I’m stuck on the following problem.
There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$:
$$ S_A = W_A + I_A $$
$$ S_B = W_B + I_B $$
Two receivers are co-located with the sources. The first one records the signal
$$ X_A = S_A + S_B H_{AB} $$
the second one records the signal
$$ X_B = S_A H_{AB} + S_B $$
where the channel $H_{AB}$ is known and introduces a delay and an attenuation
$$ H_{AB} = exp(-j 2\pi f \frac{x}{v})exp(-\alpha x) $$
In this case one can estimate $S_A$ and $S_B$ simply by:
$$ S_A = \frac{X_A - X_B H_{AB}}{1 - H_{AB}^2} $$
$$ S_B = X_B - S_A H_{AB} $$
The problem is when an anomalous event $N$ appears somewhere in the middle of the channel and propagates towards A and B:
The recorded signals become
$$ X_A = S_A + S_B H_{AB} + N H_{NA} $$
$$ X_B = S_A H_{AB} + S_B + N H_{NB} $$
where both $N$, $H_{NA}$ and $H_{NB}$ are unknown.
The knowledge I have about $N$ is:
- $N$ is a single impulsive event,
- Its power is below the power of the noise components of $S_A$ and $S_B$.
If I apply the previous method, $N$ is spread on both $S_A$ and $S_B$.
I need to estimate either the sources $S_A$, $S_B$, or $N H_{NA}$ or $N H_{NB}$.
Any suggestion?