Causal system
554087
216203099
2008-05-31T15:46:00Z
Tomer shalev
6831139
/* Examples of causal systems */
{{Cleanup|date=May 2008}}
A '''causal system''' (also known as a [[physical system|physical]] or '''nonanticipative system''') is a [[system]] where the output <math>y(t)</math> at some specific instant <math>t_{0}</math> only depends on the input <math>x(t)</math> for values of <math>t</math> less than or equal to <math>t_{0}</math>. Therefore these kinds of systems have outputs and internal states that depend only on the current and previous input values.
The idea that the output of a function at any time depends only on past and present values of input is defined by the property commonly referred to as [[causality]]. A system that has ''some'' dependence on input values from the future (in addition to possible dependence on past or current input values) is termed a non-causal or [[acausal system]], and a system that depends ''solely'' on future input values is an [[anticausal system]]. Note that some authors have defined an anticausal system as one that depends solely on future ''and present'' input values or, more simply, as a system that does not depend on past input values.
Classically, [[nature]] or physical reality has been considered to be a causal system. Physics involving [[special relativity]] or [[general relativity]] require more careful definitions of causality, as described in [[causality (physics)]].
The causality of systems also plays an important role in [[digital signal processing]], where [[LTI system theory|filters]] are often constructed so that they are causal. For more information, see [[causal filter]].
Note that the systems may be discrete or continuous. Similar rules apply to both kind of systems.
== Mathematical definitions ==
Definition 1: A system mapping <math>x</math> to <math>y</math> is causal if and only if, for any pair of input signals <math>x_{1}(t)</math> and <math>x_{2}(t)</math> such that
:<math>x_{1}(t) = x_{2}(t), \quad \forall \ t \le t_{0},</math>
the corresponding outputs satisfy
:<math>y_{1}(t) = y_{2}(t), \quad \forall \ t \le t_{0}.</math>
Definition 2: Suppose <math>h(t)</math> is the impulse response of the system <math>H</math>.
:<math>h(t) = 0, \quad \forall \ t <0 </math>
then the system <math>H</math> is causal, otherwise it is anti causal.
==Examples==
The following examples are for systems with an input <math>x</math> and output <math>y</math>.
=== Examples of causal systems ===
* Memoryless system
:<math>y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)</math>
* Autoregressive filter
:<math>y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau</math>
=== Examples of non-causal (acausal) systems ===
*
:<math>y(t)=\int_{-\infty}^{\infty } \sin (t+\tau) x(\tau)\,d\tau</math>
* Central moving average
:<math>y_{n}=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}</math>
=== Examples of anti-causal systems ===
*
:<math>y(t) =\int _{0}^{\infty }\sin (t+\tau) x(\tau)\,d\tau</math>
*Look-ahead
:<math>y_{n}=x_{n+1}</math>
== References ==
{{Unreferenced|date=April 2007}}
[[Category:Control theory]]
[[Category:Digital signal processing]]
[[Category:Systems theory]]
[[Category:Physical systems]]
[[Category:Dynamical systems]]
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