A system is said to be causal system if its output at all the instants depends on the past value and/or present value of the input. A system is said to be a non-causal system if its output at any instant depends on the future value of the input.
Consider the following system. x(t) is the input and y(t) is the output.
How to check whether a system is causal or non-causal?
Step1: Put different time values possibly positive as well as negative values in the input-output relationship.
Step2: After putting values of time, Check the output expression y(t) whether it depends on present, past or future value of the input x(t) for a particular instant of time. [check the only expression inside the parenthesis of input]
If at all the instants output y(t) depends on the past value and/or present value of the input x(t) then the system is a causal system.
If at any instant output y(t) depends on the future value of the input x(t) then the system is a non-causal system.
Examples
Example 1: y(t) = 2 x(t)
putting time, t values
at t = 0, y(0) = 2 x(0)
the output depends on the present value of the input x(t)
at t = 1, y(1) = 2 x(1)
the output depends on the present value of the input x(t)
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at t = −4, y(−4) = 2 x(−4)
the output depends on the present value of the input x(t)
Since the output y(t) of the system depends only on the present value of the input x(t) then the given system is the causal system.
Example 2: y(t) = 2 x(t) + 5
putting time, t values
at t = 3, y(3) = 2 x(3) + 5
the output depends on present value of the input x(t)
at t = 1, y(1) = 2 x(1) + 5
the output depends on present value of the input x(t)
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at t = −2, y(−2) = 2 x(−2) + 5
the output depends on present value of the input x(t)
Since, the output y(t) of the system depends only on the present value of the input x(t) then the given system is causal system.
Example 3: y(t) = (t+2) x(t−2)
putting time, t values
at t = 0, y(0) = (0+2) x(0−2) = 2 x(−2)
the output depends on past value of the input x(t)
at t = 1, y(1) = (1+2) x(1−2) = 3 x(−1)
the output depends on past value of the input x(t)
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at t = −3, y(−3) = (−3+2) x(−2−2) = − x(−4)
the output depends on past value of the input x(t)
Since, the output y(t) of the system depends only on the past value of the input x(t) then the given system is causal system.
Example 4: y(t) = (t−2) x(t+2)
putting time, t values
at t = 0, y(0) = (0−2) x(0+2) = −2 x(2)
the output depends on future value of the input x(t)
at t = 1, y(1) = (1−2) x(1+2) = − x(3)
the output depends on future value of the input x(t)
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at t = −4, y(−4) =(−4−2) x(−4+2) = −6 x(−2)
the output depends on future value of the input x(t)
Since, the output y(t) of the system depends only on the future value of the input x(t) then the given system is non-causal system.
Example 5: y(t) = 2 x(−t)
putting time, t values
at t = 1, y(1) = 2 x(−1)
the output depends on past value of the input x(t)
at t = 2, y(0) = 2 x(−2)
the output depends on past value of the input x(t)
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at t = −2, y(−2) = 2 x(2)
the output depends on future value of the input x(t)
Since, the output y(t) of the system depends on the past and future value of the input x(t) then the given system is non-causal system.
Example 6: y(t) = 2 x(2t+3) + 9
putting time, t values
at t = 0, y(0) = 2 x(2 x 0+3) + 9 = 2 x(3) + 9
the output depends on future value of the input x(t)
at t = 1, y(1) = 2 x(2 x 1+3) + 9 = 2 x(5) + 9
the output depends on future value of the input x(t)
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at t = −3, y(−3) = 2 x(2 x (−3)+3) + 9 = 2 x(−3) + 9
the output depends on present value of the input x(t)
Since, the ouput y(t) of the system depends on the present and future value of the input x(t) then the given system is non-causal system.
Note: If in the input-output relationship of the system, scaling is present then system will definitely be non-causal.
