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Unformatted text preview: ECSE2410 SIGNALS AND SYSTEMS FALL 2006
Rensselaer Polytechnic institute EXAM #3 (1 hour and 50 minutes ) November 29, 2006 NAME: @L—QT l O A j$ Section: 1 2 Do all work on these sheets. A Table of Integrals and Identities will be handed out seiaarately.
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. Use the deﬁnition sinc(x) = gm“) .
x Total Grades for Exam: nts Score w Grades for P0 \DOONIONUlIBMNi—l j—l
a 11
12
13
14 MQQNJQOOQQMIQQGNKIOO l TOTAL 100 Exam #3, p.1 1(8). Sketch the spectra A(a)),8(a)), C(m), Y(a)) for the communication system shown. 005(31‘) Q19?) 2;. $55”) {oﬁzﬁw Aim}g£i§f$fgg§ as KE§Q£F “2?) ngﬂ ﬁéww¢)gm§ A(w) Exam #3. p2 2(7). Consider the foilowing mixing scheme used in the frontend of superheterodyne receivers. Let’s
assume the receiver is able to tune to the following three AM stations, namely, A,B & C, as shown below. When 5(t) is tuned to one of the three stations (at a) m100,110 or 120 ), the tunable local oscillator is
automatically adjusted to a frequency we , which shifts at] stations to center frequency 50 rad/sec of the
bandpass ﬁlter. The bandpass ﬁlter has a passband of 10 rad/sec. Tune to station to rest of receiver Find the range of tuning of we, i.e., the a) com in the expression com S (00 S a) that must be provided min ’ max 3 by the tunable local oscillator in order for the receiver to play the three given stations. “mean, “it: Slaw (53. ﬁg): 39,81 swsﬁwﬁlwgmk “$1; gﬁfda—N‘f: tat(”3)
M W
tag} 7 3(6). The signal x(t) =103inc(1027 t) is to be sampied at a sampling interval, T3 . Find the range of vaEues of TS so that x(r) is uniquely represented by the éisereEetime sequence, x[n} m x0211). aw; EEEE Sahtéiawﬁ mﬂ‘iJ TY" Exam #3. 13.4 4(7). Film~based movies for Video are produced by recording 30 still frames of a scene every second. Hence the sampling interval of the VideO portion of a movxe IS if; = — . Suppose we make a movze of a movmg 30 . l .
car. Assume the car wheels have a radlus:— meters, and four spokes, 1.e., 4 When the wheels rotate at a rate of a) rad/sec, the car moves across the scene at a linear velocity of v = a) r
meters/sec. What is the slowest forward speed of the car and wheels (in miles/hour) to make the wheels
appear in the movie as if they are not rotating at all or just beginning to rotate backwards? Recall that E meter/se022.24 miles/hour. Exam #3. p5 5(7). The straightline Bode magnitude piot of H (19%” //// j Co{ Find unknowns a,b & K. {We}! fix Exam #3. 13.6 Hm: 6(7). The straightline Bode magnitude plot for a given transfer function is shown in the ﬁgure. Assume that
the system transfer function is minimum phase. (a) Sketch the corresponding phase shift curve, as a nice continuous curve. Indicate key vaiues. Label and
scale plot. ~20 dB/dec Corrections —40 dB/dec 0.1 100 (b) Determine the system transfer function. ._ MW?)
W33“ SCiewﬁgz' a? Hexm
f; swab“ i Exam #3. 9.7 7(7). Use Laplace transforms to solve the given differential equation for ya). 2
:{f+3%+2y=0, y(0')=2, y(0)=o. Exam #3. 13.8 a”, 0 s z .<. 3
8(7). Use grogerties to ﬁnd the Lapiace transfonn of x(r) ={ 0, else Exam #3. p.9 252 +53+12 9(8). Find the inverse transform, x0), of X (s): 1 .
is“ +25+10is~i~2) " 2§2+§$H2m 3 9+!
in??? , 5;; ”I“ M
@14 24S+ 39$»? 2.) §+L @419???“ ﬁ @WT ¢3 M) =. 2319+ e £09139 aw Exam #3. p.10 10(7). Derive the transfer function 3(3) for a second order Butterworéh lowpass ﬁiter with andwidthml
starting from the concept that the poles are distributed uniformly on a circle. Express the denomin tor of
3(5) in descending powers of s. Exam #3. p.11 War?“ 33 + a)
p 0 to transform a Butterworth lowpass 11(7). Use the lowpasstobandpass transformation, — z B
a) s C lter, H LP of order n = 1 and bandwidthxl to a bandpass ﬁlter H 5;) with a center frequencymﬁ and a passbandmi. Express the polynomiais in HEP in descending powers. Recall H LP (3) m ——1—— 3+1.
3 $2.7ng 32’s“; ‘5‘" %‘ss”5 Exam #3. p.12 12(7). Given the feeébaek system, Xg) Y(s) ﬁné K so that the damping factor of the cioseduioop, complex conjugate poles is 4' = m. J5
w Ligt at ._ 2 3+ 2a mg;
dwi‘ an R KSCSmi} & “m S +3) ¢ Exam #3. p.13 13(7). For the feedback system X(s) Y(s) (a) Sketch the root locus. No need to calculate any asymptotes, breakwaway 0r break—in (reentry) points,
unless you need to do so. Just a rough sketch showing What is going on in general will sufﬁce. (b) Find K and the closed 100 poles when the system exhibits a pair of complex conjugate poles that have a
real part equal to ~2. waif: gig/M if: {+ﬂgC3 =5? $3+Cg+gj$+2¢<w~§ 564:0 N 335;" g: “2“?“3 ‘3? clame4“; QWE @w’wa’ gzeeéwéélewzleQ W
; if“; girl apléiél
22¢:4ét‘k} “=7 éw‘g't: a}
Wl epwe‘ Exam #3. p.14 _ aKvbmjm wargaéfwwwj... .. _ .. _ _
“EW‘ +2. +3w+jwzmwam0  zmwa’wwiﬁ $5.2 iwﬁiﬁi 14(8). $0: the feedback system X“) (b) Sketch the root locus. Calculate and indicate on your plot any asymptotes, asymptotes locations
(centroids), asymptote angles and ,breakuaway points (hint. See part (31)), if they occur. C" ‘Sm saplane PﬁE '5' Z“
' AL. 1 “Q” "LsF ’2. AA :«j; {.35 me Wet if?” mffﬂ3' 4+ “3. Exam #3. p.15 ...
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