# Does this system of equations have a solution?

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 $$E\{d_k\} = E\{d_{k-2}\}=0$$.

• Please write the recurrence relation in matrix form. – Rodrigo de Azevedo Oct 14 at 20:32
• Where did you get such model that you need to estimate the error value? – Royi Oct 18 at 22:39