Time-invariant system

A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a 'time-varying system'.To demonstrate how to determine if a system is time-invariant, consider the two systems:A more formal proof of why systems A and B above differ is now presented.To perform this proof, the second definition will be used.We can denote the shift operator by T r {displaystyle mathbb {T} _{r}}   where r {displaystyle r}   is the amount by which a vector's index set should be shifted. For example, the 'advance-by-1' system