Quiz3Solutions

Quiz3Solutions - EECS 20N Structure and Interpretation of...

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Unformatted text preview: EECS 20N: Structure and Interpretation of Signals and Systems QUIZ 3 Department of Electrical Engineering and Computer Sciences 2 December 2008 UNIVERSITY OF CALIFORNIA BERKELEY LAST Name FIRST Name 6P 0 9 Lab Time o (5 Points) Print your name and lab time in legible, block lettering above. 0 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 through 6. 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 six numbered pages. If you find a defect in your copy, notify the staff immediately. 0 Please write neatly and legibly, because 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. Q3.1 (10 Points) In this problem, assume that a and B are complex-valued scalars, and m, y, and z are complex-valued vectors (e. g., in (C3). (a) (5 Points) Express (as: + y, z) in terms of the inner products (9:, z) and (y, z). T _ — v <o<xya>z) :. (xx+a§ z*:(oc 913‘ \Zx: ocxler—r §z*=o<<x)z>+<a,z> (b) (5 Points) True or False? ’77; Q (00, By) = Wm, 1/)- Explain your reasoning succinctly, but clearly and convincingly. < X,Pa>: 23—033sz xT (first) ; FYXTJR' : Fr<xqf> Q32 (30 Points) Throughout this problem, we will live in the universe of discrete- time signals that are periodic with fundamental period 3. An arbitrary signal :5 in this universe may be real-valued or complex-valued. We want to decompose m as a linear combination of three mutually—orthogonal signals 1/20, 1/21, and #22, two of which are shown in the figure below. One 1 Peri°d\ / 1/)” = 1 P23:d\ / 1/)1 2 $001) I 1 1(1)} 1M”) 01234577» We represent a single period of these signals with the vectors 1,!)0, flu, and 1,!)2, respectively. Two of these are shown in the figure. (a) (4 Points) Verify that the vectors 1,!)0 and 1,1)1 are mutually orthogonal. That is, ShOW ($0, ’4’1) = .r <’1";)’LY|> :«YI—WT‘ Z (meow\>i\ :0 => W0)WIE\R3 O W Lw ' \ (b) (8 Points) Determine numerical values for ||¢0||2 and Hz/JIHZ. tlwotfixrwowl'wf :[\ WWW—sum wk \W‘WI: [\ ~\ OEWB‘X =_\l\_(.\f* O in (c) (9 Points) As you know, we’ve been neglecting a third signal 7,122 whose values in one period can be stacked into a vector We want 1,110, 1,111, and 1,112 to be mutually orthogonal. Assume ||¢2|| = “1,110” and determine the constants a, b, and c as specifically as you can. Which of these constants is, and which is not, determined uniquely? Explain your work. M «31' Lawfi '. l W :9 :5. {a L c-S ‘— «ML—.0 < 5H1¥0> L >> 0\:\3'—O OQ‘Q (d) (9 Points) Consider a signal at whose values in one period can be expressed by the vector Determine the coefficients X0, X1, and X2 in the expansion w: [\DD—‘Cfl = Xo’lbom) + X1¢1(”) + 99111201)- You should be able to determine X2, at least to within a :l:1 scale factor, even if you could not determine 1&2 in part (c). ,\ g \ Q X: {El/Z lLfi \ oz) \ -_—, 2:“ =3 O <VON’OB 9‘ Q §—\ 0 X I <X’W‘v : ‘— lS’ \ .2] ’\ : 01+ :92 I <1y|lxh> ; o X -. 9121,: L1: \ a] a} 4;; u, A <1¥Aiwfiz> :1 E or 9 x92: <Xlfil’2) v ~J—l— ‘J‘r a“? 21+ fifi'firnqfi Wale“; Arc “th‘nud fie QMMQQ oi:— & V960“ 0L 4 IMPRESSIVE TRIVIA TO SPOUT OFF AT YOUR NEXT SOCIAL EVENT Inner Products 0 If no and y are in (CF, then p—l Cray>:::BTy*z= 2E:1%y;7 [=0 where we and ye denote the 8”“ entries of ac and y, respectively. 0 If m and y are periodic signals with period p, then p—l (w, y) = 296(01/‘(0- e=o Magnitude-Squared o If ac is a vector, then (93,93) = 9993* = llwllg- o If m is a periodic signal with period p, then H lflcl2 = ($796) = $(n)$*(n) = Z |$(n)|2. 7L '6 II o Parseval’s Identity Suppose 7,121, . . . 7,121,-1 are mutually-orthogonal periodic signals with period p, each having magnitude-squared value [1mg = A. Let :c be a periodic signal having period p. Then p—l p—l Z |96(n)|2 = A: lel2 n: k=0 ...
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This note was uploaded on 01/23/2012 for the course EE 20 taught by Professor Edwarda.lee during the Spring '08 term at Berkeley.

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Quiz3Solutions - EECS 20N Structure and Interpretation of...

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