z変換の公式 証明

■正弦波:$x_k=sin(k\omega T)$

$X(z)=\displaystyle\sum_{k=0}^{+\infty} sin(k\omega T) z^{-k}=\displaystyle\sum_{k=0}^{+\infty}
[ \displaystyle\frac{e^{j k \omega T}-e^{-jk\omega T}}{2j} ]z^{-k}
=\displaystyle\lim_{\zeta \to \infty}\displaystyle\frac{1}{2j} \displaystyle\sum_{k=0}^{\zeta}[(e^{j \omega T}\cdot z^{-1})^k -( e^{-j \omega T}\cdot z^{-1})^k ] $

$=\displaystyle\frac{1}{2j} [\displaystyle\frac{1}{1-e^{j \omega T}z^{-1}} – \displaystyle\frac{1}{1-e^{-j \omega T}z^{-1}}](|z|>1)
= \displaystyle\frac{1}{2j} [\displaystyle\frac{ (e^{j\omega T}-e^{-j \omega T})z^{-1}}{ 1- (e^{j\omega T}+e^{-j \omega T})z^{-1}+z^{-2}} ]$

$=\displaystyle\frac{1}{2j} [\displaystyle\frac{2j sin(\omega T)z^{-1}}{ 1- 2cos(\omega T)z^{-1} +z^{-2}} ]=\displaystyle\frac{ sin(\omega T)z^{-1}}{ 1- 2cos(\omega T)z^{-1} +z^{-2}}$

■余弦波:$x_k=cos(k\omega T)$

$X(z)=\displaystyle\sum_{k=0}^{+\infty} cos(k\omega T) z^{-k}=\displaystyle\sum_{k=0}^{+\infty}
[ \displaystyle\frac{e^{j k \omega T}+e^{-jk\omega T}}{2} ]z^{-k}
=\displaystyle\lim_{\zeta \to \infty}\displaystyle\frac{1}{2} \displaystyle\sum_{k=0}^{\zeta}[(e^{j \omega T}\cdot z^{-1})^k +( e^{-j \omega T}\cdot z^{-1})^k ] $

$=\displaystyle\frac{1}{2} [\displaystyle\frac{1}{1-e^{j \omega T}z^{-1}} + \displaystyle\frac{1}{1-e^{-j \omega T}z^{-1}}] (|z|>1)
= \displaystyle\frac{1}{2} [\displaystyle\frac{2-(e^{j\omega T}+e^{-j \omega T})z^{-1}}{ 1- (e^{j\omega T}+e^{-j \omega T})z^{-1}+z^{-2}} ]$

$=\displaystyle\frac{1}{2} [\displaystyle\frac{2-2 cos(\omega T)z^{-1}}{ 1- 2 cos(\omega T)z^{-1} +z^{-2}} ]=\displaystyle\frac{1-cos(\omega T)z^{-1}}{ 1- 2cos(\omega T) z^{-1} +z^{-2}}$

■指数関数的に振幅が変化する正弦波:$x_k=e^{-akT}sin(k\omega T)$

$X(z)=\displaystyle\sum_{k=0}^{+\infty} e^{-akT}sin(k\omega T) z^{-k}=\displaystyle\sum_{k=0}^{+\infty} e^{-akT}[\displaystyle\frac{e^{j k \omega T}-e^{-jk\omega T}}{2j} ]z^{-k}$

$=\displaystyle\lim_{\zeta \to \infty}\displaystyle\frac{1}{2j} \displaystyle\sum_{k=0}^{\zeta}[(e^{j \omega T-aT}\cdot z^{-1})^k -( e^{-j \omega T-aT} \cdot z^{-1})^k ] $

$=\displaystyle\frac{1}{2j} [\displaystyle\frac{1}{1-e^{j \omega T-aT}\cdot z^{-1}} – \displaystyle\frac{1}{1-e^{-j \omega T-aT} \cdot z^{-1}}](|z|>|e^{-aT}|)$

$= \displaystyle\frac{1}{2j} [\displaystyle\frac{e^{-aT}(e^{j\omega T}-e^{-j \omega T})z^{-1}}{ 1- e^{-aT}(e^{j\omega T}+e^{-j \omega T})z^{-1}+e^{-2aT}z^{-2}} ]= \displaystyle\frac{1}{2j} [\displaystyle\frac{e^{-aT}2j sin(\omega T) z^{-1}}{ 1- 2e^{-aT}cos(\omega T)z^{-1} +e^{-2aT}z^{-2}} ]$

$=\displaystyle\frac{e^{-aT} sin(\omega T) z^{-1}}{ 1- 2e^{-aT}cos(\omega T)z^{-1} +e^{-2aT}z^{-2}}$

■指数関数的に振幅が変化する余弦波:$x_k=e^{-akT}cos(k\omega T)$

$X(z)=\displaystyle\sum_{k=0}^{+\infty} e^{-akT}cos(k\omega T) z^{-k}=\displaystyle\sum_{k=0}^{+\infty} e^{-akT}[\displaystyle\frac{e^{j k \omega T}+e^{-jk\omega T}}{2j} ]z^{-k}$

$=\displaystyle\lim_{\zeta \to \infty}\displaystyle\frac{1}{2} \displaystyle\sum_{k=0}^{\zeta}[(e^{j \omega T-aT}\cdot z^{-1})^k +( e^{-j \omega T-aT} \cdot z^{-1})^k ] $

$= \displaystyle\frac{1}{2} [\displaystyle\frac{2-e^{-aT} (e^{j\omega T}+e^{-j \omega T})z^{-1}}{ 1- e^{-aT}(e^{j\omega T}+e^{-j \omega T})z^{-1}+e^{-2aT}z^{-2}} ]
=\displaystyle\frac{1}{2} [\displaystyle\frac{2-e^{-aT}2 cos(\omega T) z^{-1}}{ 1- 2e^{-aT}cos(\omega T)z^{-1} +e^{-2aT}z^{-2}} ](|z|>|e^{-aT}|)$

$=\displaystyle\frac{1-e^{-aT}cos(\omega T) z^{-1}}{ 1- 2e^{-aT}cos(\omega T)z^{-1} +e^{-2aT}z^{-2}}$