Estimate Signal by Its Convolution by 2 Different Kernels

Question

I have a discrete Signal $s$ that has been convoluted with two functions $h_1$ and $h_2$. I measure the result of this convolution:

$$y_1=s*h_1, \quad y_2=s * h_2.$$

I have a short time segment (for identical times) of $y_1$ and $y_2$ (shorter than the convolution length).

If I know $h_1$ and $h_2$, can I make any conclusions about $s$ based on $y_1$ and $y_2$?

Answer 1

Let's write this in Matrix Form:

$$ \boldsymbol{y}_{1} = {H}_{1} \boldsymbol{s}, \quad \boldsymbol{y}_{2} = {H}_{2} \boldsymbol{s} $$

If we set $ \boldsymbol{y} = \begin{bmatrix} \boldsymbol{y}_{1} \\ \boldsymbol{y}_{2} \end{bmatrix} $ and $ H = \begin{bmatrix} {H}_{1} \\ {H}_{2} \end{bmatrix} $ then we have:

$$ \boldsymbol{y} = H \boldsymbol{s} $$

Solve that and you have the Least squares solution for your problem.