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Unformatted text preview: Midterm exam #3 Date: December 2, 2008
“ME Systems and Signals
r’rof. Stanislav Emelianov Grade: :' Code of Honor Statement
i have neither given nor received aid on this examination.
Student Name (Last, First) Student Signature Rules:
1. This is an open book examination.
2. Show your work when appropriate (no credit will be given for irrelevant information and no
credit will be given for the correct answer alone without relevant work.
3. You cannot answer BONUS questions unless you have answered all other questions. Instructions:
1. Write your name (last, ﬁrst) on the front page of the exam.
2. Read and sign Code of Honor Statement (your exam is not valid otherwise).
3. Do not include any additional pages — your work and answers should fit into space provided.
You could use the other side of the page or your own scratch paper if necessary.
Make sure that we can tell what is your final answer (located in the designated space).
You can solve every problem given below — did you bring your intuition with you?
Note that problems have different weighting — plan your work accordingly.
There should be no questions regarding any problems. If you have to make assumptions —
state them clearly and proceed (make sure your assumptions are valid and necessary,
however). if you have any other question — raise your hand or come to a proctor please.
8. Please keep an eye on the blackboard once in a while — there may be something useful written
there as we go through the exam.
9. Keep the pages stapled. Good luck — you will do well! N93971:“ PROBLEM 2 (6 points)
Use the Fourier transform analysis (direct integration) to calculate the Fourier transform of x0) = e’z"“‘)u(t — 1) PROBLEM 3 (17 points)
Suppose you are given the following information about a signal x(t):
1) x(t) is a real signal
2) x(t) is periodic with period To=6 and has Fourier coefficients 13,1 3) Dn=0 for n=0andn23
4) 350) : x(tﬂ— 3)
1 l 3jxm2 dr ~e
6 ,3 " 2 6) 01 is a positive real number 5 v Determine the signal x(t). PROBLEM 4 (17 points)
Suppose g0) = W) 0050)
and the Fourier transform of the g(t) is G(w):{1 cos2 0 otherwise
Determine x(t) using convolution progeny of the Fourier transform. PROBLEM 5 (10 points)
Consider an LTI system S with impuEse response ' 4 t—1
ha) 2 sm( ( ))
EU —1)
determine the output of S for the following input
__sh1(4(t%1)) x“) xa+n PROBLEM 6 (6 points)
The band—limited signals X1(t) and xga‘) X1 (w) = 0, ml 2 691 X290) = 0, lw 2 mg are multiplied together WU = er‘xzﬂ)
and the product y(t) is sampled by a periodic impulse train. Determine the maximum sampling interval Tsuch that y(t) is recoverable from sampled signal yp(t) through the use of
ideal Iowpass filter. PROBLEM 7 (3*7=21 points)
A signal x(t) with Fourier transform X(a)) undergoes impulsetrain sampling to generate xp (0 z 2 x(nT)5(t — Hi") there
T =1~10‘4
For each of the following sets of constrains on x(t) and/or X(w), does the sampling theorem guarantee that x(t) can be
recovered exactly from Xp(t)? (a) X(a)) = 0 for lwl > 5,00071 (b) X(a))=0 for m>15,0007r (c) Rc[X(aJ)]:0 for a2 >5,0007r (d) m) is reai and X(a))=0 for a)>5,00071 (e) m) is real and X((o)=0 for “45,0007: 1/ (f) X(a)) * X(a)) = 0 for lml (g) X(a))‘ = 0 for a) > 5,00071' PROBLEM 8 (3*5+2=17 points)
The Nyquist sampling periods for 1D band—limited signals f(t) and g(r) are Tf and T9 , respectively.
Find the Nyquist sampling periods for the following signals x'(t) (i.e., state Tx explicitly if possible): (51) x6) : for—t0) ,where to is a given constant TX: to) x(r)=f(r)*f(r) TX: 10 PROBLEM 9 (7 points)
Given the following MATLAB code, answer the following questions: t= —1D:0.001:10; y = 5*pi/pi*sin(5*pi*t)./(5*pi*t);
y(10001) m 1; Y t fft(fftshift(y)); Yn = fftshift(Y); a y = trapzlabs(y).A2)*O.OOl; :igureil)
pl0t(trY)
figurel2)
plot(real(Y))
figure(3)
plotlreallYnll Which variable represents the Fourier Transform of a sine? Plot by hand figures 2 and 3 (label your x and y axis with numbers). Write code for calculation of energy in the frequency domain (one line of code is all that‘s needed). What is the relationship between E__y and E_Y? 11 PROBLEM B1 (6 points) _
Consider three continuous—time systems 81, S; and 83 whose responses to a complex exponential input eJ5t are given
below. For each system determine whether given information is sufficient to conclude that the system is definitely n_ot LTl. V '5 '5
r126} ’ —>re’ ’ S2 :ef5r —> emf—l) S3 :eJSI —> 005(51) PROBLEM 32 (5 points)
We know that the foltowing is true F"{F[x(t)]} = W) 
However, if we apply fonivard Fourier transform twice (instead of forward FT and then inverse FT): ftﬂxﬁﬂ} e W) ,
what would be the relationship between the X(t) and y(t)? 12 ...
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 Fall '09
 Emelianov
 Nyquist sampling periods

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