Linear system

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

$x_1(t) \,$

$x_2(t) \,$

as well as their respective outputs

$y_1(t) = H \left( x_1(t) \right)$

$y_2(t) = H \left( x_2(t) \right)$

then a linear system must satisfy

$\alpha y_1(t) + \beta y_2(t) = H \left( \alpha x_1(t) + \beta x_2(t) \right)$

for any scalar values $\alpha \,$ and $\beta \,$ .

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 $x (t)$ 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.