z変換
$ x:\Z\to\Complexに対して
$ \mathcal Z(x_\bullet)(z):=\sum_{n\in\Z}x_nz^{-n}
$ \mathcal Z^{-1}(X)_n:=\frac1{2\pi i}\oint_CX(z)z^{n-1}\mathrm dz
性質
$ \mathcal F_{DN}(x_\bullet)(\omega)=\mathcal Z(x_\bullet)(e^{i\omega})
$ \mathcal Z(f(n\Delta T))\left(e^{s\Delta T}\right)=\mathcal L_B\left(f(t)\sum_{n\in\Z}\llbracket t=n\Delta T\rrbracket\right)(s)
$ \mathcal L_B(f)(s):=\mathcal L(f)(s)+\mathcal L(t\mapsto f(-t))(s)=\int_\Re^{-st}f(t)\mathrm dt：両側Laplace変換 $ \mathcal L\left(f(t)\sum_{n\in\Z}\llbracket t=nT\rrbracket\right)(s)+\mathcal L\left(f(-t)\sum_{n\in\Z}\llbracket -t=nT\rrbracket\right)(-s)=\int_\R e^{-st}f(t)\sum_{n\in\Z}\llbracket t=nT\rrbracket\mathrm dt
$ = \sum_{n\in\Z}\int_\R e^{-st}f(t)\llbracket t=nT\rrbracket\mathrm dt
$ = \sum_{n\in\Z}e^{-snT}f(nT)
$ = \mathcal Z(f(nT))\left(e^{sT}\right)