2011_exam3 - UNIVERSITY OF TEXAS AT AUSTIN BIOMEDICAL...

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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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This note was uploaded on 12/14/2011 for the course BME 343 taught by Professor Emelianov during the Spring '09 term at University of Texas.

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2011_exam3 - UNIVERSITY OF TEXAS AT AUSTIN BIOMEDICAL...

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