# 时不变系统

## 简单例子

• 系统A：${\displaystyle y(t)=t\,x(t)}$
• 系统B：${\displaystyle y(t)=10\cdot x(t)}$

## 正式例子

${\displaystyle y(t)=t\,x_{d}(t)}$
${\displaystyle y_{1}(t)=t\,x_{d}(t)=t\,x(t+\delta )}$

${\displaystyle y(t)=t\,x_{d}(t)}$
${\displaystyle y_{2}(t)=\,\!y(t+\delta )=(t+\delta )x(t+\delta )}$

${\displaystyle y(t)=10\,x_{d}(t)}$
${\displaystyle y_{1}(t)=10\,x_{d}(t)=10\,x(t+\delta )}$

${\displaystyle y(t)=10\,x_{d}(t)}$
${\displaystyle y_{2}(t)=y(t+\delta )=10\,x(t+\delta )}$

## 抽象例子

${\displaystyle x(t+1)=\,\!\delta (t+1)*x(t)}$

${\displaystyle {\tilde {x}}_{1}=\mathbb {T} _{1}\,{\tilde {x}}}$

${\displaystyle {\tilde {x}}=x(t)\,\forall \,t\in \mathbb {R} }$

${\displaystyle {\tilde {x}}_{1}=x(t+1)\,\forall \,t\in \mathbb {R} }$

${\displaystyle \mathbb {T} _{r}\,\mathbb {H} =\mathbb {H} \,\mathbb {T} _{r}\,\,\forall \,r}$

${\displaystyle {\tilde {y}}=\mathbb {H} \,{\tilde {x}}}$

${\displaystyle \mathbb {T} _{r}\,\mathbb {H} \,{\tilde {x}}=\mathbb {T} _{r}\,{\tilde {y}}={\tilde {y}}_{r}}$

${\displaystyle \mathbb {H} \,\mathbb {T} _{r}\,{\tilde {x}}=\mathbb {H} \,{\tilde {x}}_{r}}$

${\displaystyle \mathbb {H} \,{\tilde {x}}_{r}={\tilde {y}}_{r}}$