...  ...  @@ 155,7 +155,7 @@ s_2 = \alpha_{21}x_1 + \alpha_{22}x_2 


S = WX



```






All we need to do is finding the values for $`\alpha_{ij}`$ in W. Note that we need to find W when A is unknown (we cannot simply use the inverse A). What we know though is, W defines the vectors in the mixture space and each vector (e.g. $`[\alpha_{11},\alpha_{12}`$]) basically extracts one source signal (here it is $`s_{1}`$). If you look at the above sketch of ICA, we see that these vectors must be orthogonal to the samples associated with all sources except the one it describes.So, we need to find W such that each vector in W is orthogonal to all sources but one. Okay, now we are getting closer to define an optimization problem.



All we need to do is finding the values for $`\alpha_{ij}`$ in W. Note that we need to find W when A is unknown (we cannot simply use the inverse A). What we know though is, W defines the vectors in the mixture space and each vector (e.g. $`[\alpha_{11},\alpha_{12}`$]) basically extracts one source signal (here it is $`s_{1}`$). If you look at the above sketch of ICA, we see that these vectors must be orthogonal to the samples associated with all sources except the one it describes. So, we need to find W such that each vector in W is orthogonal to all sources but one. Okay, now we are getting closer to define an optimization problem.






We also said that we are after independent signals. By saying so, we assume that the sources reflect this property, than the merged signals. With this constraint, we can say, "I will find such a W that "independency" is maximized in the extracted signals.




...  ...  