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Linear system

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the systems theory concept. For the linear algebra concept, see <a href="System_of_linear_equations" title="System of linear equations">System of linear equations</a>. For the algebraic geometry concept, see <a href="Linear_system_of_divisors" title="Linear system of divisors">Linear system of divisors</a>. For the tactical formation, see <a href="Line_(formation)" title="Line (formation)">Line (formation)</a>.</div>

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In <a href="Systems_theory" title="Systems theory">systems theory</a>, a linear system is a <a href="Mathematical_model" title="Mathematical model">mathematical model</a> of a <a href="System" title="System">system</a> based on the use of a <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a>.

Linear systems typically exhibit features and properties that are much simpler than the <a href="Nonlinear" class="mw-redirect" title="Nonlinear">nonlinear</a> case.

As a mathematical abstraction or idealization, linear systems find important applications in <a href="Automatic_control" class="mw-redirect" title="Automatic control">automatic control</a> theory, <a href="Signal_processing" title="Signal processing">signal processing</a>, and <a href="Telecommunications" title="Telecommunications">telecommunications</a>. For example, the propagation medium for wireless communication systems can often be

modeled by linear systems.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A general <a href="Deterministic_system_(mathematics)" class="mw-redirect" title="Deterministic system (mathematics)">deterministic system</a> can be described by an operator, H, that maps an input, x(t), as a function of t to an output, y(t), a type of <a href="Black_box_(systems)" class="mw-redirect" title="Black box (systems)">black box</a> description.

A system is linear if and only if it satisfies the <a href="Superposition_principle" title="Superposition principle">superposition principle</a>, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)<a href="#cite_note-Phillips_2008-1">[1]</a><a href="#cite_note-Bessai_2005-2">[2]</a><a href="#cite_note-Alkin_2014-3">[3]</a><a href="#cite_note-Nahvi_2014-4">[4]</a>

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.<a href="#cite_note-Sundararajan_2008-5">[5]</a><a href="#cite_note-Roberts_2018-6">[6]</a>

In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor.<a href="#cite_note-Roberts_2018-6">[6]</a> 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.<a href="#cite_note-Roberts_2018-6">[6]</a>

Mathematically, for a continuous-time system, given two arbitrary inputs

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}(t)\\x_{2}(t)\end{aligned}}}</annotation>

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as well as their respective zero-state outputs

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y_{1}(t)&=H\left\{x_{1}(t)\right\}\\y_{2}(t)&=H\left\{x_{2}(t)\right\}\end{aligned}}}</annotation>

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then a linear system must satisfy

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<annotation encoding="application/x-tex">{\displaystyle \alpha y_{1}(t)+\beta y_{2}(t)=H\left\{\alpha x_{1}(t)+\beta x_{2}(t)\right\}}</annotation>

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for any <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> values α and β, for any input signals x1(t) and x2(t), and for all time t.

The system is then defined by the equation H(x(t)) = y(t), where y(t) is some arbitrary function of time, and x(t) is the system state. Given y(t) and H, the system can be solved for x(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.

For <a href="Time-invariant_system" title="Time-invariant system">time-invariant</a> systems this is the basis of the <a href="Impulse_response" title="Impulse response">impulse response</a> or the <a href="Frequency_response" title="Frequency response">frequency response</a> methods (see <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system theory</a>), which describe a general input function x(t) in terms of <a href="Unit_impulse" class="mw-redirect" title="Unit impulse">unit impulses</a> or <a href="Frequency_component" class="mw-redirect" title="Frequency component">frequency components</a>.

Typical <a href="Differential_equation" title="Differential equation">differential equations</a> of linear <a href="Time-invariant_system" title="Time-invariant system">time-invariant</a> systems are well adapted to analysis using the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> in the <a href="Continuous_function" title="Continuous function">continuous</a> case, and the <a href="Z-transform" title="Z-transform">Z-transform</a> in the <a href="Discrete_mathematics" title="Discrete mathematics">discrete</a> case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> which act like <a href="Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a> in the geometric sense.

A common use of linear models is to describe a nonlinear system by <a href="Linearization" title="Linearization">linearization</a>. 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 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {x} }_{1}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathbf {x} }_{1}(t)}</annotation>

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</math><img src="./7ee4877feb209effd7b4f556775dfaeacc88701a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.114ex; height:2.843ex;" alt="{\displaystyle {\mathbf {x} }_{1}(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {x} }_{2}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathbf {x} }_{2}(t)}</annotation>

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</math><img src="./4781d3f458ba06d3ea38c4f332d26cb1fdfd6330.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.114ex; height:2.843ex;" alt="{\displaystyle {\mathbf {x} }_{2}(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {y} }_{1}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathbf {y} }_{1}(t)}</annotation>

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</math><img src="./88aa88e66f2a4773ca918f0ffd72a3a4df2f2294.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.114ex; height:2.843ex;" alt="{\displaystyle {\mathbf {y} }_{1}(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {y} }_{2}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathbf {y} }_{2}(t)}</annotation>

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</math><img src="./361f177e88f0a3eed066a094575e7ebad17d713b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.114ex; height:2.843ex;" alt="{\displaystyle {\mathbf {y} }_{2}(t)}" loading="lazy">) are considered instead of input and output signals (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}(t)}">

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</math><img src="./edc245640b5794e3f754bee7b713761282b60c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle x_{1}(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x_{2}(t)}</annotation>

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</math><img src="./8821f9acaecc995a1b0bdfea31bb01605db7142c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle x_{2}(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y_{1}(t)}</annotation>

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</math><img src="./7b17efec23e0d4f93bac22db3e978eae43f9e9a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.842ex; height:2.843ex;" alt="{\displaystyle y_{1}(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y_{2}(t)}</annotation>

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</math><img src="./69d36d96452e41973c07684c23e8baf8308ad663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.842ex; height:2.843ex;" alt="{\displaystyle y_{2}(t)}" loading="lazy">.)<a href="#cite_note-Bessai_2005-2">[2]</a><a href="#cite_note-Nahvi_2014-4">[4]</a>

This definition of a linear system is analogous to the definition of a <a href="Linear_differential_equation" title="Linear differential equation">linear differential equation</a> in <a href="Calculus" title="Calculus">calculus</a>, and a <a href="Linear_map" title="Linear map">linear transformation</a> in <a href="Linear_algebra" title="Linear algebra">linear algebra</a>.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

A <a href="Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a> obeys the differential equation:

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If <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x(t))=m{\frac {d^{2}(x(t))}{dt^{2}}}+kx(t),}">

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<annotation encoding="application/x-tex">{\displaystyle H(x(t))=m{\frac {d^{2}(x(t))}{dt^{2}}}+kx(t),}</annotation>

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then H is a linear operator. Letting y(t) = 0, we can rewrite the differential equation as H(x(t)) = y(t), which shows that a simple harmonic oscillator is a linear system.

Other examples of linear systems include those described by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)=k\,x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)=k\,\int _{-\infty }^{t}x(\tau )\mathrm {d} \tau }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle y(t)=|x(t)|}</annotation>

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</math><img src="./9dd71fa9b03171821f3596dc80ffe25c283c093e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.175ex; height:2.843ex;" alt="{\displaystyle y(t)=|x(t)|}" loading="lazy">, 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.<a href="#cite_note-DeerghaRao_2018-7">[7]</a><a href="#cite_note-Chen_2004-8">[8]</a><a href="#cite_note-ElAliKarim_2008-9">[9]</a><a href="#cite_note-Apte_2016-10">[10]</a>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}</annotation>

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</math><img src="./f02c10807337be80f18c8f96906cf7c0d045c317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.608ex; height:5.843ex;" alt="{\displaystyle y(t)=k\,{\frac {\mathrm {d} x(t)}{\mathrm {d} t}}}" loading="lazy"> (such as a constant-capacitance <a href="Capacitor" title="Capacitor">capacitor</a> or a constant-inductance <a href="Inductor" title="Inductor">inductor</a>). 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 contain <a href="Harmonic_analysis" title="Harmonic analysis">harmonics</a> (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)=(1.5+\cos {(t)})\,x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)=(1.5+\cos {(t)})\,x(t)}</annotation>

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</math><img src="./c8d62ed5244e071eb902713543511a89ccf08f7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.037ex; height:2.843ex;" alt="{\displaystyle y(t)=(1.5+\cos {(t)})\,x(t)}" loading="lazy">. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=\cos {(3t)}}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)=\cos {(3t)}}</annotation>

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</math><img src="./16faf1229e2c267a509a901ebaf4af9fb0e3e1e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.387ex; height:2.843ex;" alt="{\displaystyle x(t)=\cos {(3t)}}" loading="lazy">, using <a href="List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities" title="List of trigonometric identities">product-to-sum trigonometric identities</a> it can be easily shown that the output is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)=1.5\cos {(3t)}+0.5\cos {(2t)}+0.5\cos {(4t)}}">

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<annotation encoding="application/x-tex">{\displaystyle y(t)=1.5\cos {(3t)}+0.5\cos {(2t)}+0.5\cos {(4t)}}</annotation>

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</math><img src="./68948e507dfa27fadb8bbcecd0db6279215424a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.589ex; height:2.843ex;" alt="{\displaystyle y(t)=1.5\cos {(3t)}+0.5\cos {(2t)}+0.5\cos {(4t)}}" loading="lazy">, 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 frequencies 2 rad/s and 4 rad/s; furthermore, taking the <a href="Least_common_multiple" title="Least common multiple">least common multiple</a> of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is 1 rad/s, which is different than that of the input.

<div class="mw-heading mw-heading2"><h2 id="Time-varying_impulse_response">Time-varying impulse response</h2></div>

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 <a href="Impulse_function" class="mw-redirect" title="Impulse function">impulse</a> applied at time t = t1. In other words, if the input x(t) to a linear system is

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<annotation encoding="application/x-tex">{\displaystyle x(t)=\delta (t-t_{1})}</annotation>

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where δ(t) represents the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>, and the corresponding response y(t) of the system is

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<annotation encoding="application/x-tex">{\displaystyle y(t=t_{2})=h(t_{2},t_{1})}</annotation>

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then the function h(t2, t1) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied:

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<annotation encoding="application/x-tex">{\displaystyle h(t_{2},t_{1})=0,t_{2}<t_{1}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="The_convolution_integral">The convolution integral</h2></div>

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:

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<annotation encoding="application/x-tex">{\displaystyle y(t)=\int _{-\infty }^{t}h(t,t')x(t')dt'=\int _{-\infty }^{\infty }h(t,t')x(t')dt'}</annotation>

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If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and h is a function only of the time difference τ = t − t' which is zero for τ < 0 (namely t < t' ). By redefinition of h it is then possible to write the input-output relation equivalently in any of the ways,

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<annotation encoding="application/x-tex">{\displaystyle y(t)=\int _{-\infty }^{t}h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau =\int _{0}^{\infty }h(\tau )x(t-\tau )d\tau }</annotation>

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Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is:

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<annotation encoding="application/x-tex">{\displaystyle H(s)=\int _{0}^{\infty }h(t)e^{-st}\,dt.}</annotation>

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In applications this is usually a rational algebraic function of s. Because h(t) is zero for negative t, the integral may equally be written over the doubly infinite range and putting s = iω follows the formula for the frequency response function:

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<annotation encoding="application/x-tex">{\displaystyle H(i\omega )=\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Discrete-time_systems">Discrete-time systems</h2></div>

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

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<annotation encoding="application/x-tex">{\displaystyle y[n]=\sum _{m=-\infty }^{n}{h[n,m]x[m]}=\sum _{m=-\infty }^{\infty }{h[n,m]x[m]}}</annotation>

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or equivalently for a time-invariant system on redefining h,

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<annotation encoding="application/x-tex">{\displaystyle y[n]=\sum _{k=0}^{\infty }{h[k]x[n-k]}=\sum _{k=-\infty }^{\infty }{h[k]x[n-k]}}</annotation>

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where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=n-m}">

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<annotation encoding="application/x-tex">{\displaystyle k=n-m}</annotation>

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</math> represents the lag time between the stimulus at time m and the response at time n.

<ul><li><a href="Shift_invariant_system" class="mw-redirect" title="Shift invariant system">Shift invariant system</a></li>

<li><a href="Linear_control" title="Linear control">Linear control</a></li>

<li><a href="Linear_time-invariant_system" title="Linear time-invariant system">Linear time-invariant system</a></li>

<li><a href="Nonlinear_system" title="Nonlinear system">Nonlinear system</a></li>

<li><a href="System_analysis" title="System analysis">System analysis</a></li>

<li><a href="System_of_linear_equations" title="System of linear equations">System of linear equations</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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