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# ps3_solns - Michigan State University Department of...

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Unformatted text preview: Michigan State University Department of Electrical and Computer Engineering ECE 366 INTRODUCTION TO SIGNAL PROCESSING Problem Set 3 Spring 2011 Issued: Friday, January 28, 2011 Due: Friday (8:00am), February 4, 2011 m 1. Problems 2.4-16 and 24-17 in Lathi (Use properties to solve the latter problem.) 2. Problem 2.4-18a in Lathi 3. In this problem, you will determine the impulse response of a circuit using time-domain techniques. (a) Show that the impulse response h(t) of an LTI system may be determined by differentiating its step response 3(t) (i.e., the output of the system to the input (First, use 6(t) = dZ—(tt). Then, use did—z) = limT_,o Finally, invoke linearity and time-invariance.) (b) Use the above result to compute the impulse response relating x(t) to yl (t) in the circuit of Figure P1.8—1 in Lathi. 4. For a certain LTI system, the impulse response h(t) = e“(t‘t°)u(t —- to). (a) For What values of a and to is the system causal? (b) For What values of a and to is the system BIBO stable? (c) For What values of a and to is the system marginally stable? ) (d Compute the output y(t) of the system with a = —-1 and to = 0 in response to the input f (t) = ej(2”‘+"/4). How are the input and output related? (e) Compute the output y(t) of the system with a = ——1 and to = 0 in response to the input f (t) = cos(27rt + 7r/ How are the input and output related? 3 66 501M+Ion1 ') RIVA/6 5M+ 1(4) 9(9) \ l %—> *9 + ca): xg/ me): I Mama—r) 0W ﬁCCC) WM Jcco :2 c6910 4 W“ 0 ,4 {(927722 ca): jmwv T o .6 : ‘C‘oft/o Z ‘ (.ofaz) 4'! WA“. sari—é 2? («(14% CCU 3 (I ‘ to! (16/) [MW “#69 *2 77)] +3“: (:4; “0 ca): 7",(U * fag) : j FAG7’T) J75 L 1c / ’I‘f L V a 76 (f '4' I ) 7a,; (9'77) 1mm, no our/«f [email protected]/:0 6 I > A ' + 'Y 4- &4.qu z far'hal ovc’ap/ [Ir/re FUN.) from 0 +0 3 . 69¢ ‘1 ck 1c “'96 9 7265'?) ‘u “m FAT) cow’a‘n’y, C(é) :AK U; )5 + 6H 4 of -? (Mal ovarian Hm ﬁlm} 74.». A1? +0 0 é+V 5€ 5 no War/cf) / (/60 :0 50/ 3)[email protected] 5324691»va Iﬂm/bém) Jr ‘90 1 5(6/ = m "Qua/mam 4136 T90 T We) : Mmst : rm) [hm _,,, T : I“ _L .0 , ‘° ‘_ Hi"; T‘jMACr/Méé UJT—jokﬁj/Aff T watt] 2 IT: .TL [566/ '51 (145%)] fa I'M/WILL 9f 9‘ j/f+¢h can ‘bc L/V Jrff-aln'n,’ bar/"U rarfonlc of J/ff‘zu, +0 W % Q 7 2/1, 4. F304 ofclar Chan"! {0 “lab”, 13):“ H“, 740,“, : (/90 ~)0) (I ~Q‘T/t) +ya Currm‘!’ ‘Hwouj‘ MJudor Qanhg+ Chm” :nnt‘ahklne‘ow’ll Jo XML :/V0_: . thnc+ar 5! clmrjd 50 1/00 2 Abng 1.}. A+ £=w / T'r/m Conﬁqn'f T 01‘" #1:, C;r(u;+ ;; 33%:I/Z 50/ ﬁx ‘/2 (Fiﬁ) “59-0 ~ V; [I ~67? Me) ' I'DM CAD/Va, ﬁnch! ;th’/¢ ’31 “He. ;J Win/(01% #25“ 565), + [Hwy a 63169) -n -1. 1.\n __ ﬁver-to) " 9/ Mes-(:0) @ An J/JTLaw-I IJ Caula/ , pa For t 5° ‘60>O an ~oOLm¢oO Q An LTI I/J‘I‘em w?” In. 3180 Hula/w [FF © 1 2)” “LJoIM'Ll/ ;M+LjrnLl¢ / he, 5°0(A(£)I J“, 4 00 ~00 ALO —004£vo0 and ;f 0' Jetway/{A} Qxfon¢m+;al AA LTI J71+Lm ;f Marjl'nauy f‘buue ;+ ,‘, “L/vlu’fvl/ Fh‘hjraue, ahJ glow ho‘lL How up [MPH “me, , a“! —°041£400 oblo Jo, ha) ~— M560} m: “I, 4(7:0 , 7T 4ch '77? K0: zjérrf +/v) __ a} ’ a) hay/2 aft/tie) MIL formula» : .1 . n J "1 J A 24%}: Haida L ﬂ ‘ 'W Haw) = [0 e {Me/J Um no ~0F31ré) °0 ,6 _ Afr-é _ J / HQRW/: 5% ed Jf ' g 0’ o ~I+’31T€) Do a :- I 6( J / — * I 0 ‘U I +wa " WM 3 I HJM fa “I'M; Th }1 a~ Maybi'luée "J’Cé/tcl ( Ila/e MW 0 'f" :ﬂfu‘t Virf)°l\ PG”; («0/ <37Tf f 77/9) 0oz”! / {#020 DUI/I} formda/ ya): .‘ [Haw/X cor(wé m +41%” (9K) . a I I : f [a 2 via @1 H02!”- /:L377 a mom” IMCM)‘ \//+Wrg‘ J - l . UI+Vﬁa MW“- MI’ (W P m" (31% t 0 — WI] 2 “Iii/J 10 um, GK), Z I Cof (3774: +779 ‘ll 1+9?“ The caviar} if ob M4J4}“/qc/e Ice: ’21 I f/wm ...
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