My heuristic approaches to find a solution for the following system of equations have failed so far. Does a solution exist?
$$ \left[ {\begin{array}{cc} a_k & 0 \\ a_{k-1} & a_k \\ 0 & a_{k-1} \end{array} } \right] \left[ {\begin{array}{cc} g_{k-1} \\ g_k \end{array} } \right] = \left[ {\begin{array}{cc} 0.5 \\ 0\\ 0.5 \end{array} } \right]+\left[ {\begin{array}{cc} d_{k-2} \\ 0\\ d_k \end{array} } \right] $$ such that $\{a_k\}$ are known data symbols with equal probability, chosen from L equidistant values. Unknown $\{g_k\}$ and $\{d_k\}$ are to be found such that $E\{d_k\} = E\{d_{k-2}\}=0$.
