Design a LTI system which returns DC value of the input signal
The DC value is simply the mean. Since the signal is periodic you only need to take the mean of one period. This can be simply done with

$$h(t)=\left\{\begin{matrix}
1/T,  0 &lt; t &lt; T\\ 0, {\rm elsewhere} 
\end{matrix}\right. $$


  Clearly (1) and (2) are different.


(1) will not converge, so that doesn't work

(2) is the best way for a function that's not periodic. It will work for a periodic function as well but is needlessly complicated