# Linear system

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 ${\displaystyle H}$ that maps an input ${\displaystyle x(t)}$ as a function of ${\displaystyle t}$ to an output ${\displaystyle y(t)}$, a type of black box description. Linear systems satisfy the properties of superposition and scaling: given two valid inputs

${\displaystyle x_{1}(t)\,}$
${\displaystyle x_{2}(t)\,}$

as well as their respective outputs

${\displaystyle y_{1}(t)=H\left(x_{1}(t)\right)}$
${\displaystyle y_{2}(t)=H\left(x_{2}(t)\right)}$

then a linear system must satisfy

${\displaystyle \alpha y_{1}(t)+\beta y_{2}(t)=H\left(\alpha x_{1}(t)+\beta x_{2}(t)\right)}$

for any scalar values ${\displaystyle \alpha \,}$ and ${\displaystyle \beta \,}$.

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 ${\displaystyle x(t)}$ 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).

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense.

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.