The one sided z-transform has a convenient closed form solution in its region of convergence (ROC) for most engineering applications. Whenever X(z), an infinite series in z ^-1, converges outside the circle $\vert z\vert= R$ , where R is the radius of absolute convergence, it is not needed each time to specify the values of z over which X(z) is convergent.

$\displaystyle \vert z\vert>R \Rightarrow convergent$

$\displaystyle \vert z\vert<R \Rightarrow divergent.$

In one sided z-transform theory, while sampling a discontinuous function x(t), we assume that the function is continuous from the right, i.e., if discontinuity occurs at 0 we assume that x(0) = x(0+).

2.1 Z - Transforms of some elementary functions

Unit step function is defined as:

$\displaystyle u_s(t)$	$\displaystyle = 1, \;\;$ for $\displaystyle \;\; t \ge 0$
	$\displaystyle =0, \;\;$ for $\displaystyle \;\; t < 0$

Assuming that the function is continuous from right

$\displaystyle X(z)$	$\displaystyle =$	$\displaystyle \sum_{k=0}^{\infty}u_s(kT)z^{-k}$
	$\displaystyle =$	$\displaystyle \sum_{k=0}^{\infty}z^{-k}$
	$\displaystyle =$	$\displaystyle 1+z^{-1} + z^{-2} + z^{-3}+ .........$
	$\displaystyle =$	$\displaystyle \frac{1}{1-z^{-1}}$
	$\displaystyle =$	$\displaystyle \frac {z}{z-1}$