A linear system is a model of a system based on some kind of linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. This is a mathematical abstraction very useful in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modelled by linear systems.
A general deterministic system can be described by operator that maps an input as a function of to an output , a type of black box description. Linear systems satisfy the properties of superposition and scaling: given two valid inputs
as well as their respective outputs
then a linear system must satisfy
for any scalar values and .
The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems. For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function in terms of unit impulses or frequency components.
Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).
A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.