Exam#02_Solutions - ECSE—2410 SIGNALS AND SYSTEMS SPRING...

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Unformatted text preview: ECSE—2410 SIGNALS AND SYSTEMS SPRING 2097 Rensselaer Polytechnic Institute EXAM #2 (1 hour and 50 minutes ) February 27, 2087 NAME: Section: 1 Do all work on these sheets. One page of crib notes allowed. Calculators allowed. Label and Scale axes on all sketches and indicate ail key values. Show all work for full credit. N 0 partial credit for multiple choice questions and no credit for integration of the defining formulas. Total Grades for Exam: [ Grades for I Points I Score Part II 1 TOTAL I 100 l PART I Problem Points Grade 1 11 2 6 3 6 4 6 5 15 6 6 TOTAL 50 1(12). The exponential Fourier series representation for the periodic signal x50) shown below is x¢(t)= Zakeflm‘“ . x60) k=ww b.)(1) The signal xii) is (circle one) i.) real i ii.) purely imaginary iz'i.)complex 3i. c.)(3) The signal y(t) = xc(t) — g/fis (circle one). (No partial credit.) if 1‘.) odd " at.) even iii) neither (1.) (3) suppose xc(r) is represented by its trigonometric Fourier series, x60) = a0 + ihk cos(ka)0z)+ flk sin(km0t)]. k=l Recall that the relationship between the exponential and trigonometric Fourier series is xc (I) = a0 + i2me{a§jkw9’} ha; iv, 2(6). The ac vs. [C plot shown below represents the exponential Fourier series coefficients, ask, of a real periodic signal, x(t) , with fundamental frequency, wg = r7: rad/see. Find the equation for x(t) expressed as a trigonometric Fourier series. The series should contain only the nonzero terms. x0) x 3‘ (6) Circle one answer for each question below. No partial credit. a.) (3) The exponentfial Fourier series coefficients of the two periodic waveforms shown are related by (circle one.) i.) bk =w-ak for all k iv.) bk=ew 2Vark 3. (6) Continued, Circle one answer for the question below. No partial credit. b.)(3) The exponential Fourier series coefficients of the two periodic waveforms shown x(t) e ak 4. (6) Find the exponential} Fourier series coefficients, bk , expressed in terms of ask for all k. DO NGT find the specific values for the ak terms. 5(15). x0) is a periodic signal. Express the exponential Fourier series coefficients, ak , for aEI k, of x(t) in terms of ck , the exponential Fourier series coefficients of the periodic square wave, pa), shown beiow. DO NOT evaluate the ck. x(t) <—-> ak “ i —9 —8 —7 —6 -5 -4 -3 -2 -I 0 i 2 3 4 5 6 7 8 9 l0 5'3 a gagefigem Intermediate graph if needed Simply? Wherever possibie and find (2)0 . You must use properties. No creéit for integrating the defining integral for ak . 6. (6) a.) (3) IfX (a) (I ‘ jw +1wx2 +10) ’ find [X5 (ml ECSE-2410 SIGNALS AND SYSTEMS SPRING 2007 Rensselaer Polytechnic Institute EXAM #2 (1 hour and 50 minutes) February 27, 2007 Section: 1 2 Do all work on these sheets. One page of crib notes allowed. Calculators allowed. Label and Scale axes on all sketches and indicate all key values. Show all work for full credit. No partial credit for multiple choice questions and no credit for integration ofthe defining formulas. PART II 7 10 7 (10). Find the Fourier transform X(a)) , of the signal, x(t) w (EMU) , where Note. You must use properties. No credit for integrating the defining Fourier integral} for X (60) . AISO Simplzfi/ whenever possible. We 2M 2””- 7 w ‘W g ‘ f {j '” 5 , {f f {KW M E X grog}; :. xw em “‘3 W 2 e: “h” ’"r in s r f E w as??? W. e: “if?” P 2. I"? g W 2 M f if e r g r e: w“ {1: i: {J g x“ j :3 if” 5 3 4,3;— m 5 a; 1] 8 (10). Given x(r) <—> X ((0) , express the Fourier transform, Y(a)), of 320‘) m x(1—t) in terms of X (co) . Use properties. WM 12 9 (20). Find x0), the inverse Fourier transform of X(co) = ja) e-jwz jw+1 I E3 10 (10). a.)(5) If x(t) (—9 X(w), find 2(a)), the Fourier transform of z(:) m x(t)cos(a)0t). gka? wa? 4 s . ' n .i ,» ./ “W; 2%? : a3; “5 mg; mg? 5ft? J " 3} g: b.) (5) Find the inverse Fourier transform of X (co) = 7:50;?) - 47:) + 271580) + 7:6(59 + 47a) . d/ f . g .5 vs?” ' 6”" I?" fl 2%)”: my»)? 41 -; 3f =»f:?='_.r§ 7/) {41,3 3: i; raw-3 fizzy—"€535? 5/5”” f ,J 5’? w" i - 4. 14 1} (10). —jm W —jw2 e e sin(x) a.)(5) If X (co) = , express X(w) in terms ofsinc functions, where Sinc(x)= ja’ ‘ x E5 Problem 11 (continued), b.)(5) The exponential Fourier coafficients for a given periodic square wave, x(t), are i ' E ak = zsmcp‘ 2} “0. Plot ak fork=-4, -3, -2,-1,0, 1,2, 3, 4. E, k = 0 Use the definition, sinc(x) = Sln(x) . x fi‘ End. 16 ...
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This homework help was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

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Exam#02_Solutions - ECSE—2410 SIGNALS AND SYSTEMS SPRING...

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