-1) = 3-5 = 15 3 [7 -23 = 3.5=15 S24- ) na 10 1 2 3 4 5 6 -- Paper Cect poel delationship SH) = - 44 (H+3 hence 1 Steh gaput Step verbonge le $447-434) $ 23 47 = 447 949 = -44+3) Fultted to east YO = -4.6.05 V 464-08 G eeste 4679) 6235] Lsearmed with CamScanner

Nom Hyls) - Y3C5). X3 (5) Hy(5) = -4.23 213623 Step Capeet = Ults. . X3(S) = Į H3 (5) -4.835 +26) [H₂(S) = halt) = -4e35 -4 84 +3) H3 (5) = -4035 BLS) = -4030 Y (5) = -4e35 365) ught Y,(s) 21364+a) G eorge 21314-47 H ectares Uz (ft3) - 35 426 tokios Imreoge Laplace tocrosboom 1944) = -4 23 (H + 3) gorput out heel delaticashib S = -f utt) (P) = Step called step oorpanse Step response le when the input the out but yu) - Jos 50 ann Salthie (t) = -fult) CamScanner

Splt) = Yalta - til) Noo alebo gmpulse derponge seems wherv gorpoot is Gompulse the occt but called gombalse $ C) 2466) = . XC5) = r H(S) - YCS) (5) so foo finding grapulse response directly write out but no breed to Calcalate tocortes de action to all other calcalation bесаue Cut) — Һ4) cole fro gmbalse Signal as Impat. Y(S)- HCS). Y652 Y(S) = (5): 1 Y(5) - HGS] GH) = hlt) halt) = yolt) = -5 LCA) d with amScanner
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