2007Exam3

# 2007Exam3 - 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, Fall 2007 Exam 3 December 13, 2007 Directions There are 6 problems worth a total of 57 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 or other electronic devices are allowed. Don’t forget to put your name down on the ﬁrst page and any additional pages that you use. Name: Points possible Score ——_ —-_— —._- -—— ——— ——— 10 10 15 10 57 1. Determine the Fourier transform of each of the functions shown below. For each function, your answer should be a single expression. You do not need to include a. graph of your answers. (a) (5 points) '2 '1 O 1 2 sin(t} o (b) (5 points) n2 ‘ 2. (6 points) A signal 93(t) = 4sinc2(10wt) + 2cos(407rt) is to be sampled and recorded in a computeri Determine the minimum sampling frequency, f3 in Hz, that must be used and plot the frequency spectrum of the signal sampled at f3. 3. (6 points) A particular system has the step response, 303), shown below (recall that the step response is the output due to a step input, u(t)). Determine the frequency response of this system. Give both the equation for the frequency response and sketch a graph of the frequency response. Label appropriate points on both axes of the graph. stt) ' 4. (10 points) When the input signal given by 33(t) 2 1 + cost + cos 273 + 4(cos 3t + cos 4t) is applied to a linear time invariant system, the output, y(t), contains the following frequency spectrum (ie, 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. 5. In the system depicted below, 31(2?) = ““4” p(t) = cos 2st, q(t) = M. art ’ Trt t p(t) am In addition, the response of the system H (w) is given by the following: H(a}) 1 -211 211- m (a) (5 points) Let A(w) be the the Fourier transform of the signal a(t). Sketch and clearly label A(w). (b) {5 points) Let B (to) be the the Fourier transform of the signal b(t). Sketch and clearly label B (w) (c) (5 points) Let Y(w) be the the Fourier transform of the signal y(t). Sketch and clearly label Y0»). 6. Two signals are deﬁned as 2 f1“) = Esinc(2t) Mt) = gsincﬁt) (105(215) Let ﬁt) represent the convolution of those two signals f (t) = f1 (15) * fab?) (a) (5 points) Find the Fourier transform .7: [ f (15)] = F(w) (b) (5 points) Find the expression for f (t). Simplify your answer as much as possible. (Hint: sketch F(w) ﬁrst) ...
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