A discrete-time system can be described by a difference equation relating present and past samples of the output and input. A linear constant-coefficient form is

$$a_0y[n]+a_1y[n-1]+\cdots+a_Ny[n-N] =b_0x[n]+b_1x[n-1]+\cdots+b_Mx[n-M].$$

Because delayed output samples appear, the system generally has memory.

Constant coefficients make the input-output rule time invariant, while the linear combination of signal values makes it linear. Such equations therefore describe a broad class of discrete-time LTI systems.

As with differential equations, initial stored values matter. The zero-state response is caused by the input with zero initial memory, while the zero-input response is the evolution of the stored initial state with no external input.

Difference equations are the natural recurrence-based representation behind many digital filters and sampled dynamical systems.