What is Time Invariance?
Time Invariance describes the property of a system such that the input of signal shifted k units in time equals it's respective output shifted k units in time.
An example of time invariance
Suppose a system is modeled mathematically as: $ \ f(x)= 5x + 6 \ln(x) + 9 $
Let $ \ x(t)=Ct $ be the input to this system. Then the output is $ \ y(t) = f(x(t)) $ Therefore
$ \ x(t) \longrightarrow f(t) \longrightarrow y(t)= 5Ct + 6 \ln(Ct) + 9 \, $
Now if we introduce a time delay into the input signal:
$ x(t) \longrightarrow x(t-t_0) \longrightarrow f(x) \longrightarrow 5C(t-t_0)+6\ln(C(t-t_0)) + 9 =Z(t) $
Now instead of introducing the time delay at the input, we move it to the output:
$ x(t) \longrightarrow f(x) \longrightarrow 5C(t)+6\ln(C t) + 9 = y(t) \rightarrow y(t-t_0) = 5C(t-t_0) + 6\ln(C(t-t_0)) + 9 $
Since$ \ Z(t)= y(t) $ system is time invariant
