hwk6_solns - ECE 300 Signals and Systems Homework 6 Due...

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Unformatted text preview: ECE 300 Signals and Systems Homework 6 Due Date: Tuesday October 16, 2007 at the beginning of class Exam 2I Thursday October 18I 2007 Problems: 1. Assume x(t) has the spectrum shown below (the phase is shown in radians) and a fundamental frequency a)” = 2 rad/ sec: 4 «P ,1 I r : F : : I‘ I I l I I l l ‘ r I l I ' , , I 1 § i : : : I I a) E a E s I ‘ i I 1L . , A , 7 7 r . . c 7 7 7 7 7 . s 7 7 7 r . , e , 7 7 7 , . 7 A 7 77$, A A 7 7 7 s _ 7 r , s . A . e 7 c , A 7 7 7 _ e 7 7 , . . .7 I l I o.- i , {1 -4 3 2 1 0 1 2 3 4 Harmonic 3 e . a , _ , A , , . , , , -o T, . , , , a a . , , . . , , , e A , , ,T o , _ , , , , w I 1 z . : l . I 2 v ~ ~ — — 4:4 a — I — — ~14) — — — A ~ ——' — - — — — ——'— ~ — — — — ‘i — — — — v ——' — ~ — — — — — — ~ — — “4; 3 f I I : : : : c 1 _ _ . 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Assume all angles are multiples of 45 degrees. 2 ,,,,,,,, ,.- r~ .... .77 g ..... ,7 3 ,,,,,,, .7 7 7 . . , . , . . . , w , , . 7 , . 7 , , 7 . . , , r 7 . , , ,, L I s ‘ I l’ I I I I (F I I I 1‘5‘__.l . _ _ _ -_l_‘_l___l____i___l_»_< I I I I I g 3 I I I I I I §2—~—4_7-c . _ . 7 . _ _ ——4)..._—L——-l———_( g 1 '-_l__~fA__—€I)A~‘>__Fd>‘v_+_~-l___g _, I l I I E I I g I I I I I I I 1 7 _ _ 7 7 7;. _ _ 7 _ _ t _ 7 _ _ _ _ 7 7 7 I 7 _ 7 7 (0.5 *"t‘"’l*”’”“ ’t — i r l‘ T J T I I I l , _7 ..... .. l , , , . . , , , 7 7 , 7- l ,,,, ,, I I. 7 ..... ,. , , . , , . 7 , , , , . , . 7 l ,,,,,,,,, 7 r 9% 3 -2 -1 o 1 2 ‘3 £4 o -2 1 o 1 2 a A“ Harmonic Harmonic ‘00 7 ,,,,, 77...- , 77 m ,,,,,, 7 200 ,,,,, ., , I 7,. , , , , , . , , , . . r . , ,, I I I I I I I I I ‘I' I I " I I i I I I " I I I ; I I I 977,7 1 7,. I .77... 7...}: g 7,... 1m .1 7 . p ,7 7 4‘7 7 7 7 7 .I ,. t I7 7 7 I I I ‘ I I I I I I I v I I . I I I v I l I I I $100 77.1777177755 7 7 7 7 77I7,,7L777I..,7_ 5 097 7,; .7. ' 7,7I l ,,,,,,,,, ., 4 ,,,,,, ,. I I I I I I I I I I I I a“ I I I I ; I I I 9' I ‘73 I I I I .200I7 ,,,, .7 I 4., I 7 ,,,, 7‘ 747., I ,,,, “L 100 I ,,,,,, 7 I 7..) It» ,,,,, ,.L I7. 7, 4 -3 >2 —1 0 1 2 3 4 4 3 -2 1 0 1 2 3 4 Hamil: Harmonic a) Determine (sketch) the spectrum (magnitude and phase) of the input to the system, x(t). . b) If x(t) has the fundamental period T = 2 seconds, determine an analytical expression for x(t) in terms of sine, cosines, and constants. 4. Assume two periodic signals have the Fourier series representations x(t) = ZXke’kwI' ya) = Znej’m’o‘ For the following system (input/output) relationships: 3) ya) = be — a) b) YO) = bicU - 0) anhifinmflqn(wamzfi=%fin4+XMJ) .. 2 . 1 d) y(t)+—gy(t)+—7y(t) = Km) a; a5 i) write Yk in terms of the X k ii) If possible, determine the system transfer function H ( jm) iii) A system must be both linear and time—invariant to have a transfer function. If you cannot determine the transfer function, indicate which system property is . not satisfied (L or Tl). Fall 2007 5. Assume x(t) has the Fourier series representation x(t) =2Xke’kw‘“ and fundamental period I; .The function y(t) is related to x(t) through the relationship y(t) = a) Determine the period of y(t) in terms of I; (the period Ofx(t)) and fundamental frequency for y(t) in terms ofwo (the fundamental frequency for x(t)) b) Set up the integral to determine the Fourier series coefficients Yk in terms of the parameters determined in part a (the integral should be centered at 0), and determine how Yk is related toXk c) Starting from the relationship x(t) = ZXke/W and making a simple substitution, show how we can determine the results from part b. This problem demonstrates that compression or expansion of a signal does not change the Fourier series coefficients, it only changes the fundamental frequency. Fall 2007 “OJ/Fee (200': 4. 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This note was uploaded on 11/04/2008 for the course ECE 300 taught by Professor Throne during the Fall '07 term at Rose-Hulman.

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hwk6_solns - ECE 300 Signals and Systems Homework 6 Due...

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