Editing Linear system

{{Short description|Physical system satisfying the superposition principle}}
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In [[systems theory]], 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 [[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.

==Definition==
[[File:Additivity property block diagram for a SISO system.png|thumb|Block diagram illustrating the additivity property for a deterministic continuous-time SISO system. The system satisfies the additivity property or is additive if and only if <math>y_3(t) = y_1(t) + y_2(t)</math> for all time <math>t</math> and for all inputs <math>x_1(t)</math> and <math>x_2(t)</math>. Click image to expand it.]]
[[File:Homogeneity property block diagram for a SISO system.png|thumb|Block diagram illustrating the homogeneity property for a deterministic continuous-time SISO system. The system satisfies the homogeneity property or is homogeneous if and only if <math>y_2(t) = a \, y_1(t)</math> for all time <math>t</math>, for all real constant <math>a</math> and for all input <math>x_1(t)</math>. Click image to expand it.]]
[[File:Superposition principle block diagram for a SISO system.png|thumb|Block diagram illustrating the superposition principle for a deterministic continuous-time SISO system. The system satisfies the superposition principle and is thus linear if and only if <math>y_3(t) = a_1 \, y_1(t) + a_2 \, y_2(t)</math> for all time <math>t</math>, for all real constants <math>a_1</math> and <math>a_2</math> and for all inputs <math>x_1(t)</math> and <math>x_2(t)</math>. Click image to expand it.]]
A general [[deterministic system (mathematics)|deterministic system]] can be described by an operator, {{math|''H''}}, that maps an input, {{math|''x''(''t'')}}, as a function of {{mvar|t}} to an output, {{math|''y''(''t'')}}, a type of [[Black box (systems)|black box]] description.

A system is linear if and only if it satisfies the [[superposition principle]], or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)<ref name="Phillips_2008">{{cite book | title = Signals, Systems, and Transforms | edition = 4 | first1 = Charles L. | last1 = Phillips | first2 = John M. | last2 = Parr | first3 = Eve A. | last3 = Riskin|author3-link=Eve Riskin | publisher = Pearson | year = 2008 | page = 74 | isbn = 978-0-13-198923-8}}</ref><ref name="Bessai_2005">{{cite book | title = MIMO Signals and Systems | first = Horst J. | last = Bessai | publisher = Springer | year = 2005 | pages = 27–28 | isbn = 0-387-23488-8}}</ref><ref name="Alkin_2014">{{cite book | title = Signals and Systems: A MATLAB Integrated Approach | first = Oktay | last = Alkin | publisher = CRC Press | year = 2014 | page = 99 | isbn = 978-1-4665-9854-6}}</ref><ref name="Nahvi_2014">{{cite book | title = Signals and Systems | first = Mahmood | last = Nahvi | publisher = McGraw-Hill | year = 2014 | pages = 162–164, 166, 183 | isbn = 978-0-07-338070-4}}</ref>

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.<ref name="Sundararajan_2008">{{cite book | title = A Practical Approach to Signals and Systems | first = D. | last = Sundararajan | publisher = Wiley | year = 2008 | page = 80 | isbn = 978-0-470-82353-8}}</ref><ref name="Roberts_2018">{{cite book | title = Signals and Systems: Analysis Using Transform Methods and MATLAB® | edition = 3 | first = Michael J. | last = Roberts | publisher = McGraw-Hill | year = 2018 | pages = 131, 133–134 | isbn = 978-0-07-802812-0}}</ref>

In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor.<ref name="Roberts_2018" /> 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.<ref name="Roberts_2018" />

Mathematically, for a continuous-time system, given two arbitrary inputs
<math display="block">\begin{align} x_1(t) \\ x_2(t) \end{align}</math>
as well as their respective zero-state outputs
<math display="block">\begin{align}
y_1(t) &= H \left \{ x_1(t) \right \} \\
y_2(t) &= H \left \{ x_2(t) \right \}
\end{align} </math>
then a linear system must satisfy
<math display="block">\alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math>
for any [[scalar (mathematics)|scalar]] values {{mvar|α}} and {{mvar|β}}, for any input signals {{math|''x''<sub>1</sub>(''t'')}} and {{math|''x''<sub>2</sub>(''t'')}}, and for all time {{mvar|t}}.

The system is then defined by the equation {{math|1=''H''(''x''(''t'')) = ''y''(''t'')}}, where {{math|''y''(''t'')}} is some arbitrary function of time, and {{math|''x''(''t'')}} is the system state.  Given {{math|''y''(''t'')}} and {{nowrap|{{math|''H''}},}} the system can be solved for {{nowrap|{{math|''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 [[time-invariant system|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 {{math|''x''(''t'')}} in terms of [[unit impulse]]s or [[frequency component]]s.

Typical [[differential equation]]s of linear [[time-invariant system|time-invariant]] systems are well adapted to analysis using the [[Laplace transform]] in the [[continuous function|continuous]] case, and the [[Z-transform]] in the [[discrete mathematics|discrete]] case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of [[function (mathematics)|function]]s which act like [[vector (geometric)|vector]]s in the geometric sense.

A common use of linear models is to describe a nonlinear system by [[linearization]].  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>{\mathbf x}_1(t)</math>, <math>{\mathbf x}_2(t)</math>, <math>{\mathbf y}_1(t)</math>, <math>{\mathbf y}_2(t)</math>) are considered instead of input and output signals (<math>x_1(t)</math>, <math>x_2(t)</math>, <math>y_1(t)</math>, <math>y_2(t)</math>.)<ref name="Bessai_2005" /><ref name="Nahvi_2014" />

This definition of a linear system is analogous to the definition of a [[linear differential equation]] in [[calculus]], and a [[Linear map|linear transformation]] in [[linear algebra]].

===Examples===
A [[simple harmonic oscillator]] obeys the differential equation:
<math display="block">m \frac{d^2(x)}{dt^2} = -kx.</math>

If <math display="block">H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t),</math>
then {{math|''H''}} is a linear operator.  Letting {{nowrap|{{math|1=''y''(''t'') = 0}},}} we can rewrite the differential equation as {{nowrap|{{math|1=''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>y(t) = k \, x(t)</math>, <math>y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt}</math>, <math>y(t) = k \, \int_{-\infty}^{t}x(\tau) \mathrm d\tau</math>, and any system described by ordinary linear differential equations.<ref name="Nahvi_2014" /> Systems described by <math>y(t) = k</math>, <math>y(t) = k \, x(t) + k_0</math>, <math>y(t) = \sin{[x(t)]}</math>, <math>y(t) = \cos{[x(t)]}</math>, <math>y(t) = x^2(t)</math>, <math display="inline">y(t) = \sqrt{x(t)}</math>, <math>y(t) = |x(t)|</math>, 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.<ref name="DeerghaRao_2018">{{cite book | title = Signals and Systems | first = K. | last = Deergha Rao | publisher = Springer | year = 2018 | pages = 43–44 | isbn = 978-3-319-68674-5}}</ref><ref name="Chen_2004">{{cite book | title = Signals and systems | edition = 3 | first = Chi-Tsong | last = Chen | publisher = Oxford University Press | year = 2004 | pages = 55–57 | isbn = 0-19-515661-7}}</ref><ref name="ElAliKarim_2008">{{cite book | title = Continuous Signals and Systems with MATLAB | edition = 2 | first1 = Taan S. | last1 = ElAli | first2 = Mohammad A. | last2 = Karim | publisher = CRC Press | year = 2008 | page = 53 | isbn = 978-1-4200-5475-0}}</ref><ref name="Apte_2016">{{cite book | title = Signals and Systems: Principles and Applications | first = Shaila Dinkar | last = Apte | publisher = Cambridge University Press | year = 2016 | page = 187 | isbn = 978-1-107-14624-2}}</ref>

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>y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt}</math> (such as a constant-capacitance [[capacitor]] or a constant-inductance [[inductor]]). 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 [[Harmonic analysis|harmonics]] (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by <math>y(t) = (1.5 + \cos{(t)}) \, x(t)</math>. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form <math>x(t) = \cos{(3t)}</math>, using [[List of trigonometric identities#Product-to-sum and sum-to-product identities|product-to-sum trigonometric identities]] it can be easily shown that the output is <math>y(t) = 1.5 \cos{(3t)} + 0.5 \cos{(2t)} + 0.5 \cos{(4t)}</math>, that is, the output doesn't consist only of sinusoids of same frequency as the input ({{nowrap|3 rad/s}}), but instead also of sinusoids of frequencies {{nowrap|2 rad/s}} and {{nowrap|4 rad/s}}; furthermore, taking the [[least common multiple]] of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is {{nowrap|1 rad/s}}, which is different than that of the input.

==Time-varying impulse response==
The '''time-varying impulse response''' {{math|''h''(''t''<sub>2</sub>, ''t''<sub>1</sub>)}} of a linear system is defined as the response of the system at time ''t'' = ''t''<sub>2</sub> to a single [[impulse function|impulse]] applied at time {{nowrap|{{math|1=''t'' = ''t''<sub>1</sub>}}.}}  In other words, if the input {{math|''x''(''t'')}} to a linear system is 
<math display="block">x(t) = \delta(t - t_1)</math>
where {{math|δ(''t'')}} represents the [[Dirac delta function]], and the corresponding response {{math|''y''(''t'')}} of the system is
<math display="block">y(t=t_2) = h(t_2, t_1)</math>
then the function {{math|''h''(''t''<sub>2</sub>, ''t''<sub>1</sub>)}} 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:
<math display="block"> h(t_2, t_1) = 0, t_2 < t_1</math>

==The convolution integral==

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:
<math display="block"> y(t) = \int_{-\infty}^{t}  h(t,t') x(t')dt' = \int_{-\infty}^{\infty}  h(t,t') x(t') dt' </math>

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 {{mvar|h}} is a function only of the time difference {{math|1=''τ'' = ''t'' − ''t' ''}} which is zero for {{math|''τ'' < 0}} (namely {{math|''t'' < ''t' ''}}). By redefinition of {{mvar|h}} it is then possible to write the input-output relation equivalently in any of the ways,
<math display="block"> 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 </math>

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called  the ''transfer function'' which is:
<math display="block">H(s) =\int_0^\infty  h(t) e^{-st}\, dt.</math>

In applications this is usually a rational algebraic function of {{mvar|s}}. Because {{math|''h''(''t'')}} is zero for negative {{mvar|t}}, the integral may  equally be written over the doubly infinite range and putting {{math|1=''s'' = ''iω''}} follows the formula for the ''frequency response function'':
<math display="block"> H(i\omega) = \int_{-\infty}^{\infty}  h(t) e^{-i\omega t} dt </math>

==Discrete-time systems==
The output of any discrete time linear system is related to the input by the time-varying convolution sum:
<math display="block"> y[n] = \sum_{m =-\infty}^{n} { h[n,m] x[m] }  = \sum_{m =-\infty}^{\infty} { h[n,m] x[m] }</math>
or equivalently for a time-invariant system on redefining {{math|''h''}},
<math display="block"> y[n] = \sum_{k =0}^{\infty} { h[k] x[n-k] } = \sum_{k =-\infty}^{\infty} { h[k] x[n-k] }</math>
where <math display="block"> k = n-m </math> represents the lag time between the stimulus at time ''m'' and the response at time ''n''.

==See also==
*[[Shift invariant system]]
*[[Linear control]]
*[[Linear time-invariant system]]
*[[Nonlinear system]]
*[[System analysis]]
*[[System of linear equations]]

==References==
{{Reflist}}

{{DEFAULTSORT:Linear System}}
[[Category:Systems theory]]
[[Category:Dynamical systems]]
[[Category:Mathematical modeling]]
[[Category:Concepts in physics]]