Friday, March 1, 2013
Causal and Non-Causal Systems Better Explained
You must have come across causal and non-causal systems at some point in your Engineering life. Most of us might have simply studied that Causal systems are those systems which respond only to present and past inputs whereas Non-Causal systems can also respond to future inputs. That is correct but there is a lot more to explore about it. In this article we shall discuss the fundamental ideas and philosophies involved in this concept. I am not explaining the mathematics of causal systems here as they are readily found in any related text book and also on the Internet. You may refer to your text books any other source if you do not know what it is. Let us explore more on this.
What is a causal system?
A system is said to be causal if the output of the system depends only on past or present values of the inputs but not on future values of the inputs.
Understanding a Causal System
The term "causal" is derived from the word cause. The cause is anything that gives rise to an action, phenomenon or condition (according to English dictionary). By the meaning of cause, we can understand that cause is nothing but an input. So it is understood that a causal system is the one which responds to a cause. The cause is the input of the system and the response is the output of the system. That means, if a system produces an output only because of events (causes or inputs or excitations) that have happened in the past or happening now, such systems are called as causal systems. Causal systems are also called non-anticipative systems.
Most (almost all) of the systems that we see in our daily life are all causal systems. Your mobile phone is a causal system because it rings only for the current call but it cannot ring for a call that may happen tomorrow. Examples are too many for causal systems because most of the systems that exist and that we see in our daily life fall into this category. Even we humans are causal systems. Our present actions (outputs) depend on the events (inputs) that have happened to us in the past and the events that are happening now but we cannot respond to events which will happen tomorrow. We may sometimes respond to events that we expect them to happen in future. But we can never respond to events that have happened in the future. Also, our expectation of the future is based on our past and present experiences. So, we are not that powerful enough to cross the borderline of causality.
The concept of causality is as old as human civilisation. Many philosophies and theologies mention about the causal nature of humans and the universe. The karma siddantam discussed in the Santana Dharma (Indian philosophy) is one such theology. Alright, we shall restrict ourselves to our domain and stick to the topic.
Causal systems are classified as having memory and memory less systems. Causal systems which can respond to causes in the past are all memory having causal systems. This is because, a system can remember causes in the past only if it has memory. Else, they are memory less systems.
What is a Non-Causal System?
A system whose present response depends on future values of the inputs is called as a non-causal system. There are two types of non causal systems namely, acausal and anti-causal.
Acausal systems are those whose present response depends on future values as well as past and present values of the input.
Anti-causal systems are those systems whose present output depends only on the future values of the input or excitation but not on the past and present values.
Whenever we use the term non-causal, it implicitly means acausal in our discussion unless otherwise mentioned.
Now comes the interesting part of the discussion.
In the earlier section of causality, we have mentioned that almost all the physical systems that exist are causal in nature. A non-causal system gives an output to the inputs given to the system at some time in the future. It sounds crazy right? It is really hard to imagine any physical system that responds to future inputs. It seems something godly in nature which cannot be perceived easily. This is why non-causal systems usually do not exist physically in nature. They are not natural systems and are considered as virtual in most cases.
When there is no physical system that is non-causal in nature, then why do we classify systems as causal and non-causal when they do not have any physical significance?
Although non-causal systems hardly exist in nature, they can be made possible in virtual environments where we can define our own time axis (not the actual physical time).
Mathematical Possibility of Non-Causal Systems
Let me define a system mathematically as w(t) = x(t+3). It turns to be a non-causal system. For example, take t=1sec. The system response will be
We can see that the system which we have defined gives the output of 4th second at the 1st second itself. Wonderful! As we have the right to write mathematical equations however we like irrespective of its physical existence, we can also make Non-Causal systems a mathematical reality.
Physical Possibility Of Non-Causal Systems
Alright, we have now understood that non-causal systems can be made possible on paper. But, can we make it really possible in our real world?
The answer is YES! There is a possibility.
Non-causal systems are possible when the system has recorded input and output data.
Let me illustrate it with an example. Let us consider you as a system. Now you are watching a live cricket match. You are now a causal system because your feelings and emotions depend only on what happened in the past and what is happening at present.
However, if you record it and watch the same cricket match later, you will know what will happen after some time in the match (because data of the match is already recorded in your mind). Now, your feelings and emotions concerning the match not only depends on the past and present events in the match but also depends on the future events of the match because you already know what happens. So, if we consider you as a system, the recorded cricket match as an input and your feelings as output and the time axis is fixed with respect to the recorded data as in the above example, then you are a NON-CAUSAL SYSTEM! Hurray!
From the above illustration we can conclude that non-causal systems can be made possible when the system has knowledge of recorded values of input and output values and the time axis is taken in reference to the recorded data.
People and books usually say that all physically realizable systems are causal in nature and non-causal systems do not exist naturally. That is true, this means that no system can be non-causal when it has no data about the future. So, to make a non-causal system possible, we have to provide future data to the system. As it is not really possible to do that naturally, we have to shift the time axis to the recorded data which is advance than the real time axis as we have done in the illustration above.
Final Thoughts about Causality
Causality or non-causality is not just a mathematical representation but has quite deeper physical interpretations. Also, non-causality of a system is possible in some way or the other. That is why it is still present in the literature and we still speak of it.