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# hw3soln - BM”,7 HQ 413 problem clear X =[1 l 1 h = X y =...

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Unformatted text preview: BM”; ,7]! HQ 413 % problem ] clear; X = [1, l, 1]; h = X; y = conv(h, X); i 2 [0:4]; subplot(2,2,l),stem(i,y), xlabel('n'), ylabel('Y(n)'), title('2a. output y(n)'); l]; h = [0,1,2]; X); i = [0:4]; 2),stem(i,y), Xlabel('n'), ylabel('y(n)'), title('2b. output y(n)'); n = [0:5]; X = [1, 1, 1]; h : (0.5).An; y : conv(h, X); i = [0:7]; subplot(2,2,3),stem(i,y), xlabel('n'), ylabel(’y(n)'), title('2c. output y(n)'); n = [—5:5]; X = [1, l, l]; h = (O.5).“abs(n); y = conv(h, x); i [—5:7]; sub lot(2,2,4),stem(i,y), xlabel('n'), ylabel('y(n)'), title('2d. output y(n)‘); ll 1 “output y(n) 25 1.5 3/ y(n) 05 (1. output y(n) 15 y(n) 05 5K) 2.21. (a) The desired convolution is H ii i -x- E. 2 Mn] II it 3:. E: 2 I E: kz—oo = ﬁniM/mk for n 2 0 [6:0 ~ WMJ:5HMWHMa¢ﬂ ®)nmnwx (c) For n g 6, k=0 [6:0 For n > 6, oo 1 k n~1 1 k Mn] = 4" 2(3) - 2(3) Ic=:0 k:0 Therefore, _ (8/9)(—1/8)44", n36 “M‘{(wmeumm n>6 (d) The desired convolution is 00 mm = ijwwm—M k=—oo = a:[0]h[n] + :L‘[1]h[n — 1] + x[2]h[n — 2] + :1:[3]h[n —- 3] + I[4]h[n - 4] = h[n] +h[n—- 1] +h[n ~2]+h[n — 3] +h[n-—4]. This is as shown in Figure 82.21. Figure 82.21 3'\ 2.26. (a) We have ll H a .——. if. a [O F l .32 whﬂ=zdﬂ*wuﬂ kz—OO 00 = Z(0.5)ku[n + 3 — kl k=0 This evaluates to 2 1 __ 1 2 n+4 , n 2 “3 y1[n] = \$1[n] * \$2l'nl = { 0,{ ( / ) } otherwise ' (b) Now, ylnl = \$3M * y1[n] = mlnl —- 1/1 [71 - 1]- Therefore, 2 {1 ~ (1/2)"+3} +2 {1 — (1/2)n+4} = (1/2)"+3, n 2 —2 ylnl = { 11 n = _3_ O, otherw1se Therefore, y[n] = (1/2)"+3u[n + 3]. (c) We have an]: man] * mam] = uln + 3]— um + 2] = 6m + 31. (d) From the result of part (c), we get y[n] = y2[n] * m1['n.] = 2:1[71 + 3] = (1/2)”+3u[n + 3]. {X} 2.28. (a) Causal because h[n] == 0 for n < 0. Stable because 2%)" = 5/4 < 00. 11:0 00 (b) Not causal because h[n] 75 0 for n < 0. Stable because Z (0.8)" = 5 < oo. n=——‘2 0 (c) Anti-causal because h[n] = 0 for n > 0. Unstable because 2 (1 /2)" = 00 le—CXJ 3 ((1) Not causal because h[n] 76 O for 'n < 0. Stable because 2 5" == 6—3? <' oo nz—oo 00 _8 2.29. (a) Causal because h(t) = 0 for t < 0. Stable because / |h(t)ldt = e /4 < oo. ’00 00 (b) Not causal because h(t) # O for t < 0. Unstable because / lh(t)l = 00. "00 C” 0 (c) Not causal because h(t) # 0 for t < 0. a Stable because / |h(t)ldt = e10 /2 < 00. —‘CD 00 ((1) Not causal because Mi) 96 0 for t < 0. Stable because / [h(t)|dt = 6'2/2 < 00. ~00 2:52.50. (a) The output will be axl(t) + (9152(0- (b) The output will be 931(t — 7'). "A 0. ({(%)= YUP) 9: MM = I}, [005 (m(t~°‘f)) ~ 00; (w,(-t+o.5-))] For wo=mk, ‘6 IS curvy integer Li”); 1" C[(*)= 7(<*)%1«m z [“9“ QJWOVC'TI) 0h? ’“Dtr z; €314”? XMY Q‘SUOL d‘c at“ M- 3LB=O z 55‘ [005 Educ-taut) — 005 Hoct+0,f)]+w ‘ SxYJ ”Ht—0 5') ’JH’I (,J. (ti—05)] ‘~ {3' a Go: wo(t-u.5') —oos waQt-eo‘r) : o 9 W0 , 211k , k e» rm‘tegw’ 5W wa(f~0.y)~—S\‘n Wu<f+DJ-) z o 9 we 5 (dc-01c , k e {Wager ;0 So (we) 0W L9 0. ,7... % problem '3 clear; f0: 1/50; n: [0: 100]; x = sin(2*pi*f0*n); [y no}— flux n); subpm t(3 l), stem(no, y), xlabel('n‘), ylabel('y(n)'), title('5a. y(n) f0 = 1/50'); subpm (3 stem(n, x), xlabel('n'), ylabel('x(n)'), title('5a. X(n) f0 = 1/50'); Clear; f0:l/lO; n: [0:100]; X = sin(2*pi*f0*n); [yum] = f4<x n); subplot(3 3), stem(no, y), Xlabel('n'), ylabel('y(n)'), title('5b. y(n) f0 = 1/10‘); (3 subplot 4), stem(n, x), Xlabel(‘n'), ylabel('x(n)'), title('5b. x(n) f0 : l/lO'); clear; f0: 1/5; n: [0: 100]; X : sin(2*pi*f0*n); [y,no] = f4(x, n); subplot<3 ,2, 5), stem(no, y), Xlabel('n'), ylabel('y(n)'), title('5b. y(n) f0 = 1/5‘); subplot(3 ,2 6), stem(n, x), xlabel('n'), ylabel('x(n)'), title('5b. x(n) f0 : 1/5‘); 9H“ y(n) fO=1/50 3m x(n) f0=1/50 o 50 100 150 o 20 60 80 10c 1 iﬂl (9) (f) (f) (f) (i) (i) ﬂlﬂﬂlﬂ “L on 0 & I!) G! I“)! ) (Q) (I!) (l) (D (I! I! 0 50 100 150 O 60 80 10C (9) (i) ﬁg XE?” fO— — 1/5 nlnnnngnnnnn’ 1~| 0.5L 3 E = = = = > x L mI‘u u" -g;uiuiuiuiuiuiuiuiuiuium‘ ' u; q‘IIIIIIII- v..L.LL LLLLLLLLLL L L L L LIL.» -05“ -05 ﬁ' ”L L I L L mm'ulblb'LiD! a“ : L 000000000000000000u°L I -1 _1 00000000000 0000000001! O 50 100 150 O 40 10C n n C" T721 W m LQL mm; (1 ms. 0“ WCHJ=FW= “4 l vane)“ ”(f)"? 3(0)]: lit/[h] =§ ‘ﬂfhj : FEE? _ 9%gjﬂ : g [‘ ‘°~'H%>"~MC%~J“~ ((v°~‘1(%””'_o.3lé)”")] :[3‘5 (”5" + 6%)")?ng K/v wwvv ,7, 17' It’s “ex-Whiz 170 9509 M M»; mm- 9 more Sta—M"? J YeaOﬂi’j 0‘ J’Jce‘wt‘j Howl Frumre 0/190“? 5“ hours afﬁx: #1 I“ May. % problem a part a. i i E f a clear; n = 0:15; X : ones(l’24); ~«*‘/ NM- ﬂui+é Lx'jf'f‘f,ki:4-? h = 3.5*((O.5).“n) + 6*(O.2).“n; * ‘ subplot(3,1,l), Stem(n, h), Xlabel('n‘), ylabel('h(n)'), title('6a. h(n) '); y : conv(h,x); 1 subplot(3,l,2), stem(0:1ength(n)+length(x)—2, y), xlabel(‘n'), ylabel('y(n)'), title('6a. 1‘, y(n) when X[n] = u[n] ‘); [X1 X1 X1 x1]; X1 = [6 O O O 0 0]; x2 yl = conv(h,x2); subplot(3,l,3), stem(0:length(n)+length(x2)—2, yl), xlabel( b. y(n) when x[n] 2 size 6 every 6 hours'); 'n'), ylabe1('y(n)‘), title(‘6 .z' CT a h(n) h) when x[n] = u[n] “AW 15 T I 10 E ‘S 5 0 Tﬁﬁmg 1O 15 0 5 20 25 3O 35 4O ‘ n 5i} L: y(n) when x[n] = size 6 every 6 hours 60 1 '1 I 50 40 ’E 3: 30 20 1O 0 TgﬁpaeeoooeHa—ooeoéao—e—e— 0 5 1O 15 20 25 30 35 40 n ~9JC/J W I» give 2% 00w; W “rva/v- ...
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