 Causal system

A causal system (also known as a physical or nonanticipative system) is a system where the output depends on past/current inputs but not future inputs i.e. the output y(t_{0}) only depends on the input x(t) for values of .
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 noncausal 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 elaborately in causality (physics).
The causality of systems also plays an important role in digital signal processing, where filters are often constructed so that they are causal. For more information, see causal filter. A causal system is said to be highly stable system as outputs only depend on past history of the system and not on any future input, which means, the system is not vulnerable to future failures. For a causal system, the impulse response of the system must be 0 for all t<0. That is the sole necessary as well as sufficient condition for causality of a system, linear or nonlinear.
Note that the systems may be discrete or continuous. Similar rules apply to both kind of systems.
Contents
Mathematical definitions
Definition 1: A system mapping x to y is causal if and only if, for any pair of input signals x_{1}(t) and x_{2}(t) such that
the corresponding outputs satisfy
Definition 2: Suppose h(t) is the impulse response of the system H. (only fully accurate for a system described by linear constant coefficient differential equation)
then the system H is causal, otherwise it is noncausal.
Examples
The following examples are for systems with an input x and output y.
Examples of causal systems
 Memoryless system
 Autoregressive filter
Examples of noncausal (acausal) systems
 Central moving average
 For coefficients of t
Examples of anticausal systems
 Lookahead

 y_{n} = x_{n + 1}
References
 Oppenheim, Alan V.; Willsky, Alan S.; Nawab, Hamid; with S. Hamid (1998). Signals and Systems. Pearson Education. ISBN 0138147574.
Categories: Control theory
 Digital signal processing
 Systems theory
 Physical systems
 Dynamical systems
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