Linear time-invariant systems - Signaling System Study Notes

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Classification System

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Depending on the angle, the system can be divided into different types:

Continuous and discrete systems
for real time system and the dynamic system
lumped parameter system and the distributed parameter system
reversible and irreversible system system
of linear systems
invariant system
causal system
stabilization

system skips the specific configuration of the circuit, consider only input (excitation $e(t)$ ) and the output (response $r(t)$ relation) of.
Which system function $H[n ]$ characterize the inherent characteristics of the system function, so there is no relationship between the applied excitation, only on the intrinsic structure of the system.

The default excitation applied at time zero added, the system time zero before the original storage state is the initial state.


Pour pushing out one to one is reversible, is determined not many are irreversible

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Linear Systems

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Linear refers to the uniformity (homogeneous) and superposition (additive), both met for the called linear, or is nonlinear.
Determining a linear system is:

把原来的激励信号和把原来的响应当作新的激励信号分别经过线性组合之后看是否等价，是就是线性，不是就是不是。

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时不变系统（非时变）

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以微分电路为例：


分两步：
(1) 将原来的激励信号先时移，再经过系统的响应得到的结果
(2) 将原来的激励信号先经过系统的响应，再时移得到的结果
如果（1）和（2）的结果相同就是时不变，否则为时变。

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Linear and Time-invariant System 线性时不变系统（LTI）

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LTI性质：

微积分性质可以推广至高阶

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因果系统

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系统的因果关系：从系统的因果性来看，输入（激励）是输出（响应）的原因，输出是输入的结果。系统的因果信号：借助“因果”这一关系，我们称在t=0之后对系统产生影响的信号为因果信号，换句话说，t<0时，信号取值为0的信号为因果信号。

目前物理可实现的系统都是因果系统

如果 $t$前的加权系数不为1就不能这么判断了
例： $\bm{r(t)=e(2t)}$
因为：若 $e(t)=0, 有r(t_0)=e(2t_0)=0|_{t_0=0.5},t_0<0.5t_0$，故为非因果系统。

需注意的是：因果信号与因果系统不是一个东西。

满足 $\bm{e(t) = e(t)u(t)}$（单边信号）为因果信号。
通常在信号与系统中用到的就是这种单边拉普拉斯变换（有时也将t=0_考虑进去），也就是因果信号（含有输入信号和输出信号的信号系统）的拉氏变换。
详情可参考拉普拉斯变换【直观解释】—复变函数与积分变换学习笔记

因果信号与因果系统

总之，线性时不变系统是最基础的最简单的系统，相应的非线性时变系统最为复杂，