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 formula_1 depends on only the input formula_2 for values of formula_3.
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 use only the present and past values of the input to determine the output. This requirement is a necessary and sufficient condition for a system to be causal, regardless of linearity. Note that similar rules apply to either discrete or continuous cases. By this definition of requiring no future input values, systems must be causal to process signals in real time.
Mathematical definitions.
Definition 1: A system mapping formula_4 to formula_5 is causal if and only if, for any pair of input signals formula_6, formula_7 and any choice of formula_8, such that
the corresponding outputs satisfy
Definition 2: Suppose formula_11 is the impulse response of any system formula_12 described by a linear constant coefficient differential equation. The system formula_12 is causal if and only if
otherwise it is non-causal.
Examples.
The following examples are for systems with an input formula_4 and output formula_5.