en>Dexbot: Bot: Fixing broken section link

2014-10-03T17:52:38Z

Bot: Fixing broken section link

en>Hnapel: (single) DES is no longer considered secure due to the advancement in (cheap) hardware.

2013-09-20T09:32:10Z

(single) DES is no longer considered secure due to the advancement in (cheap) hardware.

← Older revision		Revision as of 11:32, 20 September 2013
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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}}
			{{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
			:<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 -->

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			:<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]]

en>Bedelato: "key bits" led me to expect binary instead of hex; reworded

2012-07-29T00:41:15Z

"key bits" led me to expect binary instead of hex; reworded

New page

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Weak key - Revision history

en>Dexbot: Bot: Fixing broken section link

en>Hnapel: (single) DES is no longer considered secure due to the advancement in (cheap) hardware.

en>Bedelato: "key bits" led me to expect binary instead of hex; reworded