Causal system: Definition from Answers.com

This article may require cleanup to meet Wikipedia's quality standards. 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 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 $t \le t_{0}$ .

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.

1 Mathematical definitions
2 Examples
3 References

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.

Examples

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

Examples of causal systems

Memoryless system

$y \left( t \right) = 1 + x \left( t \right) \cos \left( \omega t \right)$

Autoregressive filter

$y \left( t \right) = \int_0^\infty x(t-\tau) e^{-\beta\tau}\,d\tau$

Examples of non-causal (acausal) systems

$y(t)=\int_{-\infty}^{\infty } \sin (t+\tau) x(\tau)\,d\tau$

Central moving average

$y_{n}=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}$

For coefficients of t

$y \left( t \right) =x(at)$

Examples of anti-causal systems

$y(t) =\int _{0}^{\infty }\sin (t+\tau) x(\tau)\,d\tau$

$y \left( t \right) =x(-t)$

Look-ahead

y n = x n + 1

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)

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	How do you beat causality? Read answer...
	What is causal studies? Read answer...
	What is the causal organisms in swine flu? Read answer...

	What is meant the system is stable and causal?
	Is every causal system is system with memory?
	Some example of causal system on signal?

		Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Causal system". Read more

Causal system

Contents