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电子科技大学二零 一 零 至二零 一 一 学年第 一 学期期 末 考试
SIGNALS AND SYSTEMS 课程考试题 A 大纲A 卷 （ 120 分钟） 考试形式： 一页纸开卷 考试日期 20 年 月 日
课程成绩构成：平时 10 分， 期中 20 分， 实验 10 分， 期末 60 分
Attention: Y ou must answer the following questions in English.
1.（15 points ） Suppose
()1x t
and
()2x t
are two band-limited signals, where
π
ωω200,0)(1>=for j X ,π
ωω500,
0)(2>=for j X .
Impulse-train sampling is performed on
()()()
1234/22=+-*y t x t x t to obtain
()()()p n y t y nT t nT δ+∞
=-∞
=
-∑
.Give out the expression of
)(ωj Y in terms of )
(1ωj X and
)(2ωj X ，where )(ωj Y
is the Fourier transform of
)
(t y . Specify the largest values of the sampling period T
which ensures that ()t y is recoverable from ()t y p .
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2.（20 points ）Consider a stable system illustrated in Figure 1, if we know )()(0t u e t h t
-=，()()
1sin 5ππ=
t h t t
, ()()
2sin 3ππ=
t h t t
and the input
()()/2δ+∞
=-∞
=
-∑n x t t n ，determine the output ()y t .
Figure 1
3. ( 10 points ) Determine the function of time, []x n , for the Z transform ()X z and its associated regions of convergence:
()4
11X z z
-=
- 1z >
)
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4.（15 points ）Consider a system illustrated in Figure 2(a). The input signal has the Fourier tansform ()X j ω shown in Figure 2(b), and the output signal has the Fourier tansform ()ωY j shown in Figure 2(c). Determine a possible system S.
()
t Figure 2(a)
()
x t
Figure 2(b)
Figure 2(c)
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5.(20 points) Consider a LTI system with unit impulse response ()()()β--=+t t h t e u t e u t ,where β is an unknown constant. When the input to the system is ()1=x t ,the output is ()43
=
y t .
(a) Determine the system function ()s H of the system and sketch the pole-zero pattern, then indicate the ROC of ()s H . (b) Is this system causal and stable ?
(c) Draw a block diagram representation of this system.
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6.(20 points) A stable LTI system is described by the difference equation
[][][][]12714
2
--+
-
=y n n y n x n y .
(a) Find the system function ()H z , sketch the pole-zero pattern of ()H
z , then indicate the ROC of
()H
z .
(b) Determine the unit impulse response []h n . Is this system causal? (c) Compute the output of this system, if the input signal is []cos x n n
π=.
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电子科技大学2010-2011-1期 末 考试 信号与系统（A 大纲A 卷） 参考解答
1.(15 points) Solution: ()()4
3
11134/33
ωω+←−→
j x t X j e
, 1600ωπ=M
()42212222ωω-⎛⎫-←−→ ⎪⎝⎭
j x t X j e , 2250ωπ=M
()()8
3
122()/320, for 2503
ωωωωωπ-=
=>j Y j X j X j e
2250500ωππ>⨯=s
m ax 21250
π
ω=
=
s
T (second)
2. (20 points) Solution:
(
)1 351
0 others πωπωω⎧<<⎪
+=⎨⎪⎩
j H
j ()42π+∞
=-∞
=
∑
j k t
k x t e
(
)114tan 4ππ-⎛⎫=
- ⎪⎝
⎭y t t
3. (10 points) Solution:
[]()11
1 11-←−→=
>-u n X z z z
()()4
1=X z X z
[][] 1 4,0
/4 0 4,0 0 0=>⎧⎪
==≠>⎨⎪<⎩
n m n x n u n n m n n
4. ( 15 points ) Solution:
()1 2 3 0 o t h e r s ωπω⎧>⎪
=⎨⎪⎩H j
5.(12 points) Solution:
(a) ()()()
24
13+=++s H s s s {}R e 1>-s
()
x t ()y t S
σ
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(b) The system is causal and stable.
(c)
()
x t
()
y t
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6.( 20 points ) Solution: (a) ()1
2
171142
--=
+
-
H z z
z
124
<<z
（3 points ）
(b) [][]()[]11821949
⎛⎫=---- ⎪
⎝⎭n
n
h n u n u n
The system is not causal. (c) [][]()cos 1n
x n n π==- is the eigenfuction of the system
[]()()411cos 5
π=--=-
n
y n H n。