I have collected two signals, $B_1(x)$ and $B_2(x)$. The signals start and end at the same $x$-values. Each signal $B_i(x)$ contains:
- a base signal $b(x)$, which is the same for both, and
- a signal, either $w(x)$ or $w(x+\phi)$, which are identical except for the shift $\phi$ along $x$.
Signals $b(x)$, $w(x)$, and $w(x+\phi)$ are non-periodic.
Essentially, I have:
$B_1(x) = b(x) + w(x) \\ B_2(x) = b(x) + w(x + \phi)$,
with only access to $B_1(x)$, $B_2(x)$, and $\phi$.
My question: Is is possible to use the phase-shift knowledge from my two signals to recover the base signal $b(x)$? And if so, can you point me in the direction of a method that would work?
I've read about or tried to implement methods such as blind source separation, independent component analysis, and stationary subspace analysis, but none seem to take advantage of the phase information, or work at all for my data.
Note: If it helps, I am able to collect more data with different $\phi$-values, to obtain more $B_i(x)$ signals.
I've attached a figure with toy data (my real data is messier and longer). The top plot shows $b(x)$ (what I'd like to recover), the middle plot shows $w(x)$ and $w(x+\phi)$, and the bottom plot shows the measured signals $B_1(x)$ and $B_2(x)$, representing the combination.