Difference equations are one of the few descriptions for linear time-invariant (LTI) systems that can incorporate the effects of stored energy - that is, describe systems which are not at rest when the input is applied. In this lesson you will learn how the output of a LTI system described by a difference equation can be expressed as the sum of a steady-state and a transient response. You will learn why the steady-state response takes the same form as the input and how the transient response reflects the impact of the stored energy in the system.

Difference equation descriptions for systems are widely used in signal processing to implement and model systems. Understanding how the output depends on the input and stored energy will provide you with very helpful insight for working with such systems.

## Prerequisites

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Thomas Davies says

How would this example change if the input were x[n] = n^2 ? What would z be in this case?

Barry Van Veen says

Finding the output for an input is slightly different, mainly because this input doesn't lead to an output of exactly the same form. If is the input to an LTI system, the output is . We say that is an eigenfunction of LTI systems, and this property leads to abundant simplifications. However, this property does not hold for your input since there is no way to write as , that is, there is no for which these two are equal. Thus, the solution gets more complicated.

Here are the basics: An input (generalizing slightly, your case assumes ) produces an output . So to solve for the output, you substitute and the input into the difference equation and use the coefficients of and terms that result from the substitution to solve for . The substitution and equating of coefficients will lead to three linear equations in the three unknowns which you then solve to obtain and thus arrive at the output .