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Previously, we saw that the system needs to be independent from the future and past values to become static. In this case, the condition is almost same with little modification. Here, for the system to be causal, it should be independent from the future values only. That means past dependency will cause no problem for the system from becoming causal.
Causal systems are practically or physically realizable system. Let us consider some examples to understand this much better.
Let us consider the following signals.
a) y(t) = x(t)
Here, the signal is only dependent on the present values of x. For example if we substitute t = 3, the result will show for that instant of time only. Therefore, as it has no dependence on future value, we can call it a Causal system.
b) y(t) = x(t−1)
Here, the system depends on past values. For instance if we substitute t = 3, the expression will reduce to x(2), which is a past value against our input. At no instance, it depends upon future values. Therefore, this system is also a causal system.
c) y(t) = x(t) + x(t+1)
In this case, the system has two parts. The part x(t), as we have discussed earlier, depends only upon the present values. So, there is no issue with it. However, if we take the case of x(t+1), it clearly depends on the future values because if we put t = 1, the expression will reduce to x(2) which is future value. Therefore, it is not causal.
A non-causal system is just opposite to that of causal system. If a system depends upon the future values of the input at any instant of the time then the system is said to be non-causal system.
Let us take some examples and try to understand this in a better way.
a) y(t) = x(t+1)
We have already discussed this system in causal system too. For any input, it will reduce the system to its future value. For instance, if we put t = 2, it will reduce to x(3), which is a future value. Therefore, the system is Non-Causal.
b) y(t) = x(t) + x(t+2)
In this case, x(t) is purely a present value dependent function. We have already discussed that x(t+2) function is future dependent because for t = 3 it will give values for x(5). Therefore, it is Non-causal.
c) y(t) = x(t−1) + x(t)
In this system, it depends upon the present and past values of the given input. Whatever values we substitute, it will never show any future dependency. Clearly, it is not a non-causal system; rather it is a Causal system.
An anti-causal system is just a little bit modified version of a non-causal system. The system depends upon the future values of the input only. It has no dependency either on present or on the past values.
Find out whether the following systems are anti-causal.
a) y(t) = x(t) + x(t−1)
The system has two sub-functions. One sub function x(t+1) depends on the future value of the input but another sub-function x(t) depends only on the present. As the system is dependent on the present value also in addition to future value, this system is not anti-causal.
b) y(t) = x(t+3)
If we analyze the above system, we can see that the system depends only on the future values of the system i.e. if we put t = 0, it will reduce to x(3), which is a future value. This system is a perfect example of anti-causal system.