# Time-Invariant Systems

A system is time-invariant if shifting the input sequence on time axis leads to an equivalent shift of the output sequence along the time axis, with no other changes. So if an input generates an output , then

In other words, if an input signal is delayed by some amount , output also just gets delayed by and remains exactly the same otherwise. Whether a system is time-invariant or not can be determined by the following steps.

1. Delay the input signal by samples to make it . Find the output for this input.
2. Delay the output signal by samples to make it . Call the new output .
3. If , the system is time-invariant. Otherwise, it is time-variant.
Example

Consider a system

We will follow the steps above to determine whether this system is time-invariant or not.

1. Delaying the input signal by samples, the output in response to is

2. Delaying the output signal by samples, we get

3. Since , this system is time-invariant.

Now consider another system

and apply the same steps.

1. Delaying the input signal by samples, the output in response to is

2. Delaying the output signal by samples, we get

3. Since , this system is time-variant.

In other words, a system is time-invariant if the order of delaying does not matter, i.e.,

is the same as

Linear Time-Invariant (LTI) System

A system that is both linear and time-invariant is, not surprisingly, called Linear Time-Invariant (LTI) system. It is a particularly useful class of systems that not only truly represents many real-world systems but also possesses an invaluable benefit of having a rich set of tools available for its design and analysis.