Formulation of the Problem

A typical example is the “cocktail party problem”. (See a demo.) Given the signals $x_j(t)$ from $m$ microphones recording $n$ speakers in the room ($m\ge n$), one wants to recover the voice $s_i(t)$ of each speaker. The problem can be formulated as

• Given

  $$x_i(t)=\sum_{j=1}^n a_{ij} s_j(t)\;\;\;\;\;(i=1,\cdots,m)$$ (202)

  or in matrix form

  $${\bf x}=\left[ \begin{array}{c} x_1 \\ \vdots \\ x_m \end{array} ... ...ght] \left[ \begin{array}{c} s_1 \\ \vdots \\ s_n \end{array} \right] ={\bf As}$$ (203)

• Find

  $$y_j(t)=\sum_{i=1}^m w_{ji} x_i(t)={\bf w}^T_j{\bf x}\;\;\;\;\;(j=1,\cdots,n)$$ (204)

  that best estimates the source variables $s_j(t)$, the independent components, or in matrix form

  $${\bf y}=\left[ \begin{array}{c} y_1 \\ \vdots \\ y_n \end{array} ... ...ight] \left[ \begin{array}{c} x_1 \\ \vdots \\ x_m \end{array} \right]={\bf Wx}$$ (205)

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 ${\bf s}$ or the mixing process ${\bf A}$, except that the source signals $s_i$ are statistically independent. Although this BSS problem seems severely under constrained, the method of ICA can find nearly unique solutions satisfying certain properties.

ICA can be compared with the principal component analysis (PCA) for decorrelation. Given a set of variables ${\mathbf x}$, PCA finds a matrix ${\mathbf W}$ so that the components of ${\bf y}={\bf Wx}$ are uncorrelated. Only in the special case when ${\bf y}=[y_1,\cdots,y_n]$ is gaussian, are its components also independent. Therefore the ICA is a more powerful method in the senese that it satisfies a stronger requirement for ${\mathbf W}$ so that the components of ${\mathbf y=Wx}$ are independent (and therefore are also necessarily uncorrelated).