Causal and Noncausal System (Causality Property)

Hi friends, today we will learn What is Causal and non-causal system?. These two are very important system in control systems. These systems are distinguished from their input output relationship. Let us see these systems one by one.

A)     Causal systems:

Definition: A system is said to be causal system if its output depends on present and past inputs only and not on future inputs.

Examples: The output of casual system depends on present and past inputs, it means y(n) is a function of x(n), x(n-1), x(n-2), x(n-3)…etc. Some examples of causal systems are given below:

1)      y(n) = x(n) + x(n-2)

2)      y(n) = x(n-1) – x(n-3)

3)      y(n) = 7x(n-5)

Significance of causal systems:

Since causal system does not include future input samples; such system is practically realizable. That mean such system can be implemented practically. Generally all real time systems are causal systems; because in real time applications only present and past samples are present. Since future samples are not present; causal system is memory less system.

B)      Anti causal or non-causal system:

Definition: A system whose present response depends on future values of the inputs is called as a non-causal system.

Examples: In this case, output y(n) is function of x(n), x(n-1), x(n-2)…etc. as well as it is function of x(n+1), x(n+2), x(n+3), … etc. following are some examples of non-causal systems:

1)      Y(n) = x(n) + x(n+1)

2)      Y(n) = 7x(n+2)

3)      Y(n) = x(n) + 9x(n+5)

Significance of non-causal systems:

Since non-causal system contains future samples; a non-causal system is practically not realizable. That means in practical cases it is not possible to implement a non-causal system.

  • But if the signals are stored in the memory and at a later time they are used by a system then such signals are treated as advanced or future signal. Because such signals are already present, before the system has started its operation. In such cases it is possible to implement a non-causal system.

  • Some practical examples of non-causal systems are as follows:

1)      Population growth

2)      Weather forecasting

3)      Planning commission etc.

For continuous time (C.T.) system:

A C.T. system is said to be “causal” if it produces a response y(t) only after the application of excitation x(t). That means for a causal system the response does not begin before the application of the input x(t).

The other way of defining the causal system is as follows:

A system is said to be “causal” if its output depends on present and past values of the input and not on the future inputs. If the input is applied at t = tm then the output at t = tm y(tm) will be dependent only on the values of x(t) for t = tm.

Condition for causality: y(tm) = f[x(t); t = tm]

Causal systems are physically realizable systems. The non-causal systems do not satisfy above condition. Non-causal systems are not physically realizable.

Condition for causality in terms of impulse response h(t):

The relation between y(t) and x(t) is given by,

y(t) = x(t)*h(t)

Where * represents convolution and h(t) is the impulse response of the system. The condition for causality in terms of the impulse response is as follows:

Condition for causality: h(t) = 0 for t<0

This condition states that a linear time invariant (LTI) system is “causal” if its impulse response h(t) has a zero value for negative values of time.

Solved problems on causal and non-causal system:

Determine if the systems described by the following equations are causal or non-causal.

1)      y(n) = x(n) + x(n-3)

Solution: the given system is causal because its output (y(n)) depends only on the present x(n) and past x(n-3) inputs.

2)      y(n) = x(-n+2)

Solution: this is non-causal system. This is because at n = -1 we get y(-1) = x[-(-1)+2] = x(3). Thus present output at n = -1, expects future input i.e. x(3)

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