fall02-mt1sol

fall02-mt1sol - BALAKRISHNANI’RUNDELL SEP 24, 2002 ECE...

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Unformatted text preview: BALAKRISHNANI’RUNDELL SEP 24, 2002 ECE 301 Midterm Examination #1 1. Enter your name, student ID number, e-mail address, and signature in the space provided on this page, NOW! Also circle your section. 2. This exam has two parts. Part I consists of questions for which no justification is required. Enter the answers to Part I in the spaces provided. Partial credit will not be provided for problems from Part I. Part II consists of three problems. Unless otherwise instructed, justify your answers to these problems completely. Please note that answers provided without justification to those prob- lems requiring a full justification will be given zero credit. 3. This exam is worth 100 points. You have one hour to complete it. 4. There are 11 pages in the exam booklet. Use the back of each page for rough work, if necessary. 5. Please work as neatly as you can. 6. No calculators or crib-sheets are allowed. 7". You might want to read through all of the problems first, to get a feel for how long each one might take, but don’t worry—~several of the questions are easier than they might appear when you just scan them. Good luck! IMPORTANT! Whenever a certain space is provided for the final answer, be sure to enter your answer there. Name: StudentlD#: so LUTl ON S E-mail address: Section (circle one): Section 1 Section 2 Randell Balakn'shnan Signature: Questions for Part I Do not justify your answer. Partial credit is NOT available. 1. (30 points) A continuous-time signal x(t) is shown below. (The signal is zero over the time intervals not shown in the figure.) Signal x0) 2 15- 1- l .\ 2. Em :- riolxtuli'akt — FELHLM fight M? (b) (2 points) What is the power in x(t) over the infinite interval, that is, what is Pm? A5 Em<oo’ Pm=o Six signals labeled “Signal A” through “Signal F” are shown on the next page. (The signal x0) is also shown at the top for your convenience.) Match these signals with the following six signals, and enter your answers in the appropriate spaces in the following table. Each entry must be a letter from “A” through Each answer is worth four points. (c) The even part of x0), that is, xan). (d) The odd part of x(t). that is, xoddfl). (e) x(1—t). (f) x(—1—t). (g) xll * 23)- (h) x(2— 2:). Table] xeventi) x(1—:} x(—1+t) x(l—2r) F I! 3 (619113 mevifieok) ——'—-—-—-—-—-—-——-—————I—______._____—‘____ Signal xlt) E‘ I I I I r 15- - I- 0.5— CI .0 5'- - III- _ .I_5-— _ .2 I I I I I __.___ _4 .3 .2 .1 o l 2 :3 4 Signal A Signal B 2 . I I -—I—- I I r-— 2 I I 1"_“'I I l I l!— . I5. _ 1* - i- . os- . 95. _ 0 o as _ 4,5. _ .I . _ _‘ . _ .15 - - .15 _ _ .2 L.._. I I ..._.A .3 I I I .-—rL———I——l._.._._* 4 -3 —2 — O l 2 3 4 -d -1! -2 -l U | 2 3 ‘ Signal 0 Signal D 2-—Ié—II—r—-—-—.—_._.—,— 2m . . . . 15- - Is— - 1- - I- - 05- - ua- - D 0 —DE 1 45 ~ -l-— - —1— i 1 -Is- - -15- _ .2 I I ".4. I I I _2—I___.._I_._. I I I I 4 4 -2 —| U l 2 3 It —4 -J —2 — fl 1 2 3 4 Signal E Signal F 2 . I I . I 2 I '-—'—1_“"—l—l—l—l— 15- - 15— - 1- —‘ - 1- ~ 95- I - os— - o 1 o I -05— - -05- - _I . - _. _ - .15- ~ -15 .2 I I I .._I_._..I I I _2_....___._.... I I I I I —l —:i —2 -l 0 l 2 i! I -l —3 —2 — O l 2 3 It 2. (20 points) Determine if each of the following systems (with input x and output y) is memo- ryless or with-memory; invertible or non-invertible; causal or non-causal; stable or unstable; time-invariant or time-varying; linear or nonlinear. (a) System I: I) —— /1x(r - 1):? Ta": y( D ' (b) System II: y(_t) = x(sin(t)). Enter your answers in the following table. In each entry in the table, write “Y” if you can conclude that the property listed on the left holds for the system listed at the top of the column. Enter “N” if it can be concluded that the property does not hold. (For each case, there is sufficient data to make this decision.) I l SystemI System [I I IIIl-l III-“I ll-E-I III-I lint-“I llK-nl Every entry in the table is worth 1.5 points. You will get two additional points if all the entries are correct. @ 2 @ $18M.“ I Jc -—‘C Exec—me M gem w 0 Managua? INO,‘ Beam AaL-t) ALPWOU an. {Item is Cowl} E9 Wm" H. t‘ "Lt-'5) L3H): gxme At O IN; or HLLQ CM ‘c (0+w VA n'ablcs) (at) 0‘! CM 3d: 1th) gum: 3th) + 63%) At xLJc)=uL-t) The“, usmj mam (51-) 8mm anm \"fl‘" t at“ _ a": 51006 M: ’ TM? *iKVaviawd' Take 1%): 8Ct+‘!2.), 1M «5(1) -.— o SkiH my»): to ijM by 1, MA CWM 3. (10 points) Circle whether each of the statements is true or false, following these instructions: it Do not justify your answer. 0 A statement is true if it is always true, without further qualifications. It is false other- wise. (a) (2 points) If y(r) is the output of a linear time-invariant system for an input x0), then y(—t) is the output for the input x(—t). Gaga ‘l‘o ‘FWLA CNW‘l‘EYQXkMle. True In f'ebi- ) M‘s WM M We Lm wwkl‘ (b) (2 points) For an unstable system, every bounded input x(t) yields an output that is not bounded. T—flher 0.“, LT I “M's True (False ) § SW) NW» “I’Wl' L1()u-‘l'pmi' is bmd U“ ‘9‘“ (c) (2 points) If x(r) is a periodic signal, then x(r) +x(at) is periodic for any real number a. “a” waat be Thb‘w-«fl. The (d) (2 points) If x[n] is a periodic signal, then x[n] +x[an] is periodic for any ' eer a. False The}! Period» 0} 1, [M mA at (an) Y‘a-Ho at the. is Verdi‘me . (e) (2 points) Let y(t) be the output of a linear time-invariant system for a nonzero input 1:0). It is possible to deduce the impulse response h(t) from this information. ngbOS—t lUcir—i ’6" “M l.“- ue “RAM .._._. A (6L fiv 0~M f) n -.= A Mr. Cmt’r “km” Winn A —{ .C‘C RUE) Ms 1 Questions for Part II Justify your answer completely. No credit will be given for answers without justification. Partial credit is available. 4. (15 points) Consider a linear time-invariant system with impulse response __|,l_ e’, r<0, hm—e "{e_‘,r20. (a) (10 points) Find the step response s0), i.e., the output of the system when the input is the unit step function u(t). Lel— w.) um. Laud: glUc—Fc)httt)a< #063 Jae-t) Cm £40“. / / Lm So (b) (2 points) Suppose the input to the system is x(t) = u(I)—u[r— 1). Express the output y(t) in terms of the step response s(t). gm ‘(meavifg l hmeviwaw‘auu, 5w: SL-lfi-SUc-l) (e) (3 points) Explicitly write out, as a function of time, the output y(t) that you obtained in part (b) above. t-I Kt—U‘o SLt—Ii 2 e ’ 7,—9.0“); Ric-030 SUfl— SUV!) t a: .4) e .... e ’ t 4 O I -l' * -\ 2—e—e”) Oétu -3“) ~17 5. (15 points) The impulse response of a discrete-time linear time-invariant system is given by h[n] : 5M —5[n—— 1] +5[n—2] —5[n—3] +--- = Z(—l)k5[n—k]. 1:20 (a) (5 points) Find the step response s[n] of this system, that is, the output when the input is the unit step function u[n]. LGqu 3m = 2 W3 WV“ kzfim Mir.) 6 I 1- 3 q. 5 an IDA-Y.) 0 I 1. 3 4 s . (“=4 SW 1+ 3W 19-1 cLuw find: é \ACKMLDA-LJ = 1 CAM “70, Ml, oM SAM: avammfl- M Bob-M, H’ SMLJ- lac. CW 1104' 2 kg; men-L) —_-. O S; O , vx < 0 at“: 1 vx 7,0 MML w 0 I V\ 30 wet 01% 6“ (a) AHemak m'rgd REM at kfvd = utm [m1- mm + EDA—234M} +~- ] = [MEWS - “CM-U) —r [kin-1') _u[“flnl .1. .. . 3m + Stu-2.1+ aim—43+ (Came W M arm) 5- Kb) 100-.- 5m mun—.3 W1 = JM don-13* sun-2.3.— an-s) + .. 3° 1th km = {My} + 3C“-l])* [5Lu1_3(_“_\3+3[“_z]- ) Elm- Stu-‘3 ADA—2.3 .— SLu-s) ,f ... + SUM} - 5LW-L3 + Stu—s.) - - -. 1 fl : COM “L60 SW Mg (b) (5 points) Find the output y([n] when the input is x[n] : 5[n]+5[n—l]. (c) (5 points) What is the impulse response of the inverse system? Rm» fax/«t Us) kabétEfiLwl-i'ELW-l3325tu) S; \waj} :: 606+ 50"") 6. (10 points) The Fourier series representation (FSR) of a signal x0) is x0?) = 2(0.5)kefl“. k=0 (:1) (2 points) What is the fundamental period of x(t)? wo=L (L3 iws‘acc’n‘m) .s 2n: = 2 R T w o (b) (4 points) Find the Fourier series representation of the even part of x(t), i.e., the FSR of 10".) = (c) (4 points) Find the Fourier series representation of the odd pan of x(t), i.e., the FSR of xuddrr) = gem —x(—z)). D -\< 5U: °° \: jL’c 1mm = —- ELM) e + 209 s) e :—90 [(2-0 5 4: o - i®~5§ , k < 0 an“) — O J K :0 k k K >0 ...
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fall02-mt1sol - BALAKRISHNANI’RUNDELL SEP 24, 2002 ECE...

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