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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 timeinvariance. 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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 Spring '08
 JUANG
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response

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