第三讲 2.1 Time-Domain Representations for LTI System
2.1 Time-Domain Representations for LTI System
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WONG Zinhoo
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Traditional Methods
1. 存在经典连续时间系统的系统激励与响应的时域分析方法，其本质是对线性微分方程的求解过程。
2. 对于经典的分时域分析方法不做展开，其具有一定的局限性，这是由微分方程特解的求解难度和其物理特征不明显所决定的。
1st For homogeneous equation
：
, the homogeneous solution is
.
Characteristics equation is:
Solving the CE get the Characteristic Solution:
So, the homogeneous solution is
2nd For Non-homogeneous equation
：
, the particular solution is
.
Suppose the particular solution is
Inverse obtained the c=1/3.
3rd Complete Solution:
Complete Solution = homogeneous solution + particular solution
Inverse obtained the
Solution:
Response of LTI
1. Zero-input Response:
The Zero-input Response of
is
：
Characteristic Function:
Characteristic Solution:
So:
and,
So:
结论：一旦系统的微分方程确定了，那么该系统的零输入响应的形式就确定了，且该系统的零输入响应由方程的特征根决定。
2. Zero-state Response:
原理：将任意系统信号分解为冲激信号然后以卷积求解零状态响应。
3. Impulse response
冲激响应：
给予系统一个单位脉冲激励所得到的响应：
冲激平衡法求解
h(t):
Exp 1.
System,
Find the solution of pulse response.
解
(t)
h(t)
回代系统：
A=1
解得：
h(t)=e^-3t
Exp 2.
本质上是解二阶非齐次微分方程。
4. Step response
阶跃响应
本质上是变上限积分函数关系。
5. Question
System have a step response as pic<1> when the input is step signal u(t). What’s the function f(t) of input signal when the output like pic<2>?
Pic<1>
G(t)
单位阶跃响应
Pic<2> h(t)
冲激响应
解：
f(t)= g(t)-2g(t-1)-g(t-2)
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