Linear system

A linear system is a model of a system based on some kind of linear operator.

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Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case.

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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

Related Topics:
Deterministic system - Black box - Superposition - Scaling

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::x_1(t) ,

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::x_2(t) ,

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as well as their respective outputs

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::y_1(t) = H left( x_1(t) ight)

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::y_2(t) = H left( x_2(t) ight)

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then a linear system must satisfy

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::lpha y_1(t) + eta y_2(t) = H left( lpha x_1(t) + eta x_2(t) ight)

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for any scalar values lpha , and eta ,.

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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.

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This mathematical property makes the solution of modelling equations simpler than many nonlinear systems.

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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.

Related Topics:
Time-invariant - Impulse response - Frequency response - LTI system theory - Impulses - Frequency components

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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).

Related Topics:
Differential equation - Time-invariant - Laplace transform - Continuous - Z-transform - Discrete

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

Related Topics:
Function - Vector

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

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