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exam3_sol - VIRGINIA TECH ECE 2704 Signals 85 Systems Fall...

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Unformatted text preview: VIRGINIA TECH ECE 2704: Signals 85 Systems Fall 2007 — Morning Section Exam 3, 11/16 NAME: SOLUY (0&5 SIGNATURE: Instructions: 1. Print and sign your name above. Note that the Virginia Tech Honor System applies. 2. In accordance with class policy, you are allowed to use two sheets (both sides) of notes, where one sheet is from exam 1, plus a calculator. You are NOT allowed to use your text- book or any other references. 3. The exam consists of 5 questions, some with multiple parts. The number of points per question is indicated. There is one additional question, which is extra credit. The maximum score is 105. 4. Circle your final answer! (I’m not kidding!) 5. 6 (t) and u(t) refer to the unit impulse function and unit step function, respectively, as defined in your textbook. All other notation similarly follows conventions used in your text- book. 6. You may leave factors of j, 7r, powers of 10 and e, and ratios of integers in your answer, but all other constants must be multiplied through. 7. If you run out of space while working on a particular problem, please use the back of the previous page and clearly identify the number of the problem being worked there. Some friendly suggestions: Don’t get stalled on any one question or part of a question. Always take a moment to consider the possible ways to do the problem, and seek the easiest method. Do the parts you understand & seem easy first, and save the parts that seem more difficult for last. The questions are in no particular order, and certainly not in order of difficulty. Check your answer using an independent technique if you can. 1. X (w) consists of two unit impulse functions. One impulse is multiplied by —37r and lo- cated at w : —2. The other impulse is multiplied by +37r located at w 2 +2. (a) [5] Write an expression for X (w) in terms of impulse functions. (b) [5] Write an expression for 33(t) in terms of exponential functions. (0) [5] Write an expression for m(t) in terms of sine and/ or cosine functions. (d) [5] Determine the energy in $(t). (e) [10] Determine the power in m(t). 1.3,,— ml W“) a -277 2/!» +2) + 311' 2/“ ‘2) 2. [10] Shown below is a sketch of X (w) Only the portion of X (w) for w 2 0 is shown. However, it is known that m(t) is a real-valued function. Please sketch X (w) for w S 0 on this same plot, clearly indicating the phase and magnitude using lines with labels. Wubl 2? Was) 3. Let 112(t) : rect(t/7'). Let y(t) be the same as x(t), except delayed by 3. (a) [10] Write an expression for the phase of Y(w) in the region around a) : 0. (b) [5] Say specifically the range of frequencies (w) for which your solution to (a) is valid. m) IlL ‘9 l; 21;,— 4’72 -‘3 CA) Qlflrxlf’3)¢=>‘llw)= XMeiw < Zfi W,me_xw)=o f" ’ éw'J—Y ‘Sw ‘ ‘ ‘ [14“(001 Ail z-MW/UW. {6) A—s WW, 4. [15] Consider a system H (w) for which the output is the sum of two components. The first component is the input delayed by 7'1 and multiplied by a. The second component is the input delayed by T2 and multiplied by b. Write an expression for H (w) in terms of w. = aF'Livx M49} +1; Fifi (E35)? )1 p is 5. Let :c(t) be a rectangular waveform having magnitude equal to 3 and duration equal to 2. (a) [10] Write an expression for the energy density of this waveform, in terms of w. (b) [10] Determine the lowest positive frequency a) at which the energy density is zero. (0) [10] I wish to reduce the bandwidth of the signal by a factor of two. In this case, I define “bandwidth” as frequency separation between DC (0 Hz) and the first “zero” of the energy density function. How should I modify 22(t)? XII):3MLVL§ 5 fNNou) Sue—«Mmifi, ”WW-”L (a) xw): gamete?) am». Wm WM Q wan“ = 24 mm 15) RM W’wa‘g’i M) p, out m= E}; c If] (e) To WEN ‘ b‘XZ' 6. THIS IS EXTRA CREDIT, WORTH 5 POINTS. Consider the time-domain waveform 33(t). Prove that taking inverse Fourier transform of the forward Fourier transform of $(t) gives you a:(t). To get credit you must justify each step, not simply describe or outline the solution. You may however use known transform pairs in your proof, as long as you clearly identify them as such. ...
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