Blind Signal Separation

Abstract:
The separation of independent sources from mixed observed data is a fundamental and challenging problem. In many practical situations, observations may be modelled as linear mixtures of a number of source signals, ie. a linear multi-input multi-output system. A typical example is speech recordings made in an acoustic environment in the presence of background noise and/or competing speakers. Other examples include EEG signals, passive sonar applications and cross-talk in data communications. In this paper, we propose iterative algorithms to solve the nxn linear time invariant system under two different constraints. Some existing solutions for 2x2 systems are reviewed and compared.

Here is a simple case with two sources s1(k),s2(k) and two mixing transfer functions H1(z), H2(z) to produce two mixed inputs x1(k), x2(k). The objective of the problem is to estimate separating transfer functions W1(z), W2(z) from x1(k) and x2(k) only, such that y1(k) and y2(k) would each contain s1(k) or s2(k) only.

 ie. Y1(z)=S1(z)G1(z), Y2(z)=S2(z)G2(z). A normalising process is needed.

 An iterative algorithm is developed to estimate W1(z), W2(z). Here is an example.

 a) s1(k) b) s2(k)

 c) x1(k) d) x2(k)

 e) y1(k) f) y2(k)

 The mixing transfer functions H1(z), H2(z) were 5 taps each.

 The model can be extended to multi-sources multi-inputs model. Algorithm has been devised to solve such problem.

 a) Source 1 b) Source 2 c)Source 3

 a) Observation 1 b) Observation 2 c) Observation 3

 a) Output 1 b) Output 2 c) Output 3