Time-Invariant System

The system is time invariant if the behaviour and characteristic of the system are fixed over time. In other words, a system is said to be time invariant if a time shift in the input signal causes an identical time shift in the output signal.

It means that 

for continuous time

if x(t) → y(t), then x(t-to) → y(t-to)

for discrete time

if x(t) → y(t), then x(t-to) → y(t-to)

How to check whether a system is time invariant or time varying

  1. Let y1(t) be the output corresponding to x1(t).
  2. Consider a second input x2(t) obtained by shifting x1(t) i.e  x2(t) =  x1(t-to) and find the output corresponding to the input x2(t).
  3. From the step 1, find y1(t-to) and compare with y2(t).
  4. If y2(t) =  y1(t-to), then the system is time invariant; otherwise it is a time varying system.
Time-Varying & Time-Invariant Systems

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