# Is $y(t) = \cos(t) + x(t)$ a time-invariant system?

Is $$y(t) = \cos(t) + x(t)$$ a time-invariant system?

$$y(t-k) = \cos(t-k) + x(t-k)$$

But it isn't equal to $$\cos(t) + x(t-k)$$

So, would it be time-invariant?

You've answered the question yourself already. So, does the response to a shifted input equal the shifted response to the original (non-shifted) input? If so, then the system is time-invariant. Otherwise, it's a time-varying system.

• It's a time-varying system then. – Siddharth Garg Nov 12 '19 at 12:51
• @SiddharthGarg: That's right. – Matt L. Nov 12 '19 at 13:36