This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 _
hwo 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
ﬁsta1 @2554} 1’5 O
35H) 3. 311231: O
1:15:93 5:31.?“le O
jﬁljmjzﬁ‘lka 3 
332:) «may 21 “2"
(15:31“:— ‘jltll “ S :jg‘iinlya jail} a (o
‘jgﬁSjrjamlms (fr— _
33ce3ijtq1u 2,.
3; E7} =— thgj ‘ 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 112) 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
rawv"
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. “hwyIE 3 2.; o 0 0 Q (/3 Ejayfjﬂjeng “Vii/f); “=— XCﬂl +?<£§i£a] ': 393% QCnA*(2<@J+?‘i¥~33> "'1 Qwemﬁﬁmwma ...
View
Full Document
 Spring '07
 WOZNY
 Digital Signal Processing, LTI system theory, Impulse response, array method

Click to edit the document details