Linear system
Encyclopedia
A linear system is a mathematical model of a system
System
System is a set of interacting or interdependent components forming an integrated whole....

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
Automatic control
Automatic control is the application of concepts derived from the research area of modern control theory. Automatic control is also a technology for application of control strategies. The implementing requires prior of analyzing and modeling of the subject to be controlled...

theory, signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, and telecommunications. For example, the propagation medium for wireless communication systems can often be
modeled by linear systems.

A general deterministic system
Deterministic system (mathematics)
In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...

can be described by operator, , that maps an input, , as a function of to an output, , a type of black box description. Linear systems satisfy the properties of superposition
Superposition principle
In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...

and scaling
Scaling
Scaling may refer to:* Scaling , a linear transformation that enlarges or diminishes objects* Reduced scales of semiconductor device fabrication processes...

or homogeneity. Given two valid inputs
as well as their respective outputs
then a linear system must satisfy
for any scalar
Scalar (mathematics)
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....

values and .
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
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....

systems this is the basis of the impulse response
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

or the frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

methods (see LTI system theory
LTI system theory
Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...

), which describe a general input function in terms of unit impulses or frequency components.

Typical differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s of linear time-invariant
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....

systems are well adapted to analysis using the Laplace transform in the continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

case, and the Z-transform
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

in the discrete
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...

case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s which act like vectors in the geometric sense.

A common use of linear models is to describe a nonlinear system by linearization
Linearization
In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...

. This is usually done for mathematical convenience.

## 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

where δ(t) represents the Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

, and the corresponding response y(t) of the system is

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

### Continuous time

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

or, equivalently,

### 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 m 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:

• Linear system of divisors
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....

in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

• Shift invariant system
Shift invariant system
Shift invariant system is the discrete equivalent of the time-invariant system; defined such that if y is the response of the system to x, then y is the response of the system to x.-Applications:...

• LTI system theory
LTI system theory
Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...

• Nonlinear system
• System analysis
System analysis
System analysis in the field of electrical engineering characterizes electrical systems and their properties. System Analysis can be used to represent almost anything from population growth to audio speakers, electrical engineers often use it because of its direct relevance to many areas of their...

• System of linear equations