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sample exam 3

# sample exam 3 - 1 0 N ovember 2 011 E XAM I II E E 3 50 L...

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Problem 1: (25 points) 1. J(1(2)points) ~ig.ure 1 shows a periodic signal J(t). Determine the trigonometric Fourier series representation of t by speclfymg the parameters (a o ' an, b n ). f(t} To.::: 2:tT ~ WIQ::: ~ ::=. J rD Figure 1: Periodic signal J(t). Bect<l..v~e. .f(-t:)::. - f(--t:') IS afl oJl.J- -{")L~On I .f:,me J i:J,e.. <A.V"-,"~ ., .... I_a. [~o - 01 <frtJ)L fin ~OJ f<r f)~I)L,~,--. f. T", ~o I: +(",).1' /. (nwot) -= ¥: r t-(.f;) S If) {_.-t )£L -;: 0 v~ P.>.7-7 1J V - - ~ J.2. '!: Sl~ (n -/;).P.b ~ .;,:- [s''ICOt) - n~co5(n-l.) L 2
3

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2. (13 points) Consider the periodic signal f(t) in Figure 2. As indicated in the figure, the shape of f(t) in the interval -n/4 < t < n/4 is cos(at). J(f) SJr 4 3Jr 4 Jr 4 o cos( at) ,/ 4 3Jr 4 rr 5Jr 4 Figure 2: Periodic signal f(t). (a) (2 points) Determine the fundamental frequency of the periodic signal f(t). To = 11' => tV€) = ~::::. "2.. .,.-i.t~e;... To of (f:.) = fC-t:) (a) (b) (3 points) What is the average value of f(t)? (c) (8 points) Determine the complex exponential Fourier series coefficients D n for f(t). 4
VSlif &.7- b C.O~ "f. C OSJ = "i: COJ(7'~) -4- ~ C¢~ (;<~)) Do":: ;n: of f 1 ~ co> fc IM)<." ] J-! () + +

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Problem 2: (25 Points) 1. (6 points) Consider the following three periodic si n I . gas represented uSlllg an exponential TO" " rouner senes 100 k JklOt Xl(t) ~(D e 100 X2(t) 2: cos(kJr) t?k20t k=-lOO 100 X3(t) = 2: Jsin (k1r) eJk30t. k=-IOO 2 (a) (3 points) Which signals, if any, are real-valued? Justify yOU . . r answer III one or two short sentences ')«(:1:.) rec.Q-.,-~ ~ ~~ - (b!n)~ . D~' ::: (1:.)' orJL ~o &, l-t:.) IS not rf2j)-v~ToJ((J~.) (D~()'Z..)~:= (Co~ (-I)1r»)~:= CDSVJ.,,-).= D:~ (b) (3 points) Which sigr I of sentenceso la s, r any, are an even function of time? J ustify your answer in one or two short fl (.0 j c..o~ en (1-) an Q-<!n .c.. .. ~.["u\ I "J C:<.{) '0 t~·":r;..·I-h,(r'Q. J Cl,,{l ro SI" ("'E) ~" () Q~ {...,. ~t'W'\ 6
2. (10 points) A periodic signal f(t) with fundamental frequency W o and the exponential Fourier series represen- tation JA f D = - [1 - cos(mr)] n n7T is passed through a linear time-invariant system with frequency response representation 27T WQ < Iwi < 4w o { o otherwise W .".

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