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{{About|systems as studied in systems theory|a family of linear equations|System of linear equations|the concept in algebraic geometry|Linear system of divisors}}
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{{Unreferenced|date=December 2009}}
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 general, [[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.  


A general [[deterministic system (mathematics)|deterministic system]] can be described by operator, <math>H</math>, that maps an input, <math>x(t)</math>, as a function of <math>t</math> to an output, <math>y(t)</math>, a type of [[Black box (systems)|black box]] description. Linear systems satisfy the properties of [[Superposition principle|superposition]] and [[Scaling (geometry)|scaling]] or [[Homogeneous_function|homogeneity]]. Given two valid inputs
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:<math>x_1(t) \,</math>
:<math>x_2(t) \,</math>
as well as their respective outputs
:<math>y_1(t) = H \left \{ x_1(t) \right \} </math>
:<math>y_2(t) = H \left \{ x_2(t) \right \} </math>
then a linear system must satisfy
:<math>\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 <math>\alpha \,</math> and <math>\beta \,</math>.
<!-- Insert picture depicting the superposition and scaling properties -->
 
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, x(t) can be solved for.  For example, a simple harmonic oscillator obeys the differential equation:
:<math>m \frac{d^2(x)}{dt^2} = -kx</math>
If <math>H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t)</math>, 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.
 
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)</math> in terms of unit [[Dirac's delta function|impulses]] or [[frequency components]]. 
 
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.
 
==Time-varying impulse response==
The '''time-varying impulse response''' ''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 ''t'' = ''t''<sub>1</sub>. In other words, if the input ''x''(''t'') to a linear system is
 
:<math>x(t) = \delta(t-t_1) \,</math>
 
where δ(''t'') represents the [[Dirac delta function]], and the corresponding response ''y''(''t'') of the system is
 
:<math>y(t) |_{t=t_2} = h(t_2,t_1) \,</math>
 
then the function  ''h''(''t''<sub>2</sub>,''t''<sub>1</sub>) is the time-varying impulse response of the system.
 
==Time-varying convolution integral==
=== Continuous time ===
 
The output of any continuous time linear system is related to the input by the time-varying convolution integral:
 
:<math> y(t) = \int_{-\infty}^{\infty}  h(t,s) x(s) ds </math>
 
or, equivalently,
 
:<math> y(t) = \int_{-\infty}^{\infty}  h(t,t-\tau) x(t-\tau) d \tau </math>
 
===Discrete time===
The output of any discrete time linear system is related to the input by the time-varying convolution sum:
 
:<math> y[n] = \sum_{k=-\infty}^{\infty} { h[n,k] x[k] } </math>
 
or equivalently,
 
:<math> y[n] = \sum_{m=-\infty}^{\infty} { h[n,n-m] x[n-m] } </math>
 
where
:<math> k = n-m \, </math>
 
represents the lag time between the stimulus at time ''m'' and the response at time ''n''.
 
==Causality==
A linear system is '''causal''' if and only if the system's time varying impulse response is identically zero whenever the time ''t'' of the response is earlier than the time ''s'' of the stimulus. In other words, for a causal system, the following condition must hold:
 
:<math>h(t,s) = 0\text{ for }t < s \, </math>
 
==See also==
*[[Linear system of divisors]] in [[algebraic geometry]]
*[[Shift invariant system]]
*[[LTI system theory]]
*[[Nonlinear system]]
*[[System analysis]]
*[[System of linear equations]]
 
{{DEFAULTSORT:Linear System}}
[[Category:Systems theory]]

Latest revision as of 19:52, 3 October 2014

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Landscapers have tools that you don't have naturally will have a way to conduct better job than you can. They will be which can edge your lawn, prune back anything hanging over your property whether it be within arm's reach or need a crane to get them up by the action lines, they will test your soil in order that your plants are in great condition and they will treat it if having a arises.



But as the disease progressed, he learned to humble himself and more of any receiver than a giver. But still, my mother needed to be cautious and not do everything for him as it robbed him of his self worthwhile. Being a fighter, he decided to make one of the most of a bad thing, and did his best to boost his state of mind until he died at age 79 in 1994. However, it wasn't Parkinson's' ailment that took his life since he had a sequence of strokes, losing his mobility and gradually lost fight of his life to pneumonia.

OSU posesses a dynamic offense led by potential mega-star WR Adarius Bowman but they will prefer to play awfully well november 23 Between the hedges. They actually are rrn a position to doing it, however.

The meat complex includes Live Cattle, Feeder Cattle, Lean Hogs, and the ever-popular Pork Bellies. The name, "Pork Bellies" might sound funny, it's actually frozen bacon. The meat complex has a lot of similarities amongst its men and women. However, at times their differences and situations can be very different.

Here's my analysis of what I learnt from hearing Bill Clinton in person and noting how he was introduced. You should be able to adapt at least some the hands down points to fit your own problems.

Exercise with him (or her)----Don't just tell the one you love to get some exercise. Go for it and join him. From the how my pops used merely to walk his little pint-size min pin (miniature pinscher), "Willie Boy", down their mountain road. Tony horton created a crank to leave and walks, as well as talk with my mothers. What's more, he got the possibility greet the neighbors. Linkedin profile did he get physical exercise, however, many needed socialization as skillfully.

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