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