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Unformatted text preview: KEY EE301 Signals and Systems In—Class Exam
Exam 2 Tuesday, Mar. 29, 2011 Cover Sheet Test Duration: 75 minutes.
Coverage: Chaps. 3,4 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet
Calculators NOT allowed. All work should be done on the sheets provided.
You must show all work for each problem to receive full credit.
Plot your answers on the graphs provided. Multiple Choice Question 1. Circle the best answer below. Let X (to) be the Fourier
Transform of a signal 36(t); as you vary to, a plot of X(w)2 reveals (a) how the amplitude and phase of a sinewave change as it passes thru an LTI system.
(b) how multiplying by a high—frequency cosine makes the signal radiate from an antenna.
) (0 how the energy of a periodic signal is concentrated at continuous frequencies. .how the energy of the signal is distributed as a function of frequency. (e) how two different signals can have the same energy and phase distribution Multiple Choice Question 2. Circle the best answer below.
(a) 33(t) and 90(t — to) have the same energy distribution as a function of time. (b) 90(t — to) can be obtained by passing x(t) thru an LTI system with impulse response
h(t) = 603 — to). (c) The respective Fourier Transforms of .’L‘(t) and $0? — to) differ only by a linear phase
term. (d) x(t) and x(t — to) have the same energy distribution as a function of frequency.
(e) (a), (b), (c), and (d) are all true
(f) nly (b), (c), and (d) are true Multiple Choice Question 3. Circle the best answer below. Forming the product of a
baseband signal with a highfrequency sinewave .a)7 (b), (c)7 and (d) are all true (f only (b), (c), and (d) are true (g only (c), and (d) are true Multiple Choice Question 4. Circle the best answer below. Multiplying by the indepen—
dent variable in one domain (either time or frequency) (a) keeps the energy distribution the same in both domains
(b) always causes instability @auses differentiation with respect to the independent variable in the other domain
((1) changes the frequency band that the signal occupies (e) causes multiplication by the independent variable in the other domain Multiple Choice Question 5. Circle the best answer below. Let Lt(t) be a periodic signal
whose average value over one period is zero. (a) In the frequency domain, the energy is distributed over a continuous band of frequen
cies. ® In the frequency domain, the energy is concentrated at discrete frequencies equal to
the fundamental frequency and all of its harmonics. (c) In the frequency domain, the energy is distributed over both discrete and continuous
frequencies. (d) In the frequency domain, the energy distribution is periodic as a function of frequency. (e) In the frequency domain, the energy is concentrated at w = 0. Multiple Choice Question 6. Circle the best answer below. One of the most impor
tant practical implications of the convolution (in time) property of the Fourier Transform
(convolution in time leads to multiplication in the frequency domain) is (a) that it makes multiplication as easy as convolution (b) that it makes the energy distribution be the same at both the input and output of an
LTI system (c) that it distributes the energy the same way in both the time and frequency domains .frequency selective linear ﬁltering (e) it stabilizes an LTI system by moving the poles to the right—half—plane Problem 2. Short Workout Problems Using Fourier Transform Properties. Problem 2 (a). You are given that the Fourier Transform of a Gaussian pulse $(t) : e 2
is X(w=) V27T8— 2 That is _ 2 x _w2
116(t): e7t‘ <1: X(w) = x/27r eT
. . 3 1
Determine the Fourier Transform of S < >
.. ~ 0 72
y<t>=e275§ = 6 Write your expression for Y(w) in the space directly below. “(06¢ H m\ ><Gf7> \
Tkmg: Q16: "‘ (600)2' You): 6M (0‘09 “”53 0.1. .
26m€ '7: Problem 2 (b). You are given the Fourier Transform pair below 33(75) : cos (7;) rect (g) i X(w )_ W Determine the Fourier Transform of 47rcos(t)
t = M—
:10 W2 _4t2 Write your expression for Y(w) in the space directly below. Simplify as much as possible. Dmali i3 yawn? éfctates 1 Problem 2 (0). Consider an LTI system with impulse response . 2
h(t) : 27r sm(2t)
7rt
Determine the output y(t) for the input a:(t) given by
~1 1 ‘ oo 1 ‘k
$(t)= Z _egkt+e]0t+z_eg t
k=—oo k 19:1 k 1 i 001 .
$(t)= Z Eejkt+1+zgeﬂkt
k=1 kZ—OO Show work and write your expression for y(t )in the space directly below. Htuﬁ 4 “4‘ 4 («0
aka Fregmen?‘ 5‘19on 4 (amé \'V\c\udlha ‘0
)\$ VejeL ieé k5” ‘rhe ‘Fr'l‘reu =5 no“ passe cl =4 \+L43=O
... ‘ ‘5. ‘—5y(1> : I. H\3> ﬂ __
H(o\ —4 H: Bibi —:, an “Ii3V5) .HQ 4)
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Basic. Vesul‘l': ejwo ‘9 \‘BXCW>”_> Hhkhbﬁ
Tlmmg". 2t: : Workout Problem 3 Let H Lp(w) be the Fourier Transform of the impulse response th(t)
deﬁned below. (a) (b) (f) (s) sin(5t) sin(15t)
7Tt 7rt (1) Note that h L p(t) is both realvalued and even—symmetric as a function of time. Thus,
H Lp(w) is both real—valued and symmetric as a function of frequency. Plot H Lp(w) in
the space provided. Show as much detail as possible. hLP(t) = % h(t) is deﬁned in terms of th(t) as:
h(t) = 20 th(t) sin(20t) (2) Note that h(t) is odd—symmetric as a function of time. Thus, H (w) is purely imaginary
for all frequencies. Plot H (w) in the space provided. Note that the vertical axis values
have the multiplicative scalar j = \/—1 factored into them. Consider the input signal 13(75) below. Mt) : {$11000} 7rt Determine and plot the Fourier Transform X1(w) of the signal 931(t) in the space pro
Vided. Determine a closed—form analytical expression for the output y1(t) when the signal
901(25) in part (c) is the input to the LTI system with impulse response h.(t) in part (b)
defined by eqn. (2). Write your answer in the space provided. HINT: look carefully at
the frequency response H (to) over —10 < w < 10 and relate to one of the properties of
the Fourier Transform. Next, consider the input Signal 0005) below. :1:(t) = 2—575 {51117553 )2 cos(20t) Determine and plot the Fourier Transform X (to) of the signal m(t) in the space provided. Denote the output y(t) when the signal in part (e) directly above is the input to the
LTI system with impulse response h(t) in part (b) deﬁned by eqn. (2). Determine and
plot Y(w) in the space provided. Create a complex—valued signal as z(t) = 1093(75) +jy(t) Determine and plot Z (w) in the space provided. Show as much detail as possible. Plot your answer to Problem 3 (a) here. Show work above. \ l ‘ 1 \ \ ‘ ,r _,., \ l ,J.,...MJW_ L '
0 —8 —6 —4 —2 O 2 4 6 8 1O 12 14 16 18 20 _ wwwrﬁvg \ 5,,
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—%0—18—16—14—12—10 —8 —6 —4 —2 O 2 4 6 8 10 12 14 16 18 20 ,_. lvvviJ Show your work and write your answers to Problem 3, parts ((1) below. in”? XCW> QCQ‘AYM‘QS OVQV hoe ’g/Qﬁbwnca‘ loané
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 Spring '06
 V."Ragu"Balakrishnan

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