above differ is now presented. To perform this proof, the second definition will be used. System A: Start with a delay of the input Now delay the output by Clearly , therefore the system is not time-invariant. System B: Start with a delay of the input Now delay the output by Clearly , therefore the system is time-invariant. More generally, the relationship between the input and output is and its variation with time is For time-invariant systems, the system properties remain constant with time, Applied to Systems A and B above: in general, so it is not time-invariant, so it is time-invariant. Abstract example We can denote the shift operator by where is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system can be represented in this abstract notation by where is a function given by with the system yielding the shifted output So is an operator that advances the input vector by 1. Suppose we represent a system by an operator . This system is time-invariant if it commutes with the shift operator, i.e., If our system equation is given by then it is time-invariant if we can apply the system operator on followed by the shift operator , or we can apply the shift operator followed by the system operator , with the two computations yielding equivalent results. Applying the system operator first gives Applying the shift operator first gives If the system is time-invariant, then See also Finite impulse response Sheffer sequence State space (controls) Signal-flow graph LTI system theory Autonomous system (mathematics) References Control theory Signal processing