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A#04_Solutions

# A#04_Solutions - Assignment#4 ~— Solutions ECSE—2410...

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Unformatted text preview: Assignment #4 ~— Solutions ECSE—2410 Signals & Systems ~ Fall 2006 Due Tue 09/12/06 1(20). Carry out the convofution, y(t) = v(t) =£= w(r) , of the signals shown, and sketch your result. (a) v1(t) yv1(t) h * II I , Z I e 2 e 2 121(1') For t<0, 3110) = v10) * w1(t) m 0 1 h 1' For O<z<2, . G 2 w,(z~z') i T Qﬂ : (5) mm) / at at —[email protected] w m C :: (fyéé -zx-t 't i 3 .51” 2‘4 ai‘t LE 7? if \$4 335‘; «a: ﬁfﬁgaE-Mﬁ} as} km; 2.. mag-~24) 1569 == are) 'iﬁw— 326-? = Ugéﬁ WW .._.. “#544 wait-2; = g: m; Assignment #4 ..,_ Solutions —- p.2 BOSE-2410 Signals & Systems - Fali 2006 Due Tue 09/12f06 Assignment #4 w Solutions — 9.3 ECSE—241 0 Signals & Systems — Fall 2006 Due Tue 09/ 121’06 1(a) continued. we, mealsgjﬂa P a» W dﬂwgwﬁw fweaﬁar, Nae MWﬁifi‘tﬁvi-ﬁjé—é~zﬁg e @9va “ﬁes. Q‘rﬁl 1: Wgﬁﬂmﬁfiﬁgm ogg+)%<vjija}-tlﬁ&-ZD == 078-1sz +mﬁfviﬁt"2§ Essa? ems-um aegis} 1E3}; % Ceweﬁfmgi} a L ﬁLiﬁm giiﬁ "y gsiﬁ‘a “E 3% 2(25). This example illustrates the distributive property of convolution, namely, that convolution distributes over addition: x(r) =I= (1110) + kg (0) m x(r) * 2210) + x(1)* [12 (t). For the system shown, let h, (t) = l12 (t) 2 11(5) w u(t 4*» I) . x(r)=u(t) Determine and sketch Z Qi-iw'g‘z” “” WM”) New ; 1%) QELH+~¢Q2L9 a; ‘D + E g ? ‘9 i t c) E t <23 3 ~17 (b) x(f)*(hl(t)+h2(r)) like ﬂa‘i‘iz’ LE: 35 g E if 13%;?) Assignment #4 — Solutions —- p.4 ECSE—2410 Signals & Systems .. Fail 2006 Due Tue 09/12/06 2 Continued (C) X(I)*hl(f) = 'ézﬁ'} (d) x(:) we) .2:ng 321*) 3%; M. (6)1 i f C) E (e) m * h} (r) + x0) * hz (r) —.-.~. 3*“ ﬂag.) féiL-H Mew...» "t Assignment #4 — Solutions ~— 13.5 ECSE—24EO Signais & Systems - Fail 2006 Due Tue 09/ 12/06 3(20). This example iliustrates the associative property of convolution, namely, that the order in which cascaded terms are convolved is irrelevant: x(r) * (h, (I) * kg (0) m (x(t) * h1(t))* #12 (I). For the system shown, let h, (I) = h, (t) = MU) — u(r — 1) . (a) hmmhzm slew Age} , % Assignment #4 —- Solutions — 9.6 ECSE-2410 Signais & Systems — Fall 2006 Due Tue 09/ 12/06 3(20).C0nt1nued (C) 960)“? (I) -~ 3%) ii My?) % as is ﬁg )m ) (d) (mm (x) *h (r) Assignment #4 — Solutions — [1.7 ECSE-2410 Signais & Systems — F311 2006 Due Tue 09/ 12/06 4120) w) (20(8) Sketch y(r) = x(t) =5< Mr) {A XLﬁL Hiya —£ 4:4; (b)(8) Sketch 34(1) = x(~r+2)*h(t)‘ Assignment #4 w Solutions «w p.8 ECSE-2410 Signals & Systems — Fall 2006 Due Tue 09/12/06 4.0.0) Continued. £21135 mike; +2.12» Pwéw Mmé We dﬁkﬁz'ékﬁ‘aa Fwy? “ i KG} (Maia gag; Macias) 1.011% a. i 9&9 i' i: 1 {— a 3 “j «:32 i a... 6%; mmaﬁm/ﬁgm 29 W 3%}: {aﬂwvﬂiﬂyﬁ‘ 44545 ”‘1‘ Z£Q¥%w*/£&4}Mw m) 1‘11 M £62? kamlww ‘gg: w 0 t 3- Assignment #4 — Solutions — {1.9 ECSE-2410 Signals & Systems - Fall 2006 Due Tue 09/12/06 5(15). The input 9(0) 2 e‘m is appiied to a LTI system whose impulse response is Mt) : um —u(t — E) . (a)( 10) COmputo analyticaiiy (1.6. give a formuia for) ya) : 2cm * Mt) . Use graphicai approach. 1'. g {4) s .4: e is: e ads-i-oué—é-ﬁ «4 N19 "‘1'" a «8:- M?) Assignment #4 — Soiutions — p.10 ECSE~2410 Signals & Systems - Fail 2006 Due Tue 09/ 12/06 (3) Continued. Matlab plot of 320*). 03 10.5%,} = 6.78% - s ‘ ' i a 1 : a 5/ \ tm[u4:.01:4}; g1 mt<m0; g2:t>0&t<1 ; g3ﬁt>xl ; WW-6Xp(-1))-*eXP(t))-*g1+(2~€Xp(’£*1)~eXp(*i))-*g2+((eXp(1)4)-*eXp(~i))-*g3; p10t(t,y) grid (b)(5) At what value of t does y(r) reach its maximum value? Justify your answer. (Picture will ...
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