examIII_f01 - EE 350 EXAM III 8 November 2001 Last...

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Unformatted text preview: EE 350 EXAM III 8 November 2001 Last Name (Print): First Name (Print): Keri ID number (Last 4 digits): Section: ~PW!" DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO -SO Test Form B INSTRUCTIONS . You have 2 hours to complete this exam. Calculators are not allowed. This is a closed book exam. You may use one 8.5” X 11” note sheet. Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. . DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. . If you introduce a voltage or current in the analysis of a circuit, you must clearly label the voltage (current) in the circuit diagram and indicate the reference polarity (direction). Ifyou fail to clearly define the voltages and currents used in your analysis, you will receive ZERO credit. . The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1.1 (10 points) A certain periodic signal has the trigonometric Fourier Series representation THI' f(t) :: in: nszfiz sin sin (mrt). An estimate of f (t) is obtained using a finite number of terms of the Fourier series 8A . mr . "212 am sm(n1rt) . N N) N f“) = 2 Determine the power of the estimation error e(t) 4‘) = f(t) - f“) for A = 6, N = 3, and Pf = A2/3. P5; -— R: + Pa 3 5mm mm 01; mm orflmgmfl to the, ewohelti 1.2 (5 points) Consider a periodic real-valued signal f(t) with period To = 2 that is described by N) = e‘“ l" for —Ta/2 S t g To/Z. Does f(t) exhibit any symmetry that will simplify the computation of its Fourier series representation ? If so, what are the simplifications ? :— €~Ql“‘tl 73 The‘mtegmls «Cor 00 £0“ Mag be simplified. 1.3 (10 points) Find the trigonometric Fourier series of the periodic function f(t) specified in Problem 1.2 above with a, = 1/2. 0 1 -’£ flew -g ‘L SO \l‘tl‘Ot/fi (“to owmQ {lib Cos awe-t) Rag“ Hmt‘ \Cor €\J€v’\ sl‘g‘nab 0‘“:an . .; J“ -L \~ gem Jr \~e363“\ 1 3m ‘1‘ MW _ L \“638 Ngflfi; Cs“: 3 - ‘5 3-.\.V_\LL1 fl 4 -1 I“ Z W Nojm CA :Cogvm ‘ E. 1 l -+ (\TC '3; 4 Problem 2: (25 points) 2.1 (13 points) A function f(t) is real-valued and periodic with fundamental frequency we 2 61r X 103 rad / sec. The only nonzero complex exponential Fourier series components for n. 2 0 are D1 = ——3, D3 = 7, and D5 2 —2. (a) (3 points) Find Dn for n < 0. Explain your reasoning. . 'Lon _ jLBn Remfitlwtfioo Bn‘ijelT \ UVWME 35.39,“? B_\=Sefi_fi=3€“ '0 J0 83:?63k 833:36 "3“. Dr: 16 Ba; = 7. 8 (b) (3 points) Is f(t) even, odd, or neither ? Explain your answer. Slime B“ ER Piflisohevewijuflctx‘ow (c) (7 points) Express f(t) as a sum of cosines of the form 0'” cos(wnt + (bu), and provide numeric values for all C”, can and d)”. Recs“ mat C“: mm a 9“: Us“ 1; 2 (p COSLbflXi03t‘l’ll3 + M. Coshgqmuf) JV Mosbomufit W) 2.2 (12 points) Consider the complex-valued signal f(t) = 12 sin(600 t) + 14 cos(450 t — 30°). (11) (4 points) Find the fundamental period of f(t). booznwo é 450:mwo mgnefl e a woel’SOfi-Q (b) (8 points) Find the complex exponential Fourier series coefficientsfiDn of the signal f(t). 39 - file [NSC-:26 n Ellyficfie “ -‘309 ‘ibCf D : Ali EA 0-516 '1 a 530° .330 D4 " ELL 3 3‘4“ (08 -3go° Problem 3: (25 points) 3.1 (8 points) Given that the Fourier transform of N) = is F(w)= and that the Fourier transform of t e“ u.(t) 1 (1 +Jw)§' W) = e" u“) is P(w) :— find the Fourier transform of 1 1+]w’ g(t) = (t — 2) e" u(t — 1). mm 5"“? mm 3.2 (8 points) The Fourier transform of is What is the Fourier transform of [email protected])=e—“W F(w) = a2 3:2. w W) = 4“; ? F004e> 2m~¥tw9 X 2 —o.)W\ “ 6-5 '1!" e. a1+tz 2 3.3 (9 points) A purely imaginary signal f(t) has the Fourier transform Show that F(w) = —F‘(——w). PM 3 S“: W (55%” Problem 4: (25 points) The demodulation system shown in Figure 1 uses a lowpass filter with the frequency response function 10] 0 S w 5 we H(w) 2 ~10] —wa s w < 0 0 M >w° sin(u>st) 9 x0) y(t) cos(wot) Figure 1: Demodulation system. 1. (6 points) Find an expression for the Fourier transform X (w) of :c(t), and express your result as the sum of impulse functions (your expression should not contain the convolution operator). 10 2. (4 points) Sketch X (w) using the graph provided in Figure 2. Clearly label the location and weights (areas) of each impulse function. X(co) Figure 2: A blank graph for sketching X 3. (9 points) Sketch the Fourier transform Y(w) of y(t) in Figure 3. Clearly label the location and weights (areas) of each impulse function. Y((o) Figure 3: A blank graph for sketching Y(w). 11 4. (6 points) Find y(t) by direct integration of the inverse Fourier transform integral (no credit will be given if you use a known transform pair). \(M =«S’fi (ammo-mm 8(w—wo+ws\) ‘ t “5 3m (Mame-wwflw‘wawsfie”) on ‘ 1 ~00 'H Datum it (wo'w‘ai : 57.033 + e i _ S StU’DS l“) -jttlfis moi _ 1L6, + e i 12...
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