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E#02_Solutions

# E#02_Solutions - ECSE-2410 SIGNALS AND SYSTEMS FALL 2006...

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Unformatted text preview: ECSE-2410 SIGNALS AND SYSTEMS FALL 2006 Rensselaer Polytechnic Institute EXAM #2 (1 hour and 50 minutes ) October 18, 2006 NAME: SOLJ f S {GM S Section: 1 2 Do all work on these sheets. A Table of Integrals and Identities wilI be handed out separately. One page of crib notes allowed. Calculators allowed. Labek and Scale axes on all sketches and indicate all key vakues. Show all work for fuil credit. sin(x) Use the deﬁnition sinc(x) m . x Totai Grades for Exam: Grades for P0 nts Score M- HHi—KHé—lh—dhﬂ ChUl-FkaNI-ic: TOTAL 100 Exam #1. p.1 1(7). Use properties to ﬁnd the Fourier transform, X(w) , of the waveform shown. xm <—> X(a3) ‘ Exam #1. p2 eﬂ‘, t < 7: 2(7). Use properties to ﬁnd the Fourier transform of ﬁt) m{ i ll . Simplify. 0, t > 7: Exam #1. p.3 3(7). Use properties to ﬁné the Fourier transform, X (0)), of x(1) = g;[(er3’u(z))* (e"u(t — 2))], where “ * ” is convolution. (1.3+ [12+ 243mg. a WM Elm): % 33w we we. (om-24 Kw}: a we) ﬁg kHz”? :13 a ole—.2. Exam #1. p.4 4(7). Use properties to ﬁnd the Fourier transform, X ((0) , of x(t) : 3—g—sinez (21). A sketch ofthe answer 75' will sufﬁce if you justify it. New“ m2: 4‘ \$135+) :2 e we w 69 E599 \$sz SEE:- Efw) vii-13mg) Exam #i. p.13 5(7). Sketch the Fourier transform, H ((0), for a system whose impulse response is Mt) = 29ie{m‘;ewjz’sinc(t)}. Use properties. ”3’2. Le} ﬁg»: #maﬁﬁfs‘) t L/ (m Exam #1. 13.6 x0) 6(3). The area of a nonperiodic signal, x(r) , say, for example, (a) SRe{X(a))} (b) X" (60) (c) X(0 (d) e‘1 (e) To ‘1‘” ' dr m ea where X(a))= Tx(t)e"j‘“dt. m7 Em)? ﬂag-H é‘t‘” ‘2; M m. 52> 7(4). If a real signal, x(t), is periodic, then (a) it has a zero DC value. (b) it is even. .7 Fourier transform contains delta—functions (d) its Fourier transform is periodic. (6) its Fourier transform is real. Exam #1. p.7 is ij 8(7). Use properties to ﬁnd the inverse Fourier transform, x0), of X (0)) = j i e . dco 1+ jco m 3.1“. Aw ... mt EM HM 2%W~~€uéﬁj W ”m Emu Kim) M, We (>2 ("E “”3 W “We...“ Exam #1.}18 1 6 .S' 1'. jco(1+jw)+7r (CU) Imp 1fy 9(7). Use properties to ﬁné the inverse Fourier transform, x(t) , of X (a) = Exam #1. p.9 10(7). Find the impuise response, h(r) , of a system whose frequency response is H (w) = H mm (~5anng mu) aﬂw Exam #1. p.10 (1+ jw)(2 + jag)‘ 11(7). Given a real signal, x(r) <—> X(0)), ﬁnd and sketch y(£) , where y(zf) <~—> 93e{X(a))} x(r) Exam#1.p,11 12(7). The input-output characteristics of 3 UPI system is represented by the differential equation, d)’ +2 —2xt. 7t 3/ () 13(1) ha) «a H(a)) y(t) Find the system impulse response. jug) Ewwr 2.331242} 11'» iii/w) W 3?ng m it 2.. Egg) " We) 23w Exam #1. p.12 13(3). Consider the periodic Signal shown, Where the period, T, is held constant. 260‘) I “f; T T If the pulse width, 1'", is decreased, then the DC value of x(t) will (b) increase 5&1) T} {léuemg (6) remain the same. (d) (e) Exam #1. p.13 15(8). In the ﬁgure shown, x0) is a real, periodic signal with period T 2 4 and exponential Fourier series i, k = i1 d, spectrum ak = ‘5 . The ﬁlter has an impulse response Mt) = gem} 5:2:(1) . 0, else 35(1) h(r) <—> H(a}) ya) 5% g. gt 3: Find y(t). Simpkify \$37, 2" ‘" 1“ 3’2“” L w=?z_ mm». 7%; ‘ MM. _, __1___“3W‘f ...., :1 .. \$2; jpé’ij‘wz M1) Exam #1. p.14 16(8). The exponentiai Fourier series coefﬁcients of a periodic signal with period, T = 4 , ere isincUci) ,1}: ¢ 0 I?“ ask = {4 4 . Calcuiate and sketch x(t). Label and scaie graph, 5:}; 1:. 32; ___, E 0, k =0 V1 Mead “iii- 1% Mi) +0 W ﬂying be End! Exam #1.p.15 ...
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