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| A '''causal system''' (also known as a [[physical system|physical]] or '''nonanticipative system''') is a [[system]] where the output depends on past and
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| current inputs but not future inputs i.e. the output <math> y(t_{0})</math> only depends on the input <math>x(t)</math> for values of <math>t \le t_{0}</math>.
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| 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.
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| 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 elaborately in [[causality (physics)]].
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| The causality of systems also plays an important role in [[digital signal processing]], where [[LTI system theory|filters]] are constructed so that they are causal, sometimes by altering a non-causal formulation to remove the lack of causality so that it is realizable. For more information, see [[causal filter]]. For a causal system, the [[impulse response]] of the system must be 0 for all <math>t<0</math>. That is the sole necessary as well as sufficient condition for causality of a system, linear or non-linear. Note that similar rules apply to either discrete or continuous cases.
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| == Mathematical definitions ==
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| 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
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| :<math>x_{1}(t) = x_{2}(t), \quad \forall \ t \le t_{0},</math> | |
| the corresponding outputs satisfy
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| :<math>y_{1}(t) = y_{2}(t), \quad \forall \ t \le t_{0}.</math> | |
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| Definition 2: Suppose <math>h(t)</math> is the impulse response of the system <math>H</math>. (only fully accurate for a system described by linear constant coefficient differential equation). The system <math>H</math> is causal if and only if
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| :<math>h(t) = 0, \quad \forall \ t <0 </math> | |
| otherwise it is non-causal.
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| ==Examples==
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| The following examples are for systems with an input <math>x</math> and output <math>y</math>.
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| === Examples of causal systems ===
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| * Memoryless system
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| ::<math>y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)</math>
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| * Autoregressive filter
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| ::<math>y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau</math>
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| === Examples of non-causal (acausal) systems ===
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| *
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| ::<math>y(t)=\int_{-\infty}^\infty \sin (t+\tau) x(\tau)\,d\tau</math>
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| * Central moving average
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| ::<math>y_n=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}</math>
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| * For coefficients of t
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| ::<math>y \left( t \right) =x(at)</math>
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| === Examples of anti-causal systems ===
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| *
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| ::<math>y(t) =\int _0^\infty \sin (t+\tau) x(\tau)\,d\tau</math>
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| *Look-ahead
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| ::<math>y_n=x_{n+1}</math>
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| == References ==
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| * {{cite book | author=Oppenheim, Alan V.; Willsky, Alan S.; Nawab, Hamid; with S. Hamid | title=Signals and Systems | publisher=Pearson Education | year=1998 | isbn=0-13-814757-4}}
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| [[Category:Control theory]]
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| [[Category:Digital signal processing]]
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| [[Category:Systems theory]]
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| [[Category:Physical systems]]
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| [[Category:Dynamical systems]]
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