星标变换

因为有${\displaystyle X^{*}(s)={\mathcal {L}}[x^{*}(t)]}$，对${\displaystyle x^{*}(t)}$有:

${\displaystyle x^{*}(t){\overset {\text{def}}{=}}x(t)\cdot \delta _{T}(t)=x(t)\cdot \displaystyle \sum _{n=0}^{\infty }\delta (t-nT)}$

对于卷积定理，加星标变换等同于 ${\displaystyle {\mathcal {L}}[x(t)]=X(s)}$ 和 ${\displaystyle {\mathcal {L}}[\delta _{T}(t)]={\frac {1}{1-e^{-Ts}}}}$ 的复卷积，所以有:

${\displaystyle X^{*}(s)={\frac {1}{2\pi j}}\int _{c-j\infty }^{c+j\infty }X(p)\cdot {\frac {1}{1-e^{T(s-p)}}}\cdot dp}$

此线积分等同于沿着由一条线和无限半圆组成的闭合路径在正方向上的积分，这个闭路径将 ${\displaystyle X(s)}$的极点包在${\displaystyle p}$的左半平面内。如此积分的结果(通过留数定理):

${\displaystyle X^{*}(s)=\displaystyle \sum _{\lambda =poles\,of\,X(s)}Res_{p=\lambda }[X(p){\frac {1}{1-e^{-T(s-p)}}}]}$

上面的线积分等效于沿着该线和包括p右半平面的无穷远的无限半圆形成的闭合路径在负方向上的积分，即为:

${\displaystyle X^{*}(s)={\frac {1}{T}}\displaystyle \sum _{k=\infty }^{\infty }X(s-j{\frac {2\pi }{T}}k)+{\frac {x(0)}{2}}}$

给定Z变换，${\displaystyle X(z)}$ 相应的加星标变换即是下面简单的替换:

${\displaystyle X^{*}(s)=X(z)|_{z=e^{sT}}}$

此变换保留了对T的依赖。

注意 这是可交换的

${\displaystyle X(z)=X^{*}(s)|_{e^{sT}=z}}$

${\displaystyle X(z)=X^{*}|_{s={\frac {\ln(z)}{T}}}}$

性质1: ${\displaystyle X^{*}(s)}$ 是关于 ${\displaystyle s}$ 的以 ${\displaystyle j{\frac {2\pi }{T}}}$ 为周期的周期函数

${\displaystyle X^{*}(s+j{\frac {2\pi }{T}}k)=X^{*}(s)}$

性质2:若 ${\displaystyle X(s)}$ 在 ${\displaystyle s=s_{1}}$有极点，则有 ${\displaystyle X^{*}(s)}$ 必在 ${\displaystyle s=s_{1}+j{\frac {2\pi }{T}}k(k=0,\pm 1,\pm 2,\cdots )}$ 有极点

• 拉普拉斯变换
• Z变换
• 卷积

• Bech, Michael M. ("Digital Control Theory") (PDF). AALBORG University. Retrieved 5 February 2014.

• Gopal, M. (March 1989). Digital Control Engineering. John Wiley & Sons. ISBN 0852263082.

• Phillips and Nagle, "Digital Control System Analysis and Design", 3rd Edition, Prentice Hall, 1995. ISBN 0-13-309832-X