The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.

The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the opposite direction for a time advance.

Consider a second-order difference equation with specific coefficients and initial conditions, where the input is a unit step function. Applying the z-transform to each term converts the difference equation into an algebraic expression in the z-domain. This expression involves the z-domain representations of both the input and output signals.

To solve for the z-domain output signal, this algebraic equation can be simplified, often using partial fraction decomposition. By determining the coefficients for the partial fractions, we obtain a manageable form that can be inverted back to the time domain using the inverse z-transform. The resulting time-domain response demonstrates the effectiveness of the z-transform in simplifying the analysis of discrete-time linear systems.

This process highlights the utility of the z-transform in digital signal processing and control systems. It provides a straightforward method to transition between the time and z-domains, solve complex equations, and obtain precise system responses. It is crucial to consider the role of initial conditions and the region of convergence when applying the z-transform to ensure accurate and meaningful results.