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Unformatted text preview: UNIVERSITY OF TEXAS AT AUSTIN
BIOMEDICAL ENGINEERING DEPARTMENT BME 343, BME Signal and Systems Analysis Exam 3
December, 2011 Directions There are 7 problems worth a total of 53 points. The number of points for each question is
indicated. Make sure that you show all work since partial credit will be given. This exam is closed book,
closed notes, and no calculators are allowed. You may not communicate in any way with anyone other
than exam proctors during the exam time. Don’t forget to put your name down on the ﬁrst page and
any additional pages that you use. ' Name: Points possible Score Problem 1
Problem 2 5
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9 Total 53 l...— i Useful Equations
:36 = cost?J ij sin6 X(w) : [00 m(t)e’jmdt #00 1 0° 
Mt) : X(w)e5'°”dw
—oo l. (5 points) What is the Fourier transform of :2 cos(wct + (9)? 2. (5 points) Fill in the blank: the frequency spectrum of a sampied signal contains
frequency content than the frequency spectrum of the original Signal. Brieﬂy explain your answer (no
credit Without an explanation).
(a) more
(13) less ((1) the same 3. (8 points) Suppose that g(t) m 5505) cost and the Fourier transform of 905} is GM 3 {1, M s 4 0, otherwise What is the expression for ﬁt)? 4. (5 points) Two band limited signals, 271(15) and 562“) have Fourier transforms of X1 (w) and X2(w), which
have the following properties7 0: lwl 2 WI
X2(w) = 0,. lwl 2 tag These two signals are multiplied together to produce y(t) = m1(t)m2(t), and ﬁt) is sampled by a periodic
impulse train. Determine the maximum sampling interval, T, such that ﬁt) can be reconstructed from
the sampled signal using an ideal lowpass ﬁlter. 5. Two signals are deﬁned as f1(t) = gsincﬂt)
f2(t) = gsincﬁ) 005(2t} Let ﬁt) represent the COnvolution of those two signals f0?) = f1(t) * .1503 (a) (5 points) Find the Fourier transform .ﬂf = F(w) (b) (5 points) Find the expression for f Simplify your answer as much as possible. (Hint: sketch
FM) ﬁrst) 6. (10 points) When the input signal given by
51305} = 1 + cost + cos 223 + 4(cos 313 + cos 4t) is applied to a linear time invariant system, the output, Mt}, contains the following frequency spectrum
(ie, exponential Fourier series coefﬁcients) Plot the frequency response H (jnwo) of this system (plot the magnitude and phase separately). Please
label values on both axes of both plots. 7. (10 points) Consider the systems shOwn below. The output of system 1, which has a transfer function
H1 (w), is sampled with a. sampling interval, T = 2/3 (the MM indicates the sampling process). The
sampled signal is sent through system 2, which has a transfer function H2 Draw a labeled graph of
the frequency spectrum of the output of system 2, Y(w). seas—swam c0s(101rt) ——>—® cos(87rt) 5T (:5)
H1041) (w)
i _l:l:
“871' 81'? w —971' 977 w 8. Extra credit: What year did Joseph Fourier publish the Fourier series and Fourier integral? (1 point) 9. Extra credit: Where in the BME lobby is the Fourier integral inscribed on the ﬂoor? (1 point) ...
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 Spring '09
 Emelianov

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