Causal system

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Please improve this article if you can. (May 2008)

A causal system (also known as a physical or nonanticipative system) is a system where the output y(t) at some specific instant t0 only depends on the input x(t) for values of t less than or equal to t0. 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 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.

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[edit] Mathematical definitions

Definition 1: A system mapping x to y is causal if and only if, for any pair of input signals x1(t) and x2(t) such that

x_{1}(t) = x_{2}(t), \quad \forall \ t \le t_{0},

the corresponding outputs satisfy

y_{1}(t) = y_{2}(t), \quad \forall \ t \le t_{0}.

Definition 2: Suppose h(t) is the impulse response of the system H.

h(t) = 0, \quad \forall \ t <0

then the system H is causal, otherwise it is anti causal.

[edit] Examples

The following examples are for systems with an input x and output y.

[edit] Examples of causal systems

  • Memoryless system
y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)
  • Autoregressive filter
y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau

[edit] Examples of non-causal (acausal) systems

y(t)=\int_{-\infty}^{\infty } \sin (t+\tau) x(\tau)\,d\tau
  • Central moving average
y_{n}=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}

[edit] Examples of anti-causal systems

y(t) =\int _{0}^{\infty }\sin (t+\tau) x(\tau)\,d\tau
  • Look-ahead
yn = xn + 1

[edit] References

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