variation of the source separation problem

I have four sensors which make measurements $y_k(t)$ that can be modeled as complex time series:

$y_1(t) = h_1(t) * x(t) + n_1(t)$

$y_2(t) = h_2(t) * x(t) + n_2(t)$

$y_3(t) = h_3(t) * x(t) + n_3(t)$

$y_4(t) = h_4(t) * x(t) + n_4(t)$

I am interested in extracting the $n_k(t)$ signals; the $x(t)$ is a nuisance common mode signal that I don't care about.

The $n_k(t)$ signals are uncorrelated noisy signals. $x(t)$ is band limited. Looking at the coherence between a pair of signals, it seems the $h_k(t)$ are fairly benign and can probably be treated as a scaling factor and a delay.

It looks like I can get reasonable separation of the power spectra by looking at

$S_{ii}-|S_{ij}|^2/S_{jj}$

where I computed $S_{ii}$ using matplotlib: mlab.psd(y1) and $S_{ij}$ as mlab.csd(y1,y2) etc.

However, I'd ideally like to recover the time domain $n_k(t)$ as best as I can.

Any suggestions?

Thanks, G