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Unformatted text preview: goLoTloN EE538 Digital Signal Processing I Fall 2011
Exam 1 Monday, Oct. 3, 2011 Cover Sheet“ Test Duration: 60 minutes.
Coverage: Chapters 1—5.
Open Book but Closed Notes.
Calculators NOT allowed.
This test contains two problems.
Show your work in the space provided for each problem.
You must show all work for each problem to receive full credit.
Always simplify your answers as much as possible. Prob. No. Topic(s) Points 1. DT Autocorrelation7 Cross—Correlation 60
Correlation in terms of Convolution 2. LTI Systems: Properties7 40 Transfer Functions, Firequency Response Problem 1. [60 points] (a) Consider saw to be a realvalued sequence with autocorrelation rmm. Express the
autocorrelation sequence ryyﬂ] for y[n] = :c[—n] in terms of mm Be sure to Show
clearly how you arrived at your answer. Fay (Qa‘~ va‘neé $0ckcﬂehca: rxY cm: «H? r “5‘93 C(HLE
; 44 C'tho\‘L'\’\r°"'\ C) .
coward”? (b) Determine and plot the autocorrelation sequence for deﬁned below with p : Note; This sequence is used in parts ((3) thru ). [M rm \ K “.930th
1 2 /7 of‘ allpass “*9”
mini : 1) {6M + (p — 1)p"u[n]} <1) (:3) Determine and plot the autocorrelation sequence rum for z[n] deﬁned in terms of in Eqn. (1) as below. 2M = j(%"+%)xw S 7.0% >
Q‘KaM
ProvecL Inn C\"‘? a“; a“ a” We; Plobhi‘hﬂ), (d) Determine and plot the autocorrelation sequence TH [E] for z[n] deﬁned in terms of a;
in Eqn. (1) below.
ab] = asp — n] M h “f to //@\o\§.'w\
'E'MQ  5% (f’} Aoesh 4 meeec‘} Odd OPHUQé a“ ORR exam 9
Fan mum: 1C9?) Page“
(W U3 " V‘ﬁx [IQ—J Auegy:+_ v‘)’ 7
reueé xW Pa. ‘ — u S” QM_)QCD/{Q\pv:r60h3 3L. 3
0\ “ac
V}? C1,): KW \5x3
"VXXCLB
§0wwe (RUBY as wolf); (C) (e) Consider a simple radar example Where there are two echoes such that the received
signal may be expressed in terms of in Eqn. (1) as 1 $[n — 3] + — 5] (2) W] Compute and plot the cross—correlation sequence ry$[€] given the input sequence deﬁned in Eqn. (1) above. \:V0m cl“; 304: \Qavneé
_‘ 5+ [qua Uckwe: A
VYX ti) —‘ Txx ékhv3>
+47; ll (f) Recall the simple radar example described by Eqn Compute and plot the auto
correlation sequence for the output 'ryym given the input sequence 513M deﬁned in
Eqn. Can echo delays be determined from the autocorrelation of the output mum? ("W E 513: rm DD #6 VLRCD L U?) 3+ \AC'JD ifﬂi‘ij *— J (10 mpg: : {5CSZ'Q i”..5 5Ct—S}}¥f§[£—3>ﬁSJE&*Qj ‘ ,eri) %,SJC&+§8
1 {J [2’3] he 5(953393452119 ’Q_(_S3 ‘9“%’}"L\0’L;4§ >9, CVMW‘AOV} Aeﬁ'fme'ﬂe Awe \Ayﬁ from Problem 2. [40 points] Consider the causal DT LTI system described the difference
equation below which is used for all four parts of this problem. ylnl = j yln 1l+ xlnl  mln  4] (a) Determine the transfer function H for this system and plot the polezero diagram.
Show the region of convergence. Has“ (b) Determine the impulse response of the system Mn]. You can either list the value of
Mn] for each n or write a closed—form expression for Mn] that works for all 7;. Santa H (23 : 255 <2 ~>> (E ‘V D Z‘+:)> ::::ell::acn Ag“ Ti) Clms) Mn): {55)‘5—5]
’] {MOM (0) Determine a. closedform expression for the frequency response of the system H
equal to the DTFT of Mn]. Plot the magnitude \H over —7r < w < 1r. You
can provide a. rough sketch, but you need to Clearly indicate the frequencies where
\H : O and the frequency where IH reaches its peak value.  4 w x e *J H (uﬂ= H (74> ~ 3 ((1) Determine a closed—form expression for the output 3/[71] when the input is the following
sum, of sinewaves turned—0n forever: xinJ:1+ 2H)" + 3(1)” + 4(4)” ZN?) <93)“ 4— 3 “(47:3 (53“ #03493“ V O 4 0 i6) Ch): H0931 + W
O = 51(3)“ 11 ...
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