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.

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang