Time invariance

Time-invariance is a systems property. A CT system $T$ is time-invariant if for any input $x$ with output $y$, it holds that $y_{\tau }=T\{x_{\tau }\}$ for all time shifts $\tau \in \mathbb{R}$. What this means intuitively is that any experiment on a system in the future will produce the same results as an experiment on the system today.¹ To determine whether a system is time-invariant, what we need to do is:

• Find the original output given an arbitrary input $t$.
• Shift the original output by $\tau$.
• Compute another output given a shifted input $t-\tau$.
• If the outputs are the same, then the system is time-invariant.

Observe by definition that a proof for time-invariance doesn’t do the manipulations on the system. It does the manipulations on the input only.

Footnotes

1. From Prof Simpson-Porco’s lecture notes. ↩