Editing Linear time-invariant system (section)

==Overview==
The defining properties of any LTI system are ''linearity'' and ''time invariance''.
* ''Linearity'' means that the relationship between the input <math>x(t)</math> and the output <math>y(t)</math>, both being regarded as functions, is a linear mapping: If <math>a</math> is a constant then the system output to <math>ax(t)</math> is <math>ay(t)</math>; if <math>x'(t)</math> is a further input with system output <math>y'(t)</math> then the output of the system to <math>x(t)+x'(t)</math> is <math>y(t)+y'(t)</math>, this applying for all choices of <math>a</math>'','' ''<math>x(t)</math>'', <math>x'(t)</math>. The latter condition is often referred to as the [[superposition principle]].
* ''Time invariance'' means that whether we apply an input to the system now or ''T'' seconds from now, the output will be identical except for a time delay of ''T'' seconds. That is, if the output due to input <math>x(t)</math> is <math>y(t)</math>, then the output due to input <math>x(t-T)</math> is <math>y(t-T)</math>. Hence, the system is time invariant because the output does not depend on the particular time the input is applied.<ref>{{Cite book |last1=Phillips |first1=Charles L. |title=Signals, systems, and transforms |last2=Parr |first2=John M. |last3=Riskin |first3=Eve A. |date=2003 |publisher=Prentice Hall |isbn=978-0-13-041207-2 |edition=3rd |location=Upper Saddle River, N.J |pages=89}}</ref>

Through these properties, it is reasoned that LTI systems can be characterized entirely by a single function called the system's [[impulse response]], as, by superposition, any arbitrary signal can be expressed as a superposition of time-shifted [[Unit impulse|impulses]]. The output of the system <math>y(t)</math> is simply the [[convolution]] of the input to the system <math>x(t)</math> with the system's impulse response <math>h(t)</math>. This is called a [[continuous time]] system. Similarly, a discrete-time linear time-invariant (or, more generally, "shift-invariant") system is defined as one operating in [[discrete time]]: <math>y_{i} = x_{i} * h_{i}</math>  where ''y'', ''x'', and ''h'' are [[sequences]] and the convolution, in discrete time, uses a discrete summation rather than an integral.<ref>{{Cite book |last1=Phillips |first1=Charles L. |title=Signals, systems, and transforms |last2=Parr |first2=John M. |last3=Riskin |first3=Eve A. |date=2003 |publisher=Pearson Education |isbn=978-0-13-041207-2 |edition=3rd |location=Upper Saddle River, N.J |pages=92}}</ref>

[[File:LTI.png|thumb|Relationship between the '''time domain''' and the '''frequency domain'''|right|320px]]

LTI systems can also be characterized in the ''[[frequency domain]]'' by the system's [[transfer function]], which is the [[Laplace transform]] of the system's impulse response (or [[Z transform]] in the case of discrete-time systems).  As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the transform of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain.

For all LTI systems, the [[eigenfunction]]s, and the basis functions of the transforms, are [[complex number|complex]] [[exponential function|exponentials]].  This is, if the input to a system is the complex waveform <math>A_s e^{st}</math> for some complex amplitude <math>A_s</math> and complex frequency <math>s</math>, the output will be some complex constant times the input, say <math>B_s e^{st}</math> for some new complex amplitude <math>B_s</math>.  The ratio <math>B_s/A_s</math> is the transfer function at frequency <math>s</math>.

Since [[sine wave|sinusoids]] are a sum of complex exponentials with complex-conjugate frequencies, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different [[amplitude]] and a different [[phase (waves)|phase]], but always with the same frequency upon reaching steady-state. LTI systems cannot produce frequency components that are not in the input.

LTI system theory is good at describing many important systems.  Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or [[nonlinear]] case.  Any system that can be modeled as a linear [[differential equation]] with constant coefficients is an LTI system.  Examples of such systems are [[electrical network|electrical circuits]] made up of [[resistor]]s, [[inductor]]s, and [[capacitor]]s (RLC circuits).  Ideal spring–mass–damper systems are also LTI systems, and are mathematically equivalent to RLC circuits.

Most LTI system concepts are similar between the continuous-time and discrete-time (linear shift-invariant) cases.  In image processing, the time variable is replaced with two space variables, and the notion of time invariance is replaced by two-dimensional shift invariance.  When analyzing [[filter bank]]s and [[MIMO (systems theory)|MIMO]] systems, it is often useful to consider [[matrix (mathematics)|vectors]] of signals.

A linear system that is not time-invariant can be solved using other approaches such as the [[Green's function|Green function]] method.