The
-transform of
is defined by
![\begin{displaymath}
Z[F(t)]={\mathcal L}[F^*(t)],
\end{displaymath}](z_29.gif) |
(1) |
where
![\begin{displaymath}
F^*(t)=F(t)\delta_T(t)=\sum_{n=0}^\infty F(nT)\delta(t-nT),
\end{displaymath}](z_30.gif) |
(2) |
is the Delta Function,
is the sampling period, and
is the Laplace Transform. An alternative definition is
![\begin{displaymath}
Z[F(t)]=\sum_{\rm residues} \left({1\over 1-e^{Tz}z^{-1}}\right)f(z),
\end{displaymath}](z_34.gif) |
(3) |
where
![\begin{displaymath}
f(z)=\sum_{n=0}^\infty F(nT)z^{-n}.
\end{displaymath}](z_35.gif) |
(4) |
The inverse
-transform is
![\begin{displaymath}
Z^{-1}[f(z)]=F^*(t)={1\over 2\pi i}\oint f(z)z^{n-1}\,dz.
\end{displaymath}](z_36.gif) |
(5) |
It satisfies
![$\displaystyle Z[aF(t)+bG(t)]$](z_37.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle a Z[F(t)]+b Z[F(t)]$](z_39.gif) |
(6) |
![$\displaystyle Z[F(t+T)]$](z_40.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle z Z[F(t)]-zF(0)$](z_41.gif) |
(7) |
![$\displaystyle Z[F(t+2T)]$](z_42.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle z^2 Z[F(t)]-z^2 F(0)-z F(t)$](z_43.gif) |
(8) |
![$\displaystyle Z[F(t+mT)]$](z_44.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle z^m Z[F(t)]-\sum_{r=0}^{m-1} z^{m-r}F(rt)$](z_45.gif) |
(9) |
![$\displaystyle Z[F(t-mT)]$](z_46.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle z^{-m}Z[F(t)]$](z_47.gif) |
(10) |
![$\displaystyle Z[e^{at}F(t)]$](z_48.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle Z[e^{-aT}z]$](z_49.gif) |
(11) |
![$\displaystyle Z[e^{-at}F(t)]$](z_50.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle Z[e^{aT}z]$](z_51.gif) |
(12) |
![$\displaystyle tF(t)$](z_52.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle -Tz{d\over dz} Z[F(t)]$](z_53.gif) |
(13) |
![$\displaystyle t^{-1}F(t)$](z_54.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle -{1\over T}\int_0^z {f(z)\over z}\,dz.$](z_55.gif) |
(14) |
Transforms of special functions (Beyer 1987, pp. 426-427) include
![$\displaystyle Z[\delta(t)]$](z_56.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle 1$](z_57.gif) |
(15) |
![$\displaystyle Z[\delta(t-mT)]$](z_58.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle z^{-m}$](z_59.gif) |
(16) |
![$\displaystyle Z[H(t)]$](z_60.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {z\over z-1}$](z_61.gif) |
(17) |
![$\displaystyle Z[H(t-mT)]$](z_62.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {z\over z^m(z-1)}$](z_63.gif) |
(18) |
![$\displaystyle Z[t]$](z_64.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {Tz\over(z-1)^2}$](z_65.gif) |
(19) |
![$\displaystyle Z[t^2]$](z_66.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {T^2z(z+1)\over (z-1)^3}$](z_67.gif) |
(20) |
![$\displaystyle Z[t^3]$](z_68.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {T^3z(z^2+4z+1)\over(z-1)^4}$](z_69.gif) |
(21) |
![$\displaystyle Z[a^{\omega t}]$](z_70.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {z\over z-a^{\omega T}}$](z_71.gif) |
(22) |
![$\displaystyle Z[\cos(\omega t)]$](z_72.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {z\sin(\omega T)\over z^2-2z\cos(\omega T)+1}$](z_73.gif) |
(23) |
![$\displaystyle Z[\sin(\omega t)]$](z_74.gif) |
![$\textstyle =$](z_38.gif) |
![$\displaystyle {z[z-\cos(\omega T)]\over z^2-2z\cos(\omega T)+1},$](z_75.gif) |
(24) |
where
is the Heaviside Step Function. In general,
where the
are Eulerian Numbers. Amazingly, the
Z-transforms of
are therefore generators for Euler's Triangle.
See also Euler's Triangle, Eulerian Number
References
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 424-428, 1987.
Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 257-262, 1965.
© 1996-9 Eric W. Weisstein
1999-05-26