Most practical discrete-time systems can be represented by linear difference equations, making the z-transform a particularly useful tool.

Knowing the input signal and N initial conditions is necessary for solving an Nth-order difference equation.

For delayed or advanced signals, the z-transform shifts the signal in the z-domain by including multiplication of the inverse of z or z, respectively.

Consider a second-order difference equation characterized by specific coefficients and initial conditions. The input is the unit step function.

Taking the z-transform of each term, the equation transforms into an algebraic expression involving the z-domain representation of the input and output signals.

Solving this algebraic equation for the z-domain output signal provides an expression that can be simplified using partial fraction decomposition.

The coefficients are calculated, and the system's time-domain response is given by the inverse z-transform of the partial fractional expression.

This process demonstrates the power of the z-transform in simplifying the analysis and solution of discrete-time linear systems, making it an essential tool in various digital signal processing and control systems fields.