Assuming we have sparse vector of length $N$ such as $X = [0,1,0,-1,0,1,1,0]$ which has some non-zeros values. The vector $x = iFFT(X)$ is convoluted with another vector $h$ resluting $y = h*x$. Suppose that $h$ and $y$ are known, is it possible to recover the sparse vector $X$ using such methods of sparse vector estimation, such as compressive sensing or any other method?
$NP$: I have suggested to take $FFT$ for $y$ resulting $Y = FFT(y) = HoX$, where $o$ is the element-wise multiplication between $H$ and $X$. Then based on $Y$ I think it's easier to recover sparse vector $X$. But the issue when using element-wise division, I can't get the position of each non-zeros element in $X$. On the other hand, I don't know how to implement compressive sensing to recover $X$.
Thank you in advance