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The $Z$-transform of $F(t)$ is defined by

Z[F(t)]={\mathcal L}[F^*(t)],
\end{displaymath} (1)

F^*(t)=F(t)\delta_T(t)=\sum_{n=0}^\infty F(nT)\delta(t-nT),
\end{displaymath} (2)

$\delta(t)$ is the Delta Function, $T$ is the sampling period, and ${\mathcal L}$ is the Laplace Transform. An alternative definition is
Z[F(t)]=\sum_{\rm residues} \left({1\over 1-e^{Tz}z^{-1}}\right)f(z),
\end{displaymath} (3)

f(z)=\sum_{n=0}^\infty F(nT)z^{-n}.
\end{displaymath} (4)

The inverse $Z$-transform is
Z^{-1}[f(z)]=F^*(t)={1\over 2\pi i}\oint f(z)z^{n-1}\,dz.
\end{displaymath} (5)

It satisfies
$\displaystyle Z[aF(t)+bG(t)]$ $\textstyle =$ $\displaystyle a Z[F(t)]+b Z[F(t)]$ (6)
$\displaystyle Z[F(t+T)]$ $\textstyle =$ $\displaystyle z Z[F(t)]-zF(0)$ (7)
$\displaystyle Z[F(t+2T)]$ $\textstyle =$ $\displaystyle z^2 Z[F(t)]-z^2 F(0)-z F(t)$ (8)
$\displaystyle Z[F(t+mT)]$ $\textstyle =$ $\displaystyle z^m Z[F(t)]-\sum_{r=0}^{m-1} z^{m-r}F(rt)$ (9)
$\displaystyle Z[F(t-mT)]$ $\textstyle =$ $\displaystyle z^{-m}Z[F(t)]$ (10)
$\displaystyle Z[e^{at}F(t)]$ $\textstyle =$ $\displaystyle Z[e^{-aT}z]$ (11)
$\displaystyle Z[e^{-at}F(t)]$ $\textstyle =$ $\displaystyle Z[e^{aT}z]$ (12)
$\displaystyle tF(t)$ $\textstyle =$ $\displaystyle -Tz{d\over dz} Z[F(t)]$ (13)
$\displaystyle t^{-1}F(t)$ $\textstyle =$ $\displaystyle -{1\over T}\int_0^z {f(z)\over z}\,dz.$ (14)

Transforms of special functions (Beyer 1987, pp. 426-427) include
$\displaystyle Z[\delta(t)]$ $\textstyle =$ $\displaystyle 1$ (15)
$\displaystyle Z[\delta(t-mT)]$ $\textstyle =$ $\displaystyle z^{-m}$ (16)
$\displaystyle Z[H(t)]$ $\textstyle =$ $\displaystyle {z\over z-1}$ (17)
$\displaystyle Z[H(t-mT)]$ $\textstyle =$ $\displaystyle {z\over z^m(z-1)}$ (18)
$\displaystyle Z[t]$ $\textstyle =$ $\displaystyle {Tz\over(z-1)^2}$ (19)
$\displaystyle Z[t^2]$ $\textstyle =$ $\displaystyle {T^2z(z+1)\over (z-1)^3}$ (20)
$\displaystyle Z[t^3]$ $\textstyle =$ $\displaystyle {T^3z(z^2+4z+1)\over(z-1)^4}$ (21)
$\displaystyle Z[a^{\omega t}]$ $\textstyle =$ $\displaystyle {z\over z-a^{\omega T}}$ (22)
$\displaystyle Z[\cos(\omega t)]$ $\textstyle =$ $\displaystyle {z\sin(\omega T)\over z^2-2z\cos(\omega T)+1}$ (23)
$\displaystyle Z[\sin(\omega t)]$ $\textstyle =$ $\displaystyle {z[z-\cos(\omega T)]\over z^2-2z\cos(\omega T)+1},$ (24)

where $H(t)$ is the Heaviside Step Function. In general,
$\displaystyle Z[t^n]$ $\textstyle =$ $\displaystyle (-1)^n \lim_{x\to 0}{\partial^n\over\partial x^n}\left({z\over z-e^{-xT}}\right)$ (25)
  $\textstyle =$ $\displaystyle {T^n z\sum_{k=1}^n \left\langle{\begin{array}{c}n\\  k\end{array}}\right\rangle{} z^{k-1} \over(z-1)^{n+1}},$ (26)

where the $\left\langle{\matrix{n\cr k\cr}}\right\rangle{}$ are Eulerian Numbers. Amazingly, the Z-transforms of $t^n$ are therefore generators for Euler's Triangle.

See also Euler's Triangle, Eulerian Number


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.

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© 1996-9 Eric W. Weisstein