Assume we have a matrix $x$ of size $(8,8)$ where each column is considered to be sparse with degree of sparsity equals to $4$. it means that every column can have $4$ zeros and $4$ non-zeros values distributed randomly. The matrix $x$ can be written as follows:
Inverse Fourier transformation $(iFFT)$ is performed for every column in the matrix $x$. It means the matrix $F$ representing the $FFT$ matrix of size $(8,8)$ is multiplied with every column in matrix $x$, So $X = F^Hx$, where $X$ is the $iFFT$ column-wise of matrix $x$. The resulted matrix $X$ is read in row-wise leading to have a new row $X'$ of size $(1,64)$.(I mean the matrix $X$-transposed is reshaped to have one column).
The vector $X'$ is convoluted with such channel $h$ resulting $y$, so $y = HX'$, where $H$ is the toeplitz matrix gotten based on the channel $h$.
My question is to recover the spare vectors in matric $x$ based on the resulted vector $y$ and matrix $H$, I'm using compressive sensing, i.e OMP algorithm. My question is how to build the measurement matrix in that way.
$NP$: The measurement matrix can't be built straightforward since we reshaped the matrix $X$ in row-wise way. If not, I think we can build it to be $HF^H$ as used in the paper Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals.
Thank you in advance.