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(g) Causal because Mn] = 0 for n < 0. Stable because 2 = 1 < oo. TL=00 ) 00
2.29. (a) Causal because h(t) = O for t < 0. Stable because / Ih(t)dt = 9—8/4 < oo. —00
00 (b) Not causal because h(t) aé 0 for t < 0. Unstable because / h(t) z oo. —00
00 (c) Not causal because h(t) 75 O for t < 0. a Stable because / h(t)dt = 6100/2 < oo. —oo 00
(d) Not causal because h(t) 54$ 0 for t < 0. Stable because / h(t)dt = 62/2 < oo. —oo 00
(e) Not causal because h(t) 79 D for t < 0. Stable because / h(t)dt = 1/3 < oo. —oo 00
(f) Causal because h(t) = U for t < 0. Stable because / lh(t)dt = 1 < 00. “W 00
(g) Causal because h(t) = 0 for t < 0. Unstable because / h(t)dt = 00. —00 We need to ﬁnd the output of the system when the input is :c[n] = 6 Since we are asked
to assume initial rest, we may conclude that y[n] = 0 for n < 0. Now, 2.30. yl'n] = wlnl  Zyln  1]
.Therefore,
3/[01 = 340]  2yl~1l = 1, ylll = wlll  22/[0] = 2, W] = Il2l + 2yl2] = —4 and so on. In closed form,
ylnl : (—2)"u[n1.
This is the impulse response of the system. 2.31. Initial rest implies that y[n] = 0 for n < —2. Now
y[n] = $[n] + 2$[n — 2] — 2y[n — 1].
Therefore, yl—Zl : 1) y[4] = 56,y[5] = —110, y[n] = —110(—2)“‘5 forn 2 5. 2.32. (a) If yh[n] = A(1/2)", then we need to verify 1 n. 1 1 n—l
AG) 74(5) =°' Clearly this is true. 48 ylll Figure 82.58 (c) The ﬁgures corresponding to the remaining parts of this problem are shown in Figure
82.59. 2.60. (a) Integrating the given differential equation once and simplifying, we get _al 1: an t 1'
ya) = ——— y(r)dT—/ f mowed?
‘12 —oo 0’2 oo —00 t 1’ t
+ b—O/ / :1:(a)dad7' + :z:(T)dT + 22376).
“2 oo —oo “2 —oo “1 Therefore, A = ~a1/a2, B = —ao/a2, C = (lg/(11, D = b1/a2,E = bo/ag. (b) Realizing that x2(t) = y1(t), we may eliminate these from the two given integral equa
tions. (c) The ﬁgures corresponding to the remaining parts of this problem are shown in Figure
S2.60. 2.61. (a) (i) From Kirchoﬂ’s voltage law, we know that the input voltage must equal the sum
of the voltages across the inductor and capacitor. Therefore, fMﬂ ﬁw=LC +ma 72 ...
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 Spring '08
 Enright
 Volt, initial rest

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