Independent component analysis: a tutorial

Imagine that you are in a room where two people are speaking simultaneously. You have two microphones, which you hold in di erent locations. The microphones give you two recorded time signals, which we could denote by x1(t) and x2(t), with x1 and x2 the amplitudes, and t the time index. Each of these recorded signals is a weighted sum of the speech signals emitted by thetwo speakers, which we denote by s1(t) and s2(t). We could express this as a linear equation: x1(t) =a11s1 + a12s2 x2(t) =a21s1 + a22s2 where a11�a12�a21, and a22 are some parameters that depend on the distances of the microphones from the speakers. It would be very useful if you could now estimate the two original speech signals s1(t) and s2(t), using only the recorded signals x1(t) and x2(t). This is called the cocktail-party problem. For the time being, we omit any time delays or other extra factors from our simpli ed mixing model. As an illustration, consider the waveforms in Fig. 1 and Fig. 2. These are, of course, not realistic speech signals, but su ce for this illustration. The original speech signals could look something like those in Fig. 1 and the mixed signals could look like those in Fig. 2. The problem is to recover the data in Fig. 1 using only the data in Fig. 2