Design a LTI system which returns DC value of the input signal Suppose that $h(t)$ is the impulse response of LTI system. The input signal $x(t)$ is periodic with period $T$. Determine $h(t)$ so that the output signal $y(t)$ only be the DC component of $x(t)$. Is $h(t)$ necessarily unique?

My attempt:
It's known that LTI system response to the periodic input is periodic. So I think the only possible $h(t)$ is constant function. If we let $h(t) = 1$ then: $$y(t) = \int_{-\infty}^{\infty}x(\lambda)d\lambda \tag{1}$$
According to Matt L.'s answer, definition of DC value is:
$$\bar{x}=\lim_{T_0\rightarrow\infty}\frac{1}{T_0}\int_{-T_0/2}^{T_0/2}x(t)dt\tag{2}$$
Clearly $(1)$ and $(2)$ are different. So what's the appropriate $h(t)$? Maybe question uses different definition for DC value?
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