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1 62 Signals and Systems Chap.’“
(3) 12(1) = .101 — 2) + x12 — 1*) (W111) = [120313019110 1
1c) 1211) = 13:. 1:11-1:11 11111-111 = i :11) + m _ 2), :1 (a) W) — iii-111+ x(t— 2) :13 :00 1111-11) = W3) l 1g) 1111— — 15.5.” 1.28. Determine which of the properties listed in Problem 1.27 hold and which do Iii
hold for each of the following discrete-time systems. Justify your answers. In ea F- example. y[n] denotes the system output and x[1-1] is the system input. ‘1a) 1111] = 11-11] (1)) 11111— “ scin — 2] — 21111 — 8]
1 t1?) yin] = 1111111] (d) 1111] — 3v{x[ [11 — 1]}
ache]. n E 1 Mn] 11. E l
(B) ﬂit] = 0, 11 = 0 (f) y[n] = 0, .11 = 0
.1:[11+ l], 11 E ‘—1 :1:[rt], 11 E —1 -(g) yie] = «11411 +1] 1.29. (a) Show that the discrete- time system whose input .1:[11] and output y[a] are relatai
by y[11]= Gie{;c[a]} 1s additive. Does this system remain additive if its input-2
output relationship is changed to y[11]= Gide“ “”14x[11]}‘? (Do not assume thai x[1-1] is real 1n this problem.) (b) In the test we discussed the fact that the property of linearity for a system 11
equivalent to the system possessing both the additivity property and homo gent-f.
ity property. Determine whether each of the systems deﬁned below is addititi-
and/or homogeneous. Justify your answers by providing a proof for each pro
erty if it holds or acounterexample if it does not. x[rt].r[11—2]
1 _ 1 11.11.1312 -- _ _ , x[a — 1] a5 0
1 1‘ — — 11 11 1. .1111 1]
011) Wig] ()11] {0} x[n_1]:0
1. 30. Determine if each of the following systems 1s 1nvertible If 1t IS construct the 1nve111 system. If it is not ﬁnd two input signals to the system that have the same output - 11a) 111)- - x11 — 11 -- 1b) 111)— 1 cos[x(r)]
(1:) yin] = 1111111] (11) 12(1) = 111-15de
x[11 — 1]. rt 2 1
(11) yin] = 0. 11 z 0 (f) yin] = X[n]x[11 - 1]
1111]. n E —1
1g) 12?] = 111— 11 111112111: 1; [tree-11:11-
11111111 = Z.-_.1— 1*- 1x1u 111 1211)— _ 1;}?
2 0
1k) 111:] = 13:; ”g 2 E _1 11) 11:) = 1121) -
1m) yin] = 1:12:11 111) 1111 = 13‘1”” 21- 12:33“ 1..31 In this problem we illustrate one of the most important consequences of the prop-
erties of linearity and time invariance. Speciﬁcally. once we know the responti
of a linear system or a linear time- invariant (LTI) system to a single input or thlE
responses to several inputs we can directly compute the responses to many othtiL ...

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- Spring '09
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