Is there a way to reduce the covariance matrix of several source signals to the dominant source signal?

Question

The problem I have can be seen in the context of DoA estimation or blind source signal separation and similar fields, where several source signals are observed by several antennas (or by an antenna array, respectively).

The basic model is something like $$\mathbf{y}(t) = \sum_{i=1}^{n_s} s_i \mathbf{a}\left(\theta _i\right) + \mathbf{n}\left(t\right),$$ where $s_i$ describes the source signals, $\mathbf{a}\left(\theta _i\right)$ the mixing vector and $\mathbf{n}\left(t\right)$ the additive noise term. Note that neither of these terms should be considered known.

From this, I have available the sample covariance matrix $$\mathbf{\hat{C}} = E\{\mathbf{y}(t)\mathbf{y}^H(t)\} = \frac{1}{T} \sum_{t=1}^T\mathbf{y}(t)\mathbf{y}^H(t)$$ with dimensions $N \times N $, if $N$ is the number of antennas, $n_s$ the number of source signals and $T$ the number of samples.

Now, what I want to do is the following: I don't want to estimate all DoAs nor all source signals. I am looking for a method that is able to reduce the covariance matrix containing information about $n_s$ source signals to a covariance matrix containing just information about the dominant source. Actually, it is not even so important that it is the dominant source, the main goal is the reduction to just one signal.

Is there any method to achieve this kind of cleaning, of signal elimination, of source reduction, of clutter elimination or whatever you want to call it?

EDIT: I think I have not made clear a very important side fact: I am not particularly interested in the amplitudes or powers $\mathbf{s}$ themselves, but more in their geometry/directions $\mathbf{\theta}$.

Answer 1

What you want to do is dimension reduction. The most basic, yet very powerful and commonly used, technique to do it is principal component analysis (PCA). PCA operates on the covariance matrix. You can look it up on Wikipedia for a throughout tutorial.

PCA decomposes your data $y$, having dimension $d$, into $d$ principal components. The first principal component explains the most variance in your data, the next component the second most variance, and so on. Throw off the last $k$ principal components $-$ and ta-daa $-$ you reduce dimension. To answer your question, take $k = d - 1$ and then you get the most dominant component (in terms of explained variance).

(Tip: MATLAB and similar software usually come with an easy-to-use function to perform PCA).

Answer 2

If you want to do source separation, then independent component analysis (ICA) is the correct technique, which you mention in the comments.

ICA decomposes data as $x = As$ where $x$ are mixtures of some source signals $s$, and $A$ is the mixing matrix. Neither $A$ or $s$ is assumed to be known, ICA will find both of them given some $x$. The only assumptions for ICA are:

1) the source signals $s$ are statistically independent,
2) the mixing matrix $A$ is stationary (does not change over time),
3) there can be maximally one gaussian source.

Usually one also assumes $A$ to be a square matrix. So, if you have more sensors than sources, you would first reduce dimension using PCA and some model order selection method (i.e. dimension estimation method), such as minimum description length (MDL).

I would recommend you to use FastICA by Hyvärinen and Oja. It's the most famous algorithm to do ICA (the second one would be Infomax ICA by Bell and Sejnowski).

You can find some nice publications at http://www.cs.helsinki.fi/u/ahyvarin/papers/ICA.shtml and a MATLAB implementation of FastICA from http://research.ics.aalto.fi/ica/fastica/

Since the model in your question has parametric $A$, which neglects the stationarity assumption, you could do ICA after taking the Fourier transform of $x$. This is called Fourier-ICA. Taking the Fourier transform effectively removes the parameter of $A$, but you will need to handle complex-valued data.