ex3_solns - Name Exam 3 363M CM 1(25 points Finding the...

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Unformatted text preview: Name Exam 3 363M CM 1. (25 points) Finding the energy in a given bandwidth For the signal x(t) with spectrum shown below: . _ . _ _ _ _ _ _ -_. . _ . _ _ _ _ . _ . _ . _ _ . -g O -100 £0 -60 40 -20 0 20 4O 60 80 100 Frequency (radians/sec) Determine the percentage of the total energy in x(t) between 20 and 60 radians/ sec. R AID .no '20 L”) Sro CT : ’E—ES 313)“) g ‘iz‘m 1% (9250* g Lizacvd #2 110?“) It! ~90 no ‘20 lo ‘0 7.qu 0 + 401: :‘LK‘Lquc-f-(Qp 10+ 2(9qu «HQ: 1 ‘i .m. 2H L” 0 9:0 “\ 4,3431: f0 r“ «{on CINE“ 2 .40 101'- 4T? : , X Li wing 2 u " —— Lawué 1" -810 q: i 7f? Name CM 2. (30 points) System analysis with the Fourier Transform Consider a linear time invariant system with impulse response given by h(t) = -3—sinc 22: 27: with input x(t) = 3 sinc2 cos(t — 3) Jr 71' The output of the system is y(t) . Show all of your work and draw a BOX around your final answer. a) Determine X (m) . b) Sketch the spectrum of X ((0) (magnitude and phase) accurately labeling the axes and important points. c) Determine H (w) . d) Sketch the spectrum of H ((0) (magnitude and phase) accurately labeling the axes and important points. w 2'3“) 6) Determine y(t), the output of the system. Z I‘ 3 2’ (lo Ca) For 11(9) 3 €m8<$n> Yul”) :TrJu “) D XXI») E’V‘IT K: ,3 Kim) = 2 )4 I ‘\ W Name CM 3. (25 points) Fourier Series of a Periodic Signal The following set of questions refer to the signal below (a) What is the fundamental frequency of x(t) in (rad/s)? (b) Find an expression for the Fourier Series Coeffi answer as much as possible. cients, ck, of x(t). Simplify your @ 7; .5ng w” 7 7:2 ’ r 2 Z t KW A“ (My a ‘3’“; fwd 0/é*f‘2a4 at] K 1 -2 0 ’0 W0 Kaleb o fl’t’woe‘ a") eqkwo “I ‘F e w ,I 6-0 / + e a t : i “mi—“wax - 2 #1 2 L7 Milo 1 170*“? 0 )fiwo " -‘ a“ +9? - u a .1 20045103 M2] 2 ~~Mg§gu~y fl~m w A ’ l W 4Kujo ~y<.—A W»~~ Haw/W"Hfl-“ ~ ~ ‘4.“ ¢¥ e l ws (CW>~/; ‘ lk1 2 {A )K., A 9": ‘Jm ” # A Name CM 4. (20 points) Properties of the Fourier Transform Show that if a signal, x(t), is real and even, then the Fourier Transform of the signal, X(co), is also real and even. (7/ a m i ’33 K m new 4th z / 90/5 - «(1-3 QM (wt X/w) = :5398 “’1”; .Lq’tw W d M 'T M avgv‘ don; : UJJ : {GK/k3 om/wt3Jé Cniiautmg, Over 1‘ T '30 .. :: 40?) 00qu J/yw) g {53/63 VJ (a 35/43 Jo dA/p’ah/PVVW ...
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