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01-EE20-F08-Quiz-1-Solutions

01-EE20-F08-Quiz-1-Solutions - EECS 20N Structure and...

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Unformatted text preview: EECS 20N: Structure and Interpretation of Signals and Systems QUIZ 1 Department of Electrical Engineering and Computer Sciences 18 September 2008 UNIVERSITY OF CALIFORNIA BERKELEY LAST Name L FIRST Name / Lab Time 710* n m l ”L Lelflere _ o (5 Points) Print your name and lab time in legible, block lettering above. o This quiz should take up to 20 minutes to complete. You will be given at least 20 minutes, up to a maximum of 30 minutes, to work on the quiz. 0 This quiz is closed book. Collaboration is not permitted. You may not use or access, or cause to be used or accessed, any reference in print or elec- tronic form at any time during the quiz. Computing, communication, and other electronic devices (except dedicated timekeepers) must be turned off. Noncompliance with these or other instructions from the teaching staff— including, for example, commencing work prematurely or continuing beyond the announced stop time—is a serious violation of the Code of Student Conduct. 0 We will provide you with scratch paper. Do not use your own. 0 The quiz printout consists of pages numbered 1 throughé When you are prompted by the teaching staff to begin work, verify that your copy of the quiz is free of printing anomalies and contains all of the four numbered pages. If you find a defect in your copy, notify the staff immediately. 0 Please write neatly and legibly, because if we can’t read it, we can't grade it. o For each problem, limit your work to the space provided specifically for that problem. No other work will be considered in grading your quiz. No exceptions. 0 Unless explicitly waived by the specific wording of a problem, you must ex- plain your responses (and reasoning) succinctly, but clearly and convincingly. 0 We hope you do a fantastic job on this quiz. I_-_l (21.1 (20 Points) The instantaneous position of a particle on the complex plane is described by the complex-valued function x as follows: . 1 . Vt E R, IL‘ t = ezwt + ____ez(wt+1r/2)‘ ( ) fl (a) Show that cc(t ) can be written in the form a:(t )— - A(t)ei9(t), where A and 6 )8” are real—valued functions of t. Determine an explicit, and reasonably sim- Wet ple, expression for each of A and 0. You may find it helpful to note that 9/4; cos(7r/6)= x/g/Z and sin(7r/6)- —— 1/2. 3 . 1 t' at We WV ”‘1: -_ (Z xlflzemfi— JfCLE-{e’ L =(l+-\-r-t>ew :\____Q~e {i f/c' l 3 ’ tQ‘YFi'Vé) Q t’ (It .. 3 ”m?” d W >2?sz n f “(E5 Mal? \AiC Vt wst 19W, (97%:ka \‘Vr‘ {Ra‘3 \>9 \Mr Stow/A $52?er C 14 “7G /9\ i 71 ‘ whoa, fE:S :___:—_ \lz‘fn ’3 C ””1 B (.05 \\/L W; E a ‘G A“ 4; (b) The figure on the next page shows sketches that may or may not describe the trajectory of the particle as it moves on the complex plane with the forward passage of time (in particular, as 0 —> t —> oo). Identify every sketch that is consistent with the particle’s forward trajectory. On each sketch that you so identify, determine and label the location of the particle at t = 0 and denote, with one or more arrows, the direction of the particle’s forward travel in time along the trajectory. To receive credit, you must explain all your work succinctly, but clearly and convincingly. (”0 / (IV) \ \ C000 \ g 0 Mom: 9’? tee SV‘PJS \5 We? S<©\—\n 8-K 0k COPPQQ-‘V C\f\o;ce S'or X65), a\.\§::\ ’ tGJhF-‘f/G) ‘ kCC‘K/KSQ \XG/“Zgg— , VLT XL“: —; C __, \3 an ax CIFC\(-: 0’8“ MALkS 3.1;:_ . Q12 (20 Points) The instantaneous position of a particle on the complex plane is described by the complex-valued function a: as follows: Vt E R, $(t) = elk-”0t + eiwlt, where O < wo < wl. (a) Show that 27(t) can be written in the form :c(t) = A(t)ei9(‘), where A and 0 are real-valued functions oft, which can be expressed as A(t) = 2cos (coat) and 0(t) = Wgt. Determine wa and wg in terms of can and wl. Potentially helpful trigonometric identities: 2cos (a+fl) -cos (a— ) 2 2 ' cos a + cosfl sina+sinfi = 2sin(a:fi)-cos (a; ). (b) Suppose we, = 7r and wg = 107r, so x(t) = 2cos(7rt)e“°”. The figure on the next page shows sketches that may or may not describe the trajectory of the particle as it moves on the complex plane with the forward passage of time, from t = 0 tot = 1/2. Identify every sketch that is consistent with the particle’s forward trajectory. On each sketch that you so identify, determine and label the location of the particle att = 0,t = 1/5, t = 2/5, and t = 1/2. Also, denote, with one or more arrows, the direction of the particle’s forward travel in time along the trajectory. To receive credit, you must explain all your work succinctly, but clearly and convincingly. - \OT L XL“: ates «Wk» 6 1 For oat“ fi>227 lacoS‘l—rf—éO-s ESW‘l $0 hm 1,68%th «16:030th cuffs as l > t W Max niJRrJlQ 39m 16/) 7 wil‘h The, ma“ll&vlfl O 41 T; ZCPO of" f’,‘/9\ , ”\CCPQKS:4 ' & 7” be ox fmmfivltxmjhi’ (m: LAOM) +wa #69, figkto-J—aa COU’l 9—0—— ‘ clf‘c\e,. mg TULES‘ITOA {5/171064/ all’llolgt’onfl 0 we 5/”st I) uasisléfif will: KU’); Acosijjr) 8f 10 ? Matte, fivl/ ffie I’LASOK ail-afl CAC/‘V‘CIeS fie OFZé/I’L 3:“ Ff J ‘ ' g c . k ,‘A an 1%“ \l/‘MC/S 0~$ JamSToxS Cos (fl/TV) fl0£5 fimyk ’5"ch 66/6 chce7 |+ o ‘ s {GEQZJ almol'e/S one W‘LFl/Cf of“ L Cow/\C [977’1’ \ wwq‘l/ Ct TQS’ “1";515’ Carr/l6) oj“ KC Phajof‘ at . 7% 0101/75 ‘l’Ea/l fie parl’tk/ltlrlfinclN/lfigl 5 l'KE “I‘ll" 075’ #4455 "’1 [Oil/,1} TE 9M5 affirm/l ConslslenT NITR ”£35 {3 DC) a \ J 0 0+ TRC 0P;9?f\ - Emma mafia X04 Cannon" \,g 0x C\H‘C/\Q) m5 ALflZAC°5Wt ' JCCY‘QJKSGS ;r\ fie \\a<\/Q(‘\NK\ f€[o)‘[2j) Maw/{l ('i/ (£kaer KCFO 8 tl/(g ...
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