Causal system

In control theory, a causal system (also known as a physical or nonanticipative system) is a system where the output depends on past and 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 elaborately in causality (physics).

The causality of systems also plays an important role in digital signal processing, where 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 $t<0$. 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.

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 $H$. (only fully accurate for a system described by linear constant coefficient differential equation). The system $H$ is causal if and only if

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

otherwise it is non-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

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

Examples of anti-causal systems

$y(t) =\int _0^\infty \sin (t+\tau) x(\tau)\,d\tau$

• Look-ahead

$y_n=x_{n+1}$

References

• Oppenheim, Alan V.; Willsky, Alan S.; Nawab, Hamid; with S. Hamid (1998). Signals and Systems. Pearson Education. ISBN 0-13-814757-4.