The first order difference is defined as

The second order difference is defined as

In general, the kth order difference will need to involve .

Similar to an LTI continuous system which can be described by an LCCDE
(differential equation), a discrete LTI system can be described by an LCCDE
(difference equation):

where and are the discrete input and output, respectively. After taking the z-transform on both sides of the LCCDE, we get an algebraic equation in the domain:

and its transfer function is rational:

where is a coefficient and are the roots of the numerator polynomial and are the roots of the denominator polynomial. Note that just as the LCCDE alone does not completely specify the relationship between and (additional information such as the initial conditions is needed), the transfer function does not completely specify the system. For example, the same with different ROCs will represent different systems (e.g., causal or anti-causal).

**Example:** The input and output of an LTI system are related by

Note that without further information such as the initial condition, this equation does not uniquely specify when is given. Taking z-transform of this equation and using the time shifting property, we get

and the transfer function can be obtained

Note that the causality and stability of the system is not provided by this equation, unless the ROC of this is specified. Consider these two possible ROCs:

- If ROC is , it is outside the pole and includes
the unit circle. The system is causal and stable:

- If ROC is , it is inside the pole and does not include
the unit circle. The system is anti-causal and unstable: