
Blind source separation (BSS) is a technique for estimating individual source components from their mixtures at multiple sensors. It is called blind because we don't use any other information besides the mixtures.
For example, imagine a room with a number of persons present and a number of microphones for recording. When one or more persons are speaking at the same time, each microphone registers a different mixture of individual speaker's audio signals. It is the task of BSS to untangle these mixtures into their sources, i.e. the individual speaker's audio signals. In general, this is a difficult problem because of several complicating factors.
If the number of sensors is larger than the number of sources we speak of an overdetermined problem. If the number of sensors and the number of sources are equal we speak of a determined problem. The more difficult problem is the underdetermined one where the number of sensors is less than the number of sources.
In general two different types of mixtures are considered in the literature: instantaneous mixtures and convolutive mixtures.
y_{i} (n) = Σ_{j}^{d}Σ_{τ}^{Mij1} h_{ij}(τ)x_{j}(nτ) + N_{i}(n), for i=1..m. 
Various techniques exist for solving the blind source separation problem for instantaneous mixtures. Very popular ones make make use of second order statistics (SOS) by trying to simultaneously diagonalize a large number of crosscorrelation matrices. Other techniques like independent component analysis use higher order statistics (HOS) to find the independent components, i.e. the sources.
Given the decomposition problem Y=A·X, we can see that the solution is determined only upto a permutation and a scaling of the components. This is called the indeterminancy problem of BSS. This can be seen as follows: given a permutation matrix P, i.e. a matrix which contains only zeros except for one 1 in every row and column, and a diagonal scaling matrix D, any scaling and permutation of the independent components X_{n}=(D·P)·X can be compensated by the reversed scaling of the mixing matrix A_{n}=A·(D·P)^{1} because A·(D·P)^{1}·(D·P)·X = A·X = Y.
Solutions for convolutive mixture problems are much harder to achieve. One normally starts by transforming the problem to the frequency domain where the convolution is turned into a multiplication. The problem then translates into a separate instantaneous mixing problem for each frequency in the frequency domain. It is here that the indeterminacy problem hits us because it is not clear beforehand how to combine the independent components of each frequency bin.
© djmw, September 7, 2012