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 <math>H</math> that maps an input <math>x(t)</math> as a function of <math>t</math> to an output <math>y(t)</math>, a type of black box description. Linear systems satisfy the properties of superposition and scaling: given two valid inputs
- <math>x_1(t) \,</math>
- <math>x_2(t) \,</math>
as well as their respective outputs
- <math>y_1(t) = H \left( x_1(t) \right)</math>
- <math>y_2(t) = H \left( x_2(t) \right)</math>
then a linear system must satisfy
- <math>\alpha y_1(t) + \beta y_2(t) = H \left( \alpha x_1(t) + \beta x_2(t) \right)</math>
for any scalar values <math>\alpha \,</math> and <math>\beta \,</math>.
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 <math>x(t)</math> 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.
See also
- Linear system of divisors in algebraic geometry.
- LTI system theory
- System analysis
- System of linear equations
de:Lineares System (Systemtheorie) fr:Système linéaire he:מערכת לינארית