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]] [[ru:Причинная система]] [[zh:因果系统]]