Multichannel approach

This approach can be viewed as a generalization of the preceding method. Instead of single variables, vectors and matrices are used:  

$$X_{ir}=X_i-{\bold X}_{N-1}^T{\overline{{\bold X}}_{N-1,N-1}^{-1}\overline{{\bold X}}_{N-1,i}},$$ (5)

Here ${\bold X}_{N-1}$ is a vector whose components are Fourier transforms of all N-1 traces beyond the i-th trace, $\overline{{\bold X}}_{N-1,N-1}$is the spectral matrix of these traces, and $\overline{{\bold X}}_{N-1,i}$is the vector of cross-spectra between the i-th trace and all others. All spectra are for some particular frequency.

I don't know yet how to prove that algorithm (5) does a good job of subtracting interfering signals, but I will now try to demonstrate that this is so using some synthetic examples.