Contents
Worded Definitions
Time Invariant
A system is said to be time invariant if a time shift does not affect the output of the system. If x(t) is put through the system, then time shifted, the results should be identical to x(t) being time shifted, then put through the system.
Time Variant
A system is time variant if a time shift does affect the output of the system. In the same way as before, if x(t) is put through the system then time shifted, the results will be different than if x(t) is time shifted, then put through the system.
Example of a Time Invariant System
$ \,\!y(t)=x(t)sin(t) $
Step 1: The System
$ \,\!y(t)=x(t)sin(t) $
Step 2: Time delay
$ \,\!z(t)=y(t-k)=x(t-k)sin(t-k) $
Step 1: Time delay
$ \,\!y(t)=x(t-k) $
Step 2: The System
$ \,\!z(t)=y(t)sin(t)=x(t-k)sin(t-k) $
The functions z(t) are equal, so the system is time invariant.
Example of a Time Variant System
$ \,\!y(t)=tx(t) $
Step 1: The System
$ \,\!y(t)=tx(t) $
Step 2: Time delay
$ \,\!z(t)=y(t-k)=(t-k)x(t-k) $
Step 1: Time delay
$ \,\!y(t)=x(t-k) $
Step 2: The System
$ \,\!z(t)=ty(t)=tx(t-k) $
The results for z(t) are not equal so this system is time variant.
