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A#04_Solutions-1

# A#04_Solutions-1 - Assignment#04 ~Solutions 9.1 33832410...

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Unformatted text preview: Assignment #04 ~Solutions - 9.1 33832410 Signais & Systems - Spring 2093’ Due Tue 01/30/97 W] xin] LTI y[r:] = ﬁn} * Mn} l5 Mm). A LTI system has an impuise response, Mn} 2 {1,1,1,0,...}. (21) Find output ylfn] when the input is x,[n] m {1,1,1,0,...}. Use the array merhod. Express answer, y;[n] , if} sequence form. Sketch yin] vs. 11. \$1313 1 i i ////( "gifhlsgilz’giaﬂ‘g is _ h-w-o 2" 1 (b) Finé output 322M] whenthe inputis xﬂn} = {g,e,1,1,1,0,..§;n Use the army method. Express answer, y2 [52}, in sequence form. Sketch yz [n] vs. If. “\ L e ,‘ Hagggngaj i) 2., 3} a} \$3 (c)(3) The inputs x1 [11} and x2[n] are related by the equation x2 {n} m 2cE {n — 2}. Compare your sketches of the outputs in (a) and (b) above and ﬁnd the relationship between yin] and )22 [n]. M = e ; We 91.0%?!» 32.13/11 103 {\$593 w m r; Vi We wiﬁgﬁﬁy 31513: glmhzg Ogda’hi‘éﬂ m." 2‘2 a... “‘1 31??”1} 1&4 3? 1 Assignment #04 ~Soiuti0ns - 13.2 ECSE—2410 Signals & Systems » Spring 2007 Due Tue OHS {O7 2%) A LTI system has an impulse response, kin} m {2,1,2,0,...}. (a) Let xEEH} = {9,0,},2,1,0,...}. Piot this Signal, mm (b) Find Mn] : ma] :3 hm, when x,{n] m {g,o,1,2,1,0,...} from (21). Use the array marked. P10? yén] for - 2 g 21 S 8‘ Am a \$33“ /L 3 Z. ._ .... b 6 r; giU’ﬂﬂggto) Zgg'ib/bjz‘} (c) Now let xﬂn] = x,[n.+2]x{ ..... for —~2 S n S 8. Plot this signal. LL m '7‘ KKK-\$21 F I ybtél :3. \$3120 ! O O 0 ma: .1 m1” (:7 gbagl -.—. mﬁuﬁo Assignment #64 wSolutions u p.3 ECSE~2410 Signals & Systems — Spring 2007 Due Tue 01/30/07 2(3). Continued. A UN system has an impulse respense, [ﬁn] = {g,1,2,0,._.}. l 21 ((1) Find yin] z x291] * Mn] , with the xﬁn} m found in (c). Use the array method. Express 322M] in sequence form, i.e., yﬁn] = A513 (e) New plot y3[n] :y2[nw2} for ~2 S :2 \$8. Euler. {Ele ﬁst-a1 @2554} 1’5 O 35H) 3. 3112-31: O 1:15:93 5:31.?“le O jﬁljmjzﬁ‘lka 3 -- 332:) «may 21 “2" (15:31“:— ‘jltll “ S :jg‘iinlya jail} a- (o ‘jgﬁSjr-jamlms (fr— _ 33ce3ijtq1u 2,. 3; E7} =— th-gj ‘ O \$ 2 O (i) Campare the plots ef MW} in (b) and yin} in (e) Are they the same? l E 7’95}, ‘l’ag Samuel. 73ajgjLa_al=%iﬁs 3 Can you see the following structure in this problem? Shift original signal x,[n} to the left (advance), by two units, xiii: + 2] , to get rid of the leading zeros. New perform the simpier convolution y? {n} = x} [n + 2} * kin]. To get back to the originel problem re—insert the two leading zeros- by shifting the resulting convolution, 5J2 {rs —« 2} 3 to the right (delay) two units. This shift (i..e, replace it by 11-2) in output signal brings us right back to the eréginal output, thus yzln " 21m Jilin +2 "Zl’k lian = NH}! Assignment #04 - Solutions ~ p.41 ECSE—24IG Signals & Systems ~ Spring 2007 DueTue Oi/30/O7 3(39). Let x[n} = min + 2] — u[n — 4] and 12921;: n uEn]. (a) Use the graphica! marked to determine the value of y[2] , whim y[n} m xEn] =I= Mn}. That is, sketch graphs of x[k] mic, [in —— ic] vs. k , and x[k}- M2 —— Ar} vs}: to detennine y[2]. You only need to soive the problem for n a: 2 . Assignment #04 — Soiutions - 13.5 ECSE—ZiH 9 Siguais & Systems a Spring 200? Due Tue 0180/0? 360). Let ﬁn] m up; + 2] —z«:[n — 4} and Mn] = :1 24:2}. Centinued. (b) Use the: army method to check your answer in (a). NOTE. This methoci does not give closed form soiutions, but can be used :9 ﬁnd a single vaiue, such as in this problem, ifyou are carefw’f The Array method works only fer ﬁnite sequeaces, so let’s truncate x92] and Mn}. You must pick a sufﬁcient number of eiements in 2:34} and him}. Oiherwise you wiil get incorrect resuits! Let 11mm {g,1,2,3,4,0,...} and xgn] m {1,1,13,1,1,0,...}. Assignment #04 —- Solutions w [3.6 ECSE~2410 Signals & Systems - Spring 2007 Due Tue 01/30/07 4(30). Suppose h[n] x {1,1,2,0,...}. (2:) Find output y[n} when xin = {§,2,l,0,...}. Use array method. CH: t w: ﬂ mm my new raw-v" S 1— {t \$313-\$415} aegiszf‘ \$7" W b) Find output yin} when x1[n} = {§,2,i,0,3,2,l,0,...}. Use array method or superposition. 25/5“; :EE 5,? 5,5, 5,1 5,2?(31 (c) Find output y2{n] when x2[n] = {3,2,l,0,0,0,0,0,3,2,1,0,...}. Solve by superposition, i.e., express xzfn] in terms of x[n] and use results from (a), along with the distributive law of convolution. “hwy-IE 3 2.; o 0 0 Q (/3 Ejayfjﬂjeng “Vii/f); “=— XCﬂl +?<£§i-£a] ': 393% QCnA*(2<@J+?‘i¥~33> "'1 Qwemﬁﬁmwma ...
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