# Doubt on LTI systems (Zero input-Zero Output)

So, I have a problem where the input and output of a system are given :

\begin{align} x(t) &= \sin(10t) \\ y(t) &=5\cos(10t+\frac{\pi}{6}) \end{align}

So, I need to determine whether this system is LTI.

First, when I saw this problem, I checked for zero input-zero output condition of LTI. By substituting $$t=0$$, $$x(t)=0 , y(t) \neq 0$$. So, I guessed it was not LTI. But the answer is given as LTI.

Can someone give some insights? Specifically, is Zero input-Zero Output for LTI always satisfied?

I require just some hints, not complete answers. Particularly the answer to whether LTI systems always obey zero input/zero output condition?

• I have explained it in the second paragraph. Put t=0, sin(10t)=0 , but 5cos(pi/6)≠0 Mar 20 at 15:36
• Thank-you!!!!!!
– Peter K.
Mar 20 at 16:09

Note that for an LTI system the input can be zero at a certain time, and the output at that time can be non-zero. You can only conclude that a system is not LTI if you observe a non-zero output for zero input, i.e., $$y(t)\neq 0$$ if $$x(t)=0$$ for all $$t$$.