M2_Solution - UNIVERSITY OF CALIFORNIA SAN DIEGO Electrical...

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Unformatted text preview: UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical and Computer Engineering Department ECE 45 Circuits and Systems Fall 2007 Midterm Exam #2 No Books, No Notes, No Calculators Allowed! Show all your work clearly and neatly in order to receive full credit. Print Your Name Signature Student PID Number Problem 1 (lSpts): Problem 2 (20pts): Problem 3 (20pts): Problem 4 (20pts): Problem 5 (25pts): Problem 6 (BONUS, 30pts): TOTAL: NAME PID SOME USEFUL FORMULAS Euler’s Formula: Aej" = Acos6 +jAsin6 Fourier Series Equation: Fourier Coefficients Equation: °° 'k ,, 1 . x(t) = ZCKe’ “" CK =— pay-mow k=—oo T Fourier Transform Equation: Inverse Fourier Transform Equation: +°° . +00 X(w) = Ix(t)e"w’dt x(t) =2— jX(w)ede -00 71' -00 Instantaneous Power of Signal x(t): Energy of a Periodic Signal x(t): 2 2 i960)! j|x(t)| dt T Average Power of Periodic Signal x(t): Total Energy of a Non-Periodic Signal x(t) l 2 +°° 2 M 2 l M 2 — flx(t)| dt = Z|CK| [19cm] dt = E flX(w)\ dw T k=—00 —00 —°° Page 2 NAME PID Some Useful Fourier Transforms f(t), a > 0 F(w) e‘”’u(t) 1 a + jw eatu(_t) 1 _ a — jW te'”’u(t) 1 2 (a + J'W) e—altl 2 20 2 a + W 6(2‘) 1 1 27:6(w) Some Fourier Transform Properties Scaling 1060‘) kX (W) Time Shift 36(1‘ — 1‘0) e_jwr"X(W) Time Reversal x(_t) X(—W) 1 w Time Scaling x(at) _X[_) |al a d"x t Time Derivative ( ) OW)” X (W) 611‘" Page 3 NAME PID Problem 1 |15 ptsI (a) [2 points] Consider the following two signals: f1(t) f2(t) Does a Fourier transform representation exist for f1(t), f2(t), or both? Explain your reasoning. Note: you should be able to answer this question without doing a single calculation. ANSWER: M Stow/xv) MVQF ROWE Rwd'Gfi/N‘rcob hm OWY' *0 have a Emmer Tmfitof‘“, £0 0V\\\3 12W) NANQ O\ Vow/kw twhvm CW4 (b) [2 points] Suppose the Fourier transform of f(t) (sketched on the left) is given by sketch in the middle. f(t) A _ Ff“ ._ f(t) t [I I11 I12 113 Ill-i ISLE t w(rad/sec) Now suppose we stretch f(t) in time (as shown on the rightmost sketch). What will happen to the graph for F(w) if f(t) is stretched out in time? ANSWER: 1* ‘jow 3mm Hr) m “MK , Hw) gwmé (om/\WQCS m {5%ng Page 4 NAME PID (c) [4 points] In this class, we have been concentrating on the following block diagram: x(t) —> —> y(t) Up to now, we have learned several ways to solve for output y(t) given a specific input x(t). Describe 2 methods you can use to solve for y(t), given a specific input type. Be sure to specify the input type for each method you have chosen to describe. ANSWER: Methodl: 1-? x(H [3 PQwoArQ “)0“ (gym use FOUWZV SQW‘QS Jyo SQNQ Govxw/ a d¥wa ‘3\\’l :ZQVuQ / \{L: CLH(tWo] Method 2: if xlr) [‘3 wow ~pqrwcllc.’ \50‘4 WLQ. Fouuva l‘VOvngiOVWQ fw~cl k5 if) by \Hqu : \Mwhfiwl '—> Mom} (lele FT (d) [2 points] We already know that an LTI system can be described by its frequency response H(w). We learned recently that it can also be described by its impulse response h(t). Why is h(t) called the impulse response? ANSWER: N63 15 «mm W0. l‘mpulfiQ MWNQWMMW H- CL rm regpoww of M £5ng if M mm \‘g Om Maxim Wham (W Page 5 NAME PID (e) [5 points] Consider the following signal f(t): f(t) Now consider g(t), which is given by the sketch below. What is g(t) in terms of f(t)? What is G(w) in terms of F(w)? Make sure to show exactly how you solved for G(w). g(t) ANSWER: Page 6 NAME PID Problem 2 |20 ptsI (a) [10 points] Using Fourier transforms and Fourier transform properties, find the differential equation relating input x(t) to output y(t). . Fund. WK”) Viv/:5 \IOU‘MQ &,\\HW ’— 5/ v -.—. L X L’quw HR») 7 ;, U‘JV‘S 25' i WM waé “theJl‘Q’W‘WOJ qu“\\,\/\ ANSWER: MHMj Page 7 NAME PID (b) [10 points] Find the input x(t) that generates the following output 1‘ = 0.5e‘tu t + 0.5e3‘u —t y wkga Fouvw)! TV‘CMAQHN‘WZS HM L5 35.. flw)- fi—Jm "V gjw I “7);: 1 0).ng 059””) Saws __ 0.90%») 1—0,§C(+JN) 3‘3 (/5 ; IS ~6 SJW 149.: +0 ngQ 3 5V3 2. =— \‘NV \thJ ij 731:) Nth, L8 “(kt-+1 NAME PID Problem 3 |20 ptsI Suppose an LTI system can be described by the following differential equation relating input x(t) to output y(t): d2y(t) +5M+ 6y(t)=fl+4X(t) dt2 dt dt (a) [10 points] What is the impulse response h(t) of the system? . ugmcj Hm duo. pvoPQrw 69 FOWNV WW5 K3“ ) fitw) 1’ SN \ltw) WW”); NNMMK‘JJ Mm : W9) 4c Hm flap) / 9‘ fl: (0 i’ mm + leL (gm) (HM) NAME PID (b) [10 points] Find the output y(t) if the input is x(t) = te‘4tu(t) ' U\&Q. Fomka/V Thfijg tam/Ni I . [N 9 _. ‘L 3 Lou-m; YUM: HM WM = 4113/ , \_ @W‘DXL‘UD) Qfimyz/ . 1%: t I L V3 \ :-« 3 2. —- U3 MW) 3‘» 3 («um 1 ° [CL 7/ I A, l comma-w) 3‘9"“ UM = 1 .‘_ z : '1"— mew JW 4 HM L \hu‘)‘ > :1... L12. \(L {— ANSWER: -3’6 , y(t)= —' Q Utk‘“) tile/332$?) riLe—flcebxfl’] NAME PID Problem 3 |20 ptsI Suppose you are given the following signals: x(t) = e‘z’u(t — 2) h(t) = e"u(t +1) (a) [8 points] Fill in the blanks below specifying the range of times for when each corresponding equation is true. Also, draw a rough sketch of x(t) and h(t). Make sure to label important points on the time axis. Page 11 NAME PID (b) [12 points] Using the convolution integral, compute y(t) = x(t) * h(t). Show all your work to receive full credit. Feel free to continue your work on the next blank page. Nfiez y(t) = jxmha — ad): Taco — r)h(r)d1 $41) \(\("L\ \l\(‘l€’C) .__.i,=l>:’ LC A L L " JCH 1 Q5) HR). 53 o ; , GED 3M ”imam “Chit may ANSWER: y(t) = NAME PID NC) ‘1) tH t _ 3 ~71. —('E"C\ L _, Q ‘9, Ac Mht) 7, g M 'C, ’ ~11. -e "C JU’I ( Q, Q/ 6 AC 7/ EH 4* _ ’1 ’ Q { e A: z -— “’3 -—(_tH\ , - Q Q # Q 2, Page 13 NAME PID Problem 5 |25 ptsI (a) [10 points] Draw a rough sketch of the following function: X (w) = 3Si1’1C(2W). For your sketch, make sure to label the value of X(w) at w = 0, as well as the zero crossing (at least 3 labels for w > 0 and w < O) on the w—aXis to receive full credit. SCRATCH WORK: V _ . 3 Smut/0) l’Mc‘, N 03 - L“) XL“ ; 3M: EQMLwXWQ (oSbw) ° Fwé. 259m Creamy )UUOi : D > S‘h (2/0“ 3/ Page 14 NAME PID (b) [15 points] Suppose you are now a student in MATH 213 and are asked to compute the following integral: 2 dw +00 X:— 27r_m sin(w) W Your fellow classmate’s solution is pages and pages long! But you as a student from ECE 45 can compute the integral above in less than a page using a basic theorem you have learned in this class. Identify this important theorem and use it to compute X. Shiva a 1‘1: we. 16+ FCch: 1.— ! MW wQ MW _ \ 2 we °° / 2 [HM do /> L z X, ‘j‘ \ ’1 Pqédlvfil(§ x ilflfll 4* 81 km .1 _, Rm: :1») 4—4 1?va 1., “+41 (l ) OIOMYWUQ ' K 3 173,11? 1 — e: i _ l ,\ _{ 41A 4(b) «2’ ANSWER: Theorem: mngm\ ‘3 Page 15 NAME PID Extra Credit I30 ptsI The input to an LTI system is a periodic signal and is defined below over one period: 7:, O<t<1 X(t)= 0, 1<t<2 The LTI system is described by the following impulse response: —, Otl h(t)={” << 0, otherwise Find the output Fourier coefficients Yk when k = 0, k = even (not including k = 0), and when k = odd. Hint: You will need to calculate the frequency response to solve this problem correctly. Feel free to use the next blank page to continue your work. ' \lk.= \vaal CAR ‘W “*0“ ~> @ 1 Fwd Etc/Dy ANSWER: k=0: k= even, k7fi O: k = odd: Page 16 PID T c — ‘ ‘ 0 “I J “ WC?) ‘ -— _ ( w CL’ ’2: jn’Q—ak’ (if: wkflve oar/L, mo): 9:3); ' NOR 91‘3“?“ E‘ I ltzgven “‘1 FZOQA Page 17 ...
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