《信号与系统》第九章习题解答

shown in Figure 1. (a) Determine the system function of the system, is this system causal? (b) Determine the unit impulse response of this system. (c) If the input is x ( t ) = u ( − t ) , determine the output y ( t ) . (d) Draw a block diagram representation of this system.
17
Chapter 9
例：某连续时间 LTI 系统的系统函数为 H ( s ) =
Problem Solution
s +1 为常数。 ，其中 a, b 为常数。已知系统函 2 s + as + b
t
数 在 s = −2 有 一 个 极 点 ， 且 输 入 为 x ( t ) = e , − ∞ < t < +∞ 时 ， 系 统 的 输 出
Problem Solution
y′′(t ) − y′(t ) − 2 y (t ) = x(t )
(b) Determine h(t ) for each of the following cases: −1/ 3 1/ 3 1 1 H (s ) = 2 = = + s − s − 2 (s + 1)(s − 2 ) s + 1 s − 2 1. The system is stable. 1 −t 1 2t h(t ) = − e u (t ) − e u (− t ) − 1 < Re{s} < 2 3 3 2. The system is causal. 1 −t 1 2t h(t ) = − e u (t ) + e u (t ) Re{s} > 2 3 3 3. The system is neither stable nor causal.
x(t )
Problem Solution
h(t )
y (t ) Y (s )
X (s )
H (s )
(c) If x(t ) = e3t , − ∞ < t < +∞ , determine the output y (t ) .
H ( s) =
y(t) =
Y ( s)
X ( s)
=
( s + 1) ( s + 2 )
Homework: 9.2 9.5 9.7 9.8 9.9 9.13 9.21(a,b,i,j)
9.22(a,b,c,d) 9.28 9.31 9.32 9.33 9.35 9.45
1
Chapter 9
Problem Solution
9.7 How many signals have a Laplace transform that may be s −1 expressed as
Re { s} > −1
y′′ ( t ) + 2 y′ ( t ) + y ( t ) = x′′ ( t ) − 2 x ′ ( t ) − x ( t )
This system is stable.
11
Chapter 9 9.45 An LTI system: s+2 X (s ) = x(t ) = 0 , t > 0 s−2 2 1 y (t ) = − e 2t u (− t ) + e −t u (t ) 3 3 (a) Determine H (s ) and its ROC. (b) Determine h(t ) .
x(t ) is two sided.
3
Chapter 9 9.21 Determine the Laplace transform.
Problem Solution
( b ) x ( t ) = e −4 t u ( t ) + e −5 t ( sin 5t ) u ( t )
X ( s) =
6
因果稳定LTI系统给出如下信息：⑴ H ( s) 1 ; 系统给出如下信息： 因果稳定 系统给出如下信息 s=1
d 2h( t ) dh( t ) 输出不是绝对可积的； 输出不是绝对可积的；⑷信号 +3 + 2h( t ) 2 dt dt 是有限持续期的； 在无穷远点只有一阶零点。 是有限持续期的；⑸ H(s)在无穷远点只有一阶零点。
Problem Solution
Consider a stable LTI system with input x ( t ) = 2 and output
y ( t ) = 4 / 3 . Suppose the rational system function has the pole-pattern
输出是绝对可积的； ⑵当输入为 u( t ) 时，输出是绝对可积的；⑶当输入为 tu( t )时，
1. 试确定 H(s) ，画出其零极点图并标注收敛域； 画出其零极点图并标注收敛域； 2. 试求系统的单位冲激响应 h(t ) ； 3. 若输入 x ( t ) = e2t , −∞< t < +∞ ，试求系统的输出 y ( t ) ； 4. 写出描述该系统的常系数微分方程； 写出描述该系统的常系数微分方程； 5. 画出该系统的模拟框图。 画出该系统的模拟框图。
2
Chapter 9
Problem Solution
9.8 Let x(t ) be a signal that has a rational Laplace transform with exactly two poles,located at s = -1 and s = -3. If g (t ) = e 2t x(t ) and G ( jω ) convergence,determine whether x(t ) is left sided, right sided,or two sided.
(s + 2)(s + 3)(s 2 + s + 1)
in its region of convergence?
jω
Poles :
−
1 3 + j 2 2
s1 = −2, s1 = −3, s3, 4
1 3 =− ± j 2 2
−3 −2
− 1 3 − j 2 2
σ
There are four signals.
−2 ( s + 1 )
2 t 2 −t 4 −t y ( t ) = + e u ( − t ) + e ( cos t ) u ( t ) + e ( sin t ) u ( t ) 5 5 5
9
Chapter 9
Problem Solution
9.35 Consider a causal LTI system with the input x(t ) and output y (t ) . (b) Is this system stable?
Re{s} < −1
1 1 h(t ) = + e −t u (− t ) − e 2t u (− t ) 3 3
7
Chapter 9 9.32 Consider a causal LTI system ,
1. x(t ) = e
2t
Problem Solution
dh(t ) 2. + 2h(t ) = e −4t u (t ) + bu (t ) dt
x(t ) = e
−t
Problem Solution
s +1 s 2 + 2s + 2
, − ∞ < t < +∞
Determine the response y (t )
Y ( s) = X ( s) H ( s) =
( s + 1) ( s − 1) ( s 2 + 2 s + 2 )
−1 < Re { s} < 1
x(t )
(a) Determine a differential equation relating y (t )and x(t )
L1 -2
1 s
y (t )
-1
1 s
L2 -1
-6
10
Chapter 9
Problem Solution
s2 − 2s − 1 H ( s) = 2 s + 2s + 1
−2
②
−1
③
1
④
2
σ
① Re{s} < −2 anticausal , unstable ② -2 < Re{s} < −1 noncausal , unstable
③ -1 < Re{s} < 1
noncausal , stable Causal , unstable
6
④ Re{s} > 1
Chapter 9 9.31 Consider a continuous-time LTI system (a) Determine H (s ) .
jω
σ
−3
−2
−1
14
Chapter 9
Problem Solution
(a)
s+2 H ( s) = ( s + 1) ( s + 3 )
stable ⇒ Re { s} > −1 ⇒ causal
(b)
1 −t h ( t ) = ( e + e −3 t ) u ( t ) 2