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Insystems theory, alinear system is amathematical model of asystem based on the use of alinear operator.Linear systems typically exhibit features and properties that are much simpler than thenonlinear case.As a mathematical abstraction or idealization, linear systems find important applications inautomatic control theory,signal processing, andtelecommunications. For example, the propagation medium for wireless communication systems can often bemodeled by linear systems.

A generaldeterministic system can be described by an operator,H, that maps an input,x(t), as a function oft to an output,y(t), a type ofblack box description.

A system is linear if and only if it satisfies thesuperposition principle, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)^[1][2][3][4]

The superposition principle means that a linear combination of inputs to the system produces a linear combination of the individual zero-state outputs (that is, outputs setting the initial conditions to zero) corresponding to the individual inputs.^[5][6]

In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor.^[6] In a system that satisfies the additivity property, adding two inputs always results in adding the corresponding two zero-state responses due to the individual inputs.^[6]

Mathematically, for a continuous-time system, given two arbitrary inputs$${\displaystyle \begin{array}{r}{x}_1(t)\\ x_2(t)\end{array}}$$as well as their respective zero-state outputsthen a linear system must satisfy$${\displaystyle \alpha y_1(t)+\beta y_2(t)=H\left\{\alpha x_1(t)+\beta x_2(t)\right\}}$$for anyscalar valuesα andβ, for any input signalsx₁(t) andx₂(t), and for all timet.

The system is then defined by the equationH(x(t)) =y(t), wherey(t) is some arbitrary function of time, andx(t) is the system state. Giveny(t) andH, the system can be solved forx(t).

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.Fortime-invariant systems this is the basis of theimpulse response or thefrequency response methods (seeLTI system theory), which describe a general input functionx(t) in terms ofunit impulses orfrequency components.

Typicaldifferential equations of lineartime-invariant systems are well adapted to analysis using theLaplace transform in thecontinuous case, and theZ-transform in thediscrete case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system offunctions which act likevectors in the geometric sense.

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

The previous definition of a linear system is applicable to SISO (single-input single-output) systems. For MIMO (multiple-input multiple-output) systems, input and output signal vectors (${\displaystyle {\mathbf{x}}_1(t)}$,${\displaystyle {\mathbf{x}}_2(t)}$,${\displaystyle {\mathbf{y}}_1(t)}$,${\displaystyle {\mathbf{y}}_2(t)}$) are considered instead of input and output signals (${\displaystyle x_1(t)}$,${\displaystyle x_2(t)}$,${\displaystyle y_1(t)}$,${\displaystyle y_2(t)}$.)^[2][4]

This definition of a linear system is analogous to the definition of alinear differential equation incalculus, and alinear transformation inlinear algebra.

Asimple harmonic oscillator obeys the differential equation:$${\displaystyle m\frac{d^2(x)}{d{t}^2}=-kx.}$$

If$${\displaystyle H(x(t))=m\frac{d^2(x(t))}{d{t}^2}+kx(t),}$$thenH is a linear operator. Lettingy(t) = 0, we can rewrite the differential equation asH(x(t)) =y(t), which shows that a simple harmonic oscillator is a linear system.

Other examples of linear systems include those described by${\displaystyle y(t)=kx(t)}$,${\displaystyle y(t)=k\frac{\mathrm{d}x(t)}{\mathrm{d}t}}$,${\displaystyle y(t)=k{\int }_{-\mathrm{\infty }}^{t}x(\tau )\mathrm{d}\tau }$, and any system described by ordinary linear differential equations.^[4] Systems described by${\displaystyle y(t)=k}$,${\displaystyle y(t)=kx(t)+k_0}$,${\displaystyle y(t)=\sin [x(t)]}$,${\displaystyle y(t)=\cos [x(t)]}$,${\displaystyle y(t)=x^2(t)}$,$y(t)=\sqrt{x(t)}$,${\displaystyle y(t)=|x(t)|}$, and a system with odd-symmetry output consisting of a linear region and a saturation (constant) region, are non-linear because they don't always satisfy the superposition principle.^{[7][8][9][10]}

The output versus input graph of a linear system need not be a straight line through the origin. For example, consider a system described by${\displaystyle y(t)=k\frac{\mathrm{d}x(t)}{\mathrm{d}t}}$ (such as a constant-capacitancecapacitor or a constant-inductanceinductor). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid, the output is also a sinusoid, and so its output-input plot is an ellipse centered at the origin rather than a straight line passing through the origin.

Also, the output of a linear system can containharmonics (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by${\displaystyle y(t)=(1.5+\cos (t))x(t)}$. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form${\displaystyle x(t)=\cos (3t)}$, usingproduct-to-sum trigonometric identities it can be easily shown that the output is${\displaystyle y(t)=1.5\cos (3t)+0.5\cos (2t)+0.5\cos (4t)}$, that is, the output doesn't consist only of sinusoids of same frequency as the input (3 rad/s), but instead also of sinusoids of frequencies2 rad/s and4 rad/s; furthermore, taking theleast common multiple of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is1 rad/s, which is different than that of the input.

Thetime-varying impulse responseh(t₂,t₁) of a linear system is defined as the response of the system at timet =t₂ to a singleimpulse applied at timet =t₁. In other words, if the inputx(t) to a linear system is$${\displaystyle x(t)=\delta (t-t_1)}$$whereδ(t) represents theDirac delta function, and the corresponding responsey(t) of the system is$${\displaystyle y(t=t_2)=h(t_2,t_1)}$$then the functionh(t₂,t₁) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the followingcausality condition must be satisfied:

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:$${\displaystyle y(t)={\int }_{-\mathrm{\infty }}^{t}h(t,t^{\prime })x(t^{\prime })d{t}^{\prime }={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}h(t,t^{\prime })x(t^{\prime })d{t}^{\prime }}$$

If the properties of the system do not depend on the time at which it is operated then it is said to betime-invariant andh is a function only of the time differenceτ =t −t' which is zero forτ < 0 (namelyt <t'). By redefinition ofh it is then possible to write the input-output relation equivalently in any of the ways,$${\displaystyle y(t)={\int }_{-\mathrm{\infty }}^{t}h(t-t^{\prime })x(t^{\prime })d{t}^{\prime }={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}h(t-t^{\prime })x(t^{\prime })d{t}^{\prime }={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}h(\tau )x(t-\tau )d\tau ={\int }_0^{\mathrm{\infty }}h(\tau )x(t-\tau )d\tau }$$

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called thetransfer function which is:$${\displaystyle H(s)={\int }_0^{\mathrm{\infty }}h(t)e^{-st}dt.}$$

In applications this is usually a rational algebraic function ofs. Becauseh(t) is zero for negativet, the integral may equally be written over the doubly infinite range and puttings =iω follows the formula for thefrequency response function:$${\displaystyle H(i\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}h(t)e^{-i\omega t}dt}$$

The output of any discrete time linear system is related to the input by the time-varying convolution sum:$${\displaystyle y[n]=\sum _{m=-\mathrm{\infty }}^{n}h[n,m]x[m]=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}h[n,m]x[m]}$$or equivalently for a time-invariant system on redefiningh,$${\displaystyle y[n]=\sum _{k=0}^{\mathrm{\infty }}h[k]x[n-k]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}h[k]x[n-k]}$$where$${\displaystyle k=n-m}$$ represents the lag time between the stimulus at timem and the response at timen.

• Shift invariant system
• Linear control
• Linear time-invariant system
• Nonlinear system
• System analysis
• System of linear equations

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