A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems. For other uses, see System (disambiguation). ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... Automatic control is the research area and theoretical base for mechanization and automation, employing methods from mathematics and engineering. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Telecommunication involves the transmission of signals over a distance for the purpose of communication. ...

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 In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. ... Black box is technical jargon for a device or system or object when it is viewed primarily in terms of its input and output characteristics. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The term scaling can have several manings: Scaling can be defined as the determination of the interdependency of variables in a physical system. ...

as well as their respective outputs

then a linear system must satisfy

for any scalar values and . In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ...

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. A time-invariant system is one whose output does not depend explicitly on time. ... The Impulse response from a simple audio system. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral...

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). Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... A time-invariant system is one whose output does not depend explicitly on time. ... In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t ≥ 0) that transforms it to a function F(s) (image) with a complex argument s. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... This article is about vectors that have a particular relation to the spatial coordinates. ...

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience. Linearization in mathematics and its applications in general refers to finding the linear approximation to a function at a given point. ...

Time-Varying Impulse Response

The time-varying impulse response h(t₂,t₁) of a linear system is defined as the response of the system at time t = t₂ to a single impulse applied at time t = t₁. In other words, if the input x(t) to a linear system is For other uses, see Impulse (disambiguation). ...

where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...

then the function h(t₂,t₁) is the time-varying impulse response of the system.

Time-Varying Convolution Integral

Continuous time

The output of any continuous time linear system is related to the input by the time-varying convolution integral:

or, equivalently,

where

represents the lag time between the stimulus at time s and the response at time t.

Discrete time

The output of any discrete time linear system is related to the input by the time-varying convolution sum:

or equivalently,

where

represents the lag time between the stimulus at time k and the response at time n.

Causality

A linear system is causal if and only if the system's time varying impulse response is identically zero whenever the time t of the response is earlier than the time s of the stimulus. In other words, for a causal system, the following condition must hold:

See also

• Linear system of divisors in algebraic geometry.
• LTI system theory
• System analysis
• System of linear equations
• Nonlinear system