Let
$$\Omega = \begin{bmatrix} -1 & 1 & 0 & \ldots \\ 0 & -1 & 1 & 0 \ldots \\ \ldots \\ \ldots & 0 & -1 & 1 \end{bmatrix} \in \mathbb{R}^{N \times n} $$
be a discrete finite difference operator and let $x \in \mathbb{R}^n$ be a vector. I now want to construct $x%$ in such a way that I can guarantee that $\Omega x \in \mathbb{R}^N$ has exactly $s \in \{0,\ldots,N-1\}$ zero rows.
What I have tried so far is to select a subset $S \subset \{0,\ldots,N-1\}, |S| = s$ and choose $\Omega_S$ as the submatrix consisting of these $s$ rows. Then I draw a standard normal vector $v$ and set $$x = v - \Omega_S^T (\Omega_S \Omega_S^T)^{-1} \Omega_Sv.$$
However, this procedure fails for higher sparsities. Is there any other way to generate these signals? Thanks!