Z-Transform

The -transform of is defined by

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

(1)

where

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

(2)

$\delta(t)$ is the Delta Function,

is the sampling period, and ${\mathcal L}$ is the Laplace Transform. An alternative definition is

$\begin{displaymath} Z[F(t)]=\sum_{\rm residues} \left({1\over 1-e^{Tz}z^{-1}}\right)f(z), \end{displaymath}$

(3)

where

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

(4)

The inverse

-transform is

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

(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

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

are therefore generators for Euler's Triangle.

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.