Time-Invariant System and Time-Varaint System
A system is time invariant if for any input signal x(t)/x[n] and for any time t0$ {\in} $R, the response to the shifted input x(t-t0) is the shifted output y(t-t0). Simply put a time invariant system is one whose output does not depend explicitly on time.
A system is Time variant if its output explicitly depends on time.
Example for Time-Variant System
Let us consider the system y(t)=t.x(t)
Now,
(1).x(t)$ {\longrightarrow} $time delay$ {\longrightarrow} $y(t)=x(t-t0)$ {\longrightarrow} $system $ {\longrightarrow} $z(t)=t.x(t-t0)
Now let us see what happens when it passes through the system first and then delayed by time. (2).x(t)$ {\longrightarrow} $system$ {\longrightarrow} $y(t)=t.x(t)$ {\longrightarrow} $time delay $ {\longrightarrow} $ z(t)=(t-t0).x(t-t0)
Clearly (1) and (2) don't have the same outputs, thus the system is time variant.
Example for time-invariant system
y(t)=6.x(t)
(3).x(t)$ {\longrightarrow} $time delay$ {\longrightarrow} $y(t)=x(t-t0)$ {\longrightarrow} $system $ {\longrightarrow} $z(t)=6.x(t-t0)
(4).x(t)$ {\longrightarrow} $system$ {\longrightarrow} $y(t)=6.x(t)$ {\longrightarrow} $time delay $ {\longrightarrow} $ z(t)=6.x(t-t0)
Since (3) and (4) produce the same output the system is time invariant.
