MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.
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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed book. No otes or calculators permitted. The exam will be graded o the basis of the aswers oly. Please do t put ay explaatios or work i the aswer booklet. Five pages of scratch paper are attached at the ed of the exam. These are ot to be haded i. Let us kow if you eed additioal scratch paper.
2 Problem [8%] Cosider the discrete-time system described by the followig equatio: 2 y[] = x[ ] 2%(a) Is this system liear? 2% (b) Is this system time-ivariat? 2% (c) Is this system causal? 2% (d) Is this system stable? Problem 2 [4%] Cosider a LTI discrete time system for which the iput ad output satisfy the followig differece equatio: 2% (a) Is this system causal? 2% (b) Is this system stable? y[]+ y[ ] = x[] 2 Problem 3 [4%] Cosider the discrete time LTI system described by the followig frequecy respose: 2e ) jω H(e jω )=( 2 e jω ( 3e jω ) 2% (a) Is this system causal? 2% (b) Is this system stable? Problem 4 [6%] Determie the trasfer fuctio H xy (z) from x[] to y[] ad the trasfer fuctio H ey (z) from e[] to y[] for the followig system:
3 e[] 3? x[] + - -y[] z Z Z 3 Z Z ZZ Figure 4-: Problem 5 [5%] Determie the frequecy respose H (e jω ) of the stable LTI system whose iput ad output satisfy the differece equatio: y[] y[ ] = x[]+ x[ ] 3 2 Problem 6 [9%] We geerate a discrete time radom process x[] by drawig a sequece of i.i.d. umbers from a radom umber geerator with uiform distributio i [, ]. x[] is the processed through a LTI system with impulse respose h[] = δ[]+ δ[ ] to obtai y[], i.e. y[] = x[]+ x[ ]. 3% (a) What is the mea m y [] of the process? 3% (b) What is the autocorrelatio φ yy [m] of the process? 3% (c) What is the power spectral desity P yy (e jω ) of the process? Problem 7 [9%] We process a zero-mea, uit variace, white, discrete-time process x[] through the stable LTI system with trasfer fuctio to get a output y []. z H (z) = 2 z 4
4 4 3%(a) Fid a stable (possibly o-causal) system H 2 (z) such that if we pass y [] through H 2 (z), the output is white. 3% (b) Is your aswer to (a) uique withi a scalar factor? 3% (c) Assume we pass y [] through H 2 (z) to obtai y 2 []. Are x[] ad y 2 [] ucorrelated? (i.e. is φ xy2 [m] equal to?) Problem 8 [8%] I the system show below, two fuctios of time, x (t) ad x 2 (t), are multiplied together, ad the product w(t) is sampled by a periodic impulse trai. x (t) is badlimited to Ω, ad x 2 (t) is badlimited to Ω 2 ; that is, X (jω) =, Ω Ω X 2 (jω) =, Ω Ω 2 Determie the maximum samplig iterval T such that w(t) is recoverable from w p (t) through the use of a ideal lowpass filter. x (t) p(t) = = δ(t T )?? w(t) - - w p (t) 6 x 2 (t) X (jω) X 2 (jω) %e ' % e % e % e % e Ω Ω Ω Ω 2 Ω 2 $ Ω Figure 8-:
5 5 Problem 9 [9%] Assume that the cotiuous-time sigal x c (t) i the figure 9- has fiite eergy ad is exactly badlimited so that, X c (jω) = for Ω 2π 4 The cotiuous-time sigal x c (t) is sampled as idicated i the figure below to obtai the sequece x[], which we wat to process to estimate the total area A uder x c (t) as precisely as possible. Specifically, we defie A = x c (t)dt The discrete-time sigal calculates  = T x[] = 5%(a) Fid the largest possible value of T which will result i as accurate a estimate as possible of A for ay x c (t) cosistet with the stated assumptios. 4%(b) Uder the stated assumptios will your estimate of A be exact or approximate? - Discrete-time C/D - - x x[] system c (t) A = estimate of A 6 T Figure 9-: Problem [8%] I the system below H (e jω ) is a discrete time all-pass filter with a costat group delay of 3.7 ad cotiuous phase, i.e., H(e jω )= e jω(3.7), ω <π. x c (t) - x d [] C/D - H(e jω y d [] ) - D/C - y c (t) 6 6 T T 2
6 6 4% (a) Assumig o aliasig, describe i words (oe setece) the output of the overall system, y c (t), i terms of x c (t), x d [], T ad T 2. 4% (b) Assumig o aliasig, describe i words (oe setece) the output of the discrete time system y d [] i terms of x c (t), x d [], T ad T 2. Problem [4%] For the system show below, if the discrete-time system H(e jω )islti,is the overall system H c (jω) LTI? H c (jω) x c (t) - x d [] C/D - H(e jω y d [] ) - D/C - y c (t) 6 6 T T Problem 2 [4%] Determie the output y[] for a system with the followig iput sequece x[] ad impulse respose h[]. 2 2 x x[] x x x x Problem 3 [8%] For the pole-zero plot give below aswer the followig questios: h[] 4%(a) If the ROC is z > 2: 2%(i) Is the system stable? 2%(ii) Is the system causal? 4%(b) If the ROC z < 2: 2%(i) Is the system stable? 2%(ii) Is the system causal?
7 H(z) 7 Imagiary Part Real Part Figure 3-: Pole-Zero Diagram Problem 4 [6%] Cosider a cotiuous-time LTI system for which the magitude of the frequecy respose is ad the phase is as show below. Determie the respose of that system to the iput x(t) = s(t) cos(ω c t), where S(jΩ) = for Ω Ωc. 2 H(e jω ) φ ω φ Problem 5 [8%] The first plot i Figure 5- shows a sigal x[] that is the sum of three arrow-bad pulses which do ot overlap i time. Its trasform magitude X (e jω ) is show i the secod plot. The group delay ad frequecy respose magitude fuctios of Filter A, a discrete-time LTI system, are show i the third ad fourth plots, respectively. The remaiig plots i Figures 5-2 ad 5-3 show 8 possible output sigals, y i [] i =, 2,...8 Determie which of the possible output sigals is the output of Filter A whe the iput is x[]. Problem 6 What is your best estimate of your grade o this exam?
8 Iput Sigal x[] 8.5 x[] Fourier Trasform Magitude of Iput x[] 5 Magitude(dB) Group Delay (Samples) X(e^jw) Frequecy Normalized by pi 25 Group Delay of filter A Frequecy Normalized by pi Frequecy Respose Magitude of filter A Frequecy Normalized by pi Figure 6-: Iput ad Filter A iformatio
9 .5 Possible Output y[] 9 y[] Possible Output y2[].5 y4[] y3[] y2[] Possible Output y3[] Possible Output y4[] Figure 6-2: Outputs y y 4
10 .5 Possible Output y5[] y5[] Possible Output y6[].5 y7[] y6[] Possible Output y7[] Possible Output y8[].5 y8[] Figure 6-3: Outputs y 5 y 8
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