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# Formulation of the Problem

A typical example is the cocktail party problem''. (See a demo.) Given the signals from microphones recording speakers in the room (), one wants to recover the voice of each speaker. The problem can be formulated as

• Given

or in matrix form

• Find
• the estimation of the source variables the independent components,
• and the linear combination matrix .
This is a blind source separation (BSS) problem, i.e., to separate linearly mixed source signals. The word blind'' means that we do not assume any prior knowledge about sources or the mixing process except that the source signals are statistically independent.

Although this BSS problem seems severely under constrained, the independent component analysis (ICA) can find nearly unique solutions satisfying certain properties.

ICA can be compared with principal component analysis (PCA) for decorrelation. Given a set of variables , PCA finds a matrix so that the components of are uncorrelated. Only under the special case when are gaussian, are they also independent. In comparison, ICA is a more powerful method in the senese that it satisfies a stronger requirement of finding so that the components of are independent (and therefore are also necessarily uncorrelated).

Next: Methods of ICA Estimations Up: ica Previous: ica
Ruye Wang 2018-03-26