Blind Signal Separation (BSS) using Particle Swarm Optimization (PSO)

We are often presented with the problem of having multiple signals mixed together that we would like to isolate. BSS consists of techniques for doing this without prior knowledge of either the signals or the way in which they are mixed. The underlying assumption is that the signals are statistically independent. This allows us to analyze the statistical properties of the mixed signals in order to separate out the original signals. We will consider the case where there are m desired signals s₁,…, s_m, and they are mixed with Gaussian white noise n to form m signals,x₁,…, x_m via x_i = a_i 1 s₁ +…+ a_{i m} s_m + n, where a_{i j} represents the unknown channel information.

Our method for BSS involves two steps. First, we will perform a Linear Prediction (LP) analysis on the signals jointly. This will allow us to remove the auto-correlation from each captured signal x_i, and focus on the values of t that give x_i(l), and x_j(l+t) the highest correlation. In many situations, a correlation cutoff value would have to be carefully chosen so as to ensure detection of the lag values that should be included while preventing the inclusion of inappropriate lags. For this algorithm, we are more worried about getting a false negative than a false positive at this stage, so we set a relatively low threshold for inclusion .

In the second step, we apply PSO to determine the coefficients of optimal Finite Impulse Response (FIR) unmixing filters based upon the taps found in the first step. We will perform the optimization in such a way so as to minimize the correlation between pairs of signals x_i and x_j. This is where any incorrectly chosen lags from the first step can be dealt with. In minimizing the correlation, we will end up setting the coefficients of any unnecessary lags close to zero, thus effectively eliminating them from consideration. In fact, if we find that the swarm has converged for that coefficient, i.e. every particle has that coefficient less than a given cutoff, we can remove that lag and continue the swarm optimization in a lower dimensional space. We would simply restart the optimization in the lower dimension, and project the current positions of the particles onto the lower dimensional space for the initial seeds. Once the FIR filters have been found and applied, what will be left is ŝ_i =s_i + n’_i where n’_i is residual noise that can be diminished by using noise reduction techniques. This technique has many applications including wireless networks where it can be used to separate the different signals that are received.