Trying to solve the following problem about bandpass signal sampling. I have a signal whose Fourier transform is such that $X(w)= 0$ if $w>w_h$ or $w<w_l$. The reconstruction is done using a bandpass filter rather than a lowpass and asks me for the minimum sampling frequency $w_s$ given that $w_l > w_h - w_l$.
I followed the steps of the sampling theorem and conjectured that if $w_s> w_h-w_l$ I should be able to reconstruct the signal with a lowpass filter. Indeed, denoting with $B=(w_h-w_l)/2$ I will have a copy of the signal centred at $0$ and with extremes $-B/2, B/2$ and then another copy centred at $w_s$ and with extremes $w_s-B/2$ and $w_s+B/2$. So, if $w_s > B$ I should have no aliasing. Is it correct? A lowpass filter should recover the signal centred at $0$.
Now, for the reconstruction with a bandpass filter. I have that the filter is such that $H(w)= T $ if $w_l\leq w \leq w_h$ or $-w_l\leq w \leq -w_h$. The question is: does this filter allow me to reconstruct the original signal $x$? I think it does not. First, it would give me two copy of the signal (right?) and also, there is no guarantee that there exist an integer $k$ such that $k w_s = (w_h+w_l)/2$.
The last question is then: assuming that $w_l>w_h-w_l$ what is the smallest sampling frequency $w_s$ and largest sampling interval $T$ that allow me to reconstruct the signal? Honestly I do not understand, the issue to me remains the same, why would that assumption be enough to reconstruct the signal?