Independent Component Analysis Example (ICA)

ICA has been in existence since around the mid 1990s and is a method to separate (in this case) two or more acoustic sources which have become mixed together eg whilst recording or in speech recognition problems where one of the sources is unknown non-stationary noise or a disturbance. The assumption is made that the original sources to be recovered are statistically independent. Moreover the scaling of the original sources cannot be recovered. Originally mixture were formed in simulations which were just a constant matrix. For example for a 2 source problem the mixture matrix was a 2X2 constant mixing matrix which was assumed to be unknown. Later, the extension was made to include convolutive mixtures (which is the case in the real world due to reverberation).

Consider n random sources in a vector which are all statistically independent. Now consider that this vector has passed through a mixing transfer function matrix described by a multivariable finite impulse response (FIR) filter thus

Where is the unit delay operator. For two sources and measurements this is illustrated in Figure 1. It is assumed here that the number of inputs and output are identical so that is a square polynomial-matrix. (not always the case of course) The object of the exercise is to synthesis a cascaded multivariable FIR filter which will have the effect of nullifying the cross-talk transfer function terms (and in Figure 1). The problem is particularly difficult since is unknown and often non-minimum phase. The non-minimum phase case means that even if was known exactly then its inverse does not exist!

Figure 1. Two input-output mixture model.

Example of a non-minimum phase acoustic mixing process.

Consider a second-order non-minimum-phase multivariable polynomial mixing matrix given by:

Which has 4 zeros at and . The latter two zeros have magnitude 1.03 which are just outside the unit circle. It can be seen from Figure 2 that the algorithm is able to separate the mixture with no extra computational cost than that of a minimum-phase mixture.

Figure 2. Clean Speech signals (a),(b), Mixtures (c) and (d) and Estimates (e) and (f).

Click here to hear original speech mixture:

Click here to hear estimate of first channel:

Click here to hear estimate of second channel:

The above example uses a new unpublished method developed at Albany School of Engineering and Advanced Technology.