hw05su09 - E R IAIN TIT TE FTE HN L Y SCHOOL of ELECTRICAL...

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Unformatted text preview: E R IAIN TIT TE FTE HN L Y SCHOOL of ELECTRICAL and COMPUTER ENGINEERING E E2 2 mm r2 Problem Set #5 Assigned: 15—Jun—09 Due Date: 22—Jun—09 Please Check the T—square chat and announcements daily. All official course announcements will be posted there. ALL of the STARRED problems should be turned in for grading. Homework is due at the start of your recitation. After this assignment is handed in by everyone7 a solution to all the starred problems will be posted toihenrreba PR BLEM .1*: Suppose a particular linear time—invariant system is described by the difference equation = 7 2mm 7 1I+ 390M 7 3]. (a) Determine the response of this system to a unit impulse input; i.e., find the output 2 Mn] when the input is : 6 Make a stern plot of Mn] as a function of n. (b) This system is a causal FIR filter. Determine the filter coefficients (0) Suppose the input to this system is 0 forn<1 n+1forn:1,2 4771 forn=3,4 —1 forn25 Compute the values of Mn], over the range 0 g n g 10. Perform the computations by drawing up a “convolution table.” You will probably find it easiest to make a table with four rows that you need to sum. Make stem plots of both and PRQBLEM 52*: The subparts of this problem are unrelated. (a) Suppose an LTI system has an impulse response given by mm] = (1/2)"(u[n + 117 um i 101). Is this system causal? Explain your answer. (b) For a particular LTI system, when the input is the unit step signal: 1 fornZO 0 forn<0 mm] = ={ the corresponding output is y1[n]= 26[n + 1] i 36[n] + 36% i 1] = Determine the output when the input to the LTl system is $2M : —2u[n] + 4u[n — 4]. Give your answer as a formula expressing yg in terms of known sequences, or give a list of values for —00 < n < oo. Hint: Use the ideas of linearity and time-invariance. PROBLEM 5.3: Work Problem P—6.10 from the Signal Processing First textbook. PRQBLEM 5.4*: The diagram in Fig. 1 depicts a cascade connection of two linear time—invariant systems; i.e.7 the output of the first system is the input to the second system, and the overall output is the output of the second system. System #1 System #2 hl hg Figure 1: Cascade connection of two LTI systems. Suppose that System #1 has impulse response, 0 forn<0 h1[n]= n forOSnSfi 0 forn>7 and System #2 is described by the difference equation y n = v n + v n i 1 a Determine the difference equation of System #1; i.e., the equation that relates v[n] to b When the input signal is a unit impulse, 6%], determine the signal vlnl and make a plot. ( ) ( ) (c) Determine hglnl, the impulse response of System #2. (d) Determine the impulse response of the overall cascade system7 i.e., find y[n] when : 6 ) (e From the impulse response of the overall cascade system as obtained in part (01), obtain a single difference equation that relates directly to in Fig. 1. PROBLEM 5.5: Work Problem P—6.11 from the Signal Processing First textbook. PRQBLEM 55*: Suppose a discrete—time LTI system has the frequency response H(€J’o) : Sin eXp [—j _ Find the impulse response hlnl of this system. PRQBLEM 5.7*: Consider a discrete—time LTI system with the frequency response (a) lsin(9d)/2) . A HM“) — exp(*15w) _ 2 sin(d1 /2) Draw a labelled sketch of the magnitude of the frequency response of this system for 771' g (I) g 71' without the help of MATLAB at first. Your sketch need not be a work of art; it just needs to be good enough to convince the grader you understand the important issues. You may check your work using MATLAB, but write enough explanation to convince the grader that you know how to do it by hand. After all, you won’t have MATLAB with you on the exam: Find the output of the system, ylnl, if the input is specified by Suppose we want to implement the system with the H (6”) given above using the following fragment of MATLAB code, where the variable yy contains the output and the variable XX contains the input: hh = ????? xx = conv(hh,XX); Specify the row vector hh, i.e. hh = [something] . (Hint: The explichDl part you see above doesn’t exactly match up with the running—sum filter formula. This means that your hh vector will actually start with some zeros. Think about how a shift in the time domain corresponds to a complex exponential in the frequency domain.) Write the impulse response of the system using a difference of possibly scaled unit step functions i.e. hlnl = (whosit) >< (ulwhysitl i ulwhatsitl) ...
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This note was uploaded on 01/24/2010 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Institute of Technology.

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hw05su09 - E R IAIN TIT TE FTE HN L Y SCHOOL of ELECTRICAL...

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