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Unformatted text preview: Problem #1 (25 Points): All parts below are independent of each other. Ca) For 5, discrete—time system given below determine if it is, n21 = {0 n=0 :I:[n+1] ng—l (i)nonlinear (4 Points) 4 J ax "7/ I a‘ ‘/
Homojcmrij *3 0" XB‘] % v0 ' 0 0:0  a thnHJ "‘ 4" 0:0 7: ,
AWN}; a XanJJrYzCn] => 393:! X’ngzw A? :: (71%:ij /
MOW) ‘rXLCWJ n é" ' ) v I
. (ii) time—invariant/4 Points) Oyvfoujb WSHK. Ar a . «I )1
, A XCnno] 10/4 Xﬁm] A o
XCM‘J ‘9 Um 2 ,0 mo ma 3 0 “40:0 XOHIoH] ’14” MM] 110091 (iii) causal/n.@ 2 Points)
. chowb m XWJ “I” W"): “"3 (iv/unstable (2 Points) ﬂ
. if [MCnJK WM 2% ‘OMJéMX 2 b) A DT LTI system has an impulse response given by; h[n] é 2(a1)" ‘ u[n + a2] — 4(a3)" o u[——n + a4] + 6(a5)"  + 2] — u[n — 3]} i) Determine restrictions on a1, a2, a3, a4, and (15 so that the system is BIBO stable. (7 Points) 0
‘lﬁl‘mo ' Conﬁcler each l9?“ 231/02 .0“, ’l'lall 02%“?
0 am ‘ I
00 k ’ u a KallkL‘Mj lagl>
%\©3).u%]‘ v H09] : qﬂzw
“aw—m; at J an n; aw ii) Determine restrictions on (11, a2, a3, a4, and (15 so that the system is causal. (6 Points) k0,];0 76w n40
:3 ago 03:0 05:0 Problem #2 (28 Points): All parts below are independent of each other. . a) A DT LTI system whose exact form is unknown is tested by running some inputs into the system and then observing the output Signals. Suppose that the following input—output pairs
are the result of the tests: ﬁn] .—. 6[n] — 6[n — 1] ———> Mn] = —26[n] + 6[n —— 21+ 26[n — 3] '
W [1: (3) —+ [1:2 <3 1)
Kira/)2)”, COS 2n yn COS 2n 4 Find the output of the, system When the input is changed to: { = 26[n] — 55[n — 1] + 35[n — 2] + 58in(gn — (8 Points) 28 00.— SXCn43+356n113 : c—e 2a,an Jam] ,.  T
@4121» 952C05[%WZE> 7‘5 HEW/z)  if
mew—aw aztinewﬂk . .
30)]: 62' [:25an +801] +2 M633 ~3 {2 50m] +5932] 793011,]; 'tlofwgnﬁ—ZE b) A linear time invariant FIR system has impulse response h[n] = 35[n] — 25[n —— 1] + 45[n 9— 2] + 5[n — 4] Draw the implementation of this system as a block diagram in direct form. (5 Points) aémjstwjz—ZXM'U + afﬁrm] + X0412) . c) Suppose that when $1M [ 1% —1 1 is the input to an LTI system, the output (zero state response) is 91M = [ i 2 I . Find the system’s output (zero—state response) when the input is changeti to m2[n] = $1 * x1 I (7KPOints)
V2 (0] :: XzC/l326Mnj : X101] 2% 0x] *lacnj :. x. 04) 39(7; 0*] _ A d7 1
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Mam/ware o 2 13:9 Macz
mgj}jl£3’ﬂ:[0 ‘9 6 2 I] :) ylfajza L ingrﬂ‘ﬁ] l v1.3_ d) Assume that a DT causal LTI system is given by its impulse response h[n] = — 2].
Find the difference equation of this system. (8 Points) ﬂ ' N ’ we 7 rjwla N i t rfu/L
Misﬁt) MC“) :5) H(er):ZMk]ve =LCJ¢
‘ two :21 r W ," x P\ P no
em: t: («w E: (ewﬁ Law)
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2
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C c . 93 _wt :3 a“ z: 2.}.
he! AM an? ! .Problem #3 (15 Points): L Saga 0174. ECHO Fn (4+
. We want to design a digital oscillator to generate the sequence _ rag/U6 y[n] = Acos(27rn/N + 0)  from the algorithm y[n] = a1y[n — 1] + a2y[n — 2] Hints: cos(wn + b) = $(ejwnhejb + e‘jwne‘jb) cos(a + b) = %{cos(a) cos(b) — sin(a) sin(b)} 3) Find the necessary values of a1 and a2. (8 Points) \ r . n
. n ‘0. P _, i . n ’1? —)_Z_Z + e N 1 .
(x)[n]:ar3@’0+az(7€"'1] e710 5 (\IMOJquAaoIZ 2 O
i ‘23 ~32
C'i/‘afaC’i’G/ftsi’m (2T4: Ch (\i"az: (pi—oil) (A..AZ>:( —g N){(\Je N) .271 . A '1": EDNA—02:20 b) Demmine the initial Conditions 341] and y[2] to make A = 2 and 0 = g (7 Points)
3 2&5 (ZTHL/N 4— EL).uCn]
: 0h'fjf"”]+ Q’z‘ldCHEZJ 0 ~
30] T "O I
.'; :Zmb—Lw@— 0
8273’] : $380: a2 :(iT) g?’ [or M [OJE) 9n
30] = ChﬁKoﬂj +612 5 GI] r— 2‘ C95 .05. +55) 0 s M 7' o a": ’
We =«> —" I * 5.2/65 7:1 5"" 27"
C maﬁmlgﬁ—zjzo Q 591]— ﬁjm (U (7??) f ’aGQJ:Z&S(Z,Uﬂ)$fn(%) Problem #4 (10 Points): . Consider adiscrete—time LTI system given by its impulse response If the input signal,
a:[n], to this system, and the system impulse response, h[n], are given as shown below, use the
analytical convolution to ﬁnd zero—state response, y[n], of this system. x[n]e(l§1—3)".u[n+4] h[n]=(g)nu[—n+2]
po _ k . 3 “'k __ ‘ 7 an]: é(%}'8£ff@‘(fl'“[ew‘“ W]
' 10/4 ‘0” Problem #5 (22 Points):
. A DT system for ﬁltering CT signals is shown in the following ﬁgure: ’3, x (t) Ideal Xlnl DT Vlnl Ideal (t)
c Sampling LTI System Reconstruction y°
Fs G=T
Fc=Fs/2 a) If the DT LTI system is described by the difference equation = x[n]—4a:[n—2]+:c[n——4]
Find the frequency response of this system, H (em). (5 Points) h . "3=SCnJ—48c«.~n +gcn—Lg
‘ K — Do A
We“) = l L1" 5 + 5 Maw): 2mm)“ l4: ~99 b) Plot the magnitude and phase responses of H (6”) for —37r S w 3 31r. Keep the plot of the
phase response in the range of 7r and 7r for the phase values. (10 Points) Hid”): pjlw(edZW—£+ +2.2“) : gzw(1Co;(2w)’L+) lHle5“)l=12€os(2w>v4l [we =—1w’FTF
* WW)!
‘2 , 4 l i ‘ Cf . !? 'c)‘{1,£;a CT input signal
. r Mt) = 8 Cos(2507rt) + 6 sin(5007rt + g) is applied to this system at the sampling frequency of F, = lOOOHz, then determine the CT
output signal yc(t). (7 Points) ) ‘ ,\ 600W 1
X9]: 2 amp—50'" n>+ 65m( m Ma 1006 ...
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This note was uploaded on 10/06/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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