Causal system

From Wikipedia, the free encyclopedia

A causal system (also known as a physical or nonanticipative system) is a system where the output $y (t)$ at some specific instant $t 0$ only depends on the input $x (t)$ for values of $t$ less than or equal to $t 0$ . 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.

[edit] 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

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

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

y n = x n + 1

[edit] References


	This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2007)

Causal system

From Wikipedia, the free encyclopedia

Contents

[edit] Mathematical definitions

[edit] Examples

[edit] Examples of causal systems

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

[edit] Examples of anti-causal systems

[edit] References

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