Causal system
Encyclopedia
A causal system is a system
where the output depends on past/current inputs but not future inputs i.e. the output
only depends on the input 
for values of 
.
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 often constructed so that they are causal. For more information, see causal filter
. A causal system is said to be highly stable system as outputs only depend on past history of the system and not on any future input, which means, the system is not vulnerable to future failures.
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 the systems may be discrete or continuous. Similar rules apply to both kind of systems.
to 
is causal if and only if, for any pair of input signals 
and 
such that
the corresponding outputs satisfy
Definition 2: Suppose
is the impulse response of the system 
. (only fully accurate for a system described by linear constant coefficient differential equation)
then the system
is causal, otherwise it is non-causal.
and output 
.
System
System is a set of interacting or interdependent components forming an integrated whole....
where the output depends on past/current inputs but not future inputs i.e. the output
only depends on the input for values of .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
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
. 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
Anticausal system
An anticausal system is a hypothetical system with outputs and internal states that depend solely on future input values. Some textbooks and published research literature might define an anticausal system to be one that does not depend on past input values An anticausal system is a hypothetical...
. 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
Nature
Nature, in the broadest sense, is equivalent to the natural world, physical world, or material world. "Nature" refers to the phenomena of the physical world, and also to life in general...
or physical reality has been considered to be a causal system. Physics involving special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
or general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
require more careful definitions of causality, as described elaborately in causality (physics)
Causality (physics)
Causality is the relationship between causes and effects. It is considered to be fundamental to all natural science, especially physics. Causality is also a topic studied from the perspectives of philosophy and statistics....
.
The causality of systems also plays an important role in digital signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
, where filters
LTI system theory
Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...
are often constructed so that they are causal. For more information, see causal filter
Causal filter
In signal processing, a causal filter is a linear and time-invariant causal system. The word causal indicates that the filter output depends only on past and present inputs. A filter whose output also depends on future inputs is non-causal. A filter whose output depends only on future inputs is...
. A causal system is said to be highly stable system as outputs only depend on past history of the system and not on any future input, which means, the system is not vulnerable to future failures.
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 the systems may be discrete or continuous. Similar rules apply to both kind of systems.
Mathematical definitions
Definition 1: A system mapping to is causal if and only if, for any pair of input signals and such thatthe corresponding outputs satisfy
Definition 2: Suppose
is the impulse response of the system . (only fully accurate for a system described by linear constant coefficient differential equation)then the system
is causal, otherwise it is non-causal.Examples
The following examples are for systems with an input and output .Examples of causal systems
- Memoryless system
- Autoregressive filter
Examples of non-causal (acausal) systems
- Central moving average
- For coefficients of t
Examples of anti-causal systems
- Time reversal
- Look-ahead
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