z变换

z变换：
z变换：$X（z）=\mathcal{Z}[x(n)]=f(t)={\sum }_{n=-\infty }^{\infty }x(n)z^{-n}$
逆z变换*：$x（n）=\frac{1}{2\pi j}{\oint }_{c}X(z)z^{n-1}dz$
单边z变换：$X（z）={\sum }_{n=0}^{\infty }x(n)z^{-n}=x(0)+\frac{x(1)}{z}+\frac{x(2)}{z^2}+...$
常用单边z变换：
1、单位样值信号：$\delta (n)=\{\begin{array}{l}1（n=0）\\ 0（n̸=0）\end{array}\delta (n)$$\leftarrow$$\to 1$
2、单位阶跃信号：$\mu (n)=\{\begin{array}{l}1（n\ge 0）\\ 0（n<0）\end{array}\mu (n)$$\leftarrow$$\to \frac{z}{z-1},|z|>1$
3、矩阵序列： $R_N(n)=\{\begin{array}{l}1（0\le n\le N-1）\\ 0（n<0,n\ge N）\end{array}$
4、斜变信号： $x(n)=n\mu (n)$      $n\mu (n)$$\leftarrow$$\to \frac{z}{(z-1{)}^2},|z|>1$
5、指数信号： $x(n)=a^n\mu (n)$      $a^n\mu (n)$$\leftarrow$$\to \frac{z}{z-a},|z|>|a|\Rightarrow \{\begin{array}{l}{e}^{j{\omega }_{0}n}\mu (n)\leftarrow \to \frac{z}{z-e^{j{\omega }_0}}.|z|>1\\ e^{-j{\omega }_{0}n}\mu (n)\leftarrow \to \frac{z}{z-e^{-j{\omega }_0}}.|z|>1\end{array}$
6、正弦、余弦序列： $cos({\omega }_{0}n)\mu (n)=\frac{1}{2}(e^{j{\omega }_{0}n}+e^{-j{\omega }_{0}n})\mu (n)$$\leftarrow$$\to \frac{1}{2}(\frac{z}{z-e^{j{\omega }_0}}+$$\frac{z}{z-e^{-j{\omega }_0}})=\frac{z(z-cos{\omega }_0)}{z^2-2zcos{\omega }_0+1},|z|>1$
                  $sin({\omega }_{0}n)\mu (n)=\frac{1}{2j}(e^{j{\omega }_{0}n}-e^{-j{\omega }_{0}n})\mu (n)$$\leftarrow$$\to \frac{1}{2j}(\frac{z}{z-e^{j{\omega }_0}}-$$\frac{z}{z-e^{-j{\omega }_0}})=\frac{zsin{\omega }_0}{z^2-2zcos{\omega }_0+1},|z|>1$

$⟹\{\begin{array}{l}cos(\frac{\pi n}{2})\mu (n)\leftarrow \to \frac{z^2}{z^2+1},|z|>1\\ sin(\frac{\pi n}{2})\mu (n)\leftarrow \to \frac{z}{z^2+1},|z|>1\end{array}$

常用性质：
线性：$ax(n)+by(n)$
位移：单边：$\{\begin{array}{l}x（n-m）\mu (n-m)\leftarrow \to X(z)z^{-m}\\ x（n-m）\mu (n)\leftarrow \to z^{-m}[X(z)+{\sum }_{r=-m}^{-1}x(r)z^{-r}]\\ x（n+m）\mu (n)\leftarrow \to z^m[X(z)-{\sum }_{r=0}^{m-1}x(r)z^{-r}]\end{array}$
      双边：$\{\begin{array}{l}x（n-m）\leftarrow \to z^{-m}X(z)\\ x（n+m）\leftarrow \to z^{m}X(z)\end{array}$
      常用：$\{\begin{array}{l}x（n-1）\mu (n)\leftarrow \to z^{-1}X(z)+x(-1)\\ x（n-2）\mu (n)\leftarrow \to z^{-2}X(z)+z^{-1}x(-1)+x(-2)\end{array}$
            $\{\begin{array}{l}x（n+1）\mu (n)\leftarrow \to zX(z)-zx(0)\\ x（n+2）\mu (n)\leftarrow \to z^{2}X(z)-z^{2}x(0)-zx(1)\end{array}$
线性加权（z域微分）：$nx(n)\leftarrow \to -z\frac{d}{dz}X(z)\Rightarrow n^{m}x(n)\leftarrow \to [-z\frac{d}{dz}{]}^{m}X(z)$
指数加权（z域尺度变换）：$a^{n}x(n)\leftarrow \to X(\frac{z}{a})(R_{x1}<|\frac{z}{a}|<R_{x2})\Rightarrow \{\begin{array}{l}{a}^{-n}x(n)\leftarrow \to X(az),R_{x1}<|az|<R_{x2}\\ (-1{)}^{n}x(n)\leftarrow \to X(-z),R_{x1}<|z|<R_{x2}\end{array}$

初值定理：若x(n)为因果序列：$x（0）={\lim }_{z\to \infty }X(z)$
终值定理：若x(n)为因果序列：${\lim }_{n\to \infty }x(n)={\lim }_{z\to 1}[(z-1)X(z)]$   $只有当n\to \infty 时，x(n)收敛，才可应用终值定理（即X（z）在单位圆内）$
时域卷积定理：$x(n)\ast h(n)\leftarrow \to X(z).H（z）收敛域为两者重叠部分$

收敛域：不同收敛域对应原函数不同，如$\{\begin{array}{l}{a}^n\mu (n)\leftarrow \to \frac{z}{z-a},|z|>|a|\\ -a^n\mu (-n-1)\leftarrow \to \frac{z}{z-a},|z|<|a|\end{array}$
(1)有限长$n_1$$\le$ n $\le$$n_2$：$nx(n)\{\begin{array}{l}(a):n_1<0,n_2>0时，0<|z|<\infty \\ (b):n_2\le 0时，|z|<\infty \\ (c):n_1\ge 0时，0<|z|\end{array}$
(2)右边序列$n\ge n_1$：$\{\begin{array}{l}(a):n_1\ge 0时，R_{x1}<|z|\\ (b):n_1<0时，R_{x1}<|z|<\infty \end{array},其中R_{x1}={\lim }_{n\to \infty }\sqrt[n]{|x(n)|}$
(3)左边序列$n\le n_2$：$\{\begin{array}{l}(a):n_2\le 0时，|z|<R_{x2}\\ (b):n_2>0时，0<|z|<R_{x2}\end{array},其中R_{x2}=\frac{1}{li{m}_{n\to \infty }\sqrt[n]{|x(-n)|}}$
(4)双边序列：有X（z）=${\sum }_{n=-\infty }^{-1}x(n)z^{-n}+{\sum }_{n=0}^{\infty }x(n)z^{-n}$：$若R_{x2}<R_{x1},则X(z)不收敛；若R_{x2}>R_{x1},则R_{x1}<|z|<R_{x2}$
部分分式展开法：先将$\frac{X(z)}{z}展开，然后每个分式X（z）\Rightarrow \sum \frac{z}{z-z_n},下面常用逆z变换对：$
$X（z）$          |z|>|a|,即右序列          |z|<|a|,即左序列
$\frac{z}{z-1}$               $\mu (n)$                     $-\mu (-n-1)$
$\frac{z}{(z-1{)}^2}$             $n\mu (n)$                   $-n\mu (-n-1)$
$\frac{z}{z-a}$               $a^n\mu (n)$                   $-a^n\mu (-n-1)$
$\frac{z^2}{(z-a{)}^2}$             $(n+1)a^n\mu (n)$           $-(n+1)a^n\mu (-n-1)$
$\frac{z}{(z-a{)}^2}$             $n{a}^{n-1}\mu (n)$               $-n{a}^n\mu (-n-1)$