Editing Linear time-invariant system (section)

{{Short description|Mathematical model which is both linear and time-invariant}}
{{More footnotes needed|date=April 2009}}

[[File:Superposition principle and time invariance block diagram for a SISO system.png|320px|thumb|[[Block diagram]] illustrating the [[superposition principle]] and time invariance for a deterministic continuous-time single-input single-output system. The system satisfies the [[superposition principle]] and is time-invariant if and only if {{math|1=''y''{{sub|3}}(''t'') = ''a''{{sub|1}}''y''{{sub|1}}(''t'' – ''t''{{sub|0}}) + ''a''{{sub|2}}''y''{{sub|2}}(''t'' – ''t''{{sub|0}})}} for all time {{mvar|t}}, for all real constants {{math|''a''{{sub|1}}, ''a''{{sub|2}}, ''t''{{sub|0}}}} and for all inputs {{math|''x''{{sub|1}}(''t''), ''x''{{sub|2}}(''t'')}}.<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> Click image to expand it.]]

In [[system analysis]], among other fields of study, a '''linear time-invariant''' ('''LTI''') '''system''' is a [[system]] that produces an output signal from any input signal subject to the constraints of [[Linear system#Definition|linearity]] and [[Time-invariant system|time-invariance]]; these terms are briefly defined in the overview below. These properties apply (exactly or approximately) to many important physical systems, in which case the response {{math|''y''(''t'')}} of the system to an arbitrary input {{math|''x''(''t'')}} can be found directly using [[convolution]]: {{math|1=''y''(''t'') = (''x'' ∗ ''h'')(''t'')}} where {{math|''h''(''t'')}} is called the system's [[impulse response]] and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining {{math|''h''(''t'')}}), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any [[electrical circuit]] consisting of [[resistor]]s, [[capacitor]]s, [[inductor]]s and [[linear amplifier]]s.<ref>Hespanha 2009, p. 78.</ref>

Linear time-invariant system theory is also used in [[image processing]], where the systems have spatial dimensions instead of, or in addition to, a temporal dimension. These systems may be referred to as ''linear translation-invariant'' to give the terminology the most general reach. In the case of generic [[discrete-time]] (i.e., [[sample (signal)|sampled]]) systems, ''linear shift-invariant'' is the corresponding term. LTI system theory is an area of [[applied mathematics]] which has direct applications in [[Network analysis (electrical circuits)|electrical circuit analysis and design]], [[signal processing]] and [[filter design]], [[control theory]], [[mechanical engineering]], [[image processing]], the design of [[measuring instrument]]s of many sorts, [[NMR spectroscopy]]{{Citation needed|date=September 2020}}, and many other technical areas where systems of [[ordinary differential equation]]s present themselves.