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I'm working this summer on a very interesting, albeit difficult to understand at 100 %, project and it turns around reducing the glare effect than could be applied on a receiver when he gets signals coming from two emitters sending $X(t)$ (more energy for this signal compared to the other) and $s(t)$ (much weaker).

Let me put it in other words.

The receiver gets a signal mixture $Y(t)=H(t)*X(t)+g(t)*s(t)$ but we don't know $H$ nor $X$ nor $g$ nor $s$ nor the frequencies of the emitted signals. The only thing we know is $Y(t)$ and the fact that their spectra are dissociated in the best cases.

A picture speaks a thousand words, so...

Excuse the terrible drawing

Basically, the signal $X(t)$ overpowers $s(t)$. So I was thinking of a way to get the $s(t)$ and the $X(t)$ and with the few info I've gotten, I couldn't implement the generic solutions. However, I still have one card which is the FASTICA but that would suppose having many samples of "mixed " signal to extract the two important signals. What do you think about it? I'm open to any software and/or mechanic suggestion.

I've read an interesting paper on this but they were talking about the near-far effect for GPS signals. Thanks for the help!!

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  • $\begingroup$ If the signals aren't overlapping in frequency, like you illustrated, then you can simply apply bandpass filters to separate them from one another. Perhaps I'm not understanding the difficulty you're having. $\endgroup$ – Jason R Jul 2 '14 at 17:36
  • $\begingroup$ First of all, thank you SO much for taking the time to try to answer my question! But implementing the bandpass filters would mean that I know the frequencies of the emitted signals but I don't. Nor even the range of each one of them, the only thing we know for sure is that X(t) has more energy than s(t) to the point that it masks the latter. $\endgroup$ – Haydie Jul 2 '14 at 18:21
  • $\begingroup$ One simple method might be to analyze the signal in the frequency domain. If it really resembles what you drew, you should be able to pick out the two regions. What is the brown jagged line that goes above both of them? $\endgroup$ – Jason R Jul 2 '14 at 18:26
  • $\begingroup$ I drew the brown jagged line to show how X(t) masks s(t).What I've plotted is the best case but most often than not we can't distinguish who is who. And if the signal is sinusoidal piggy-backed by a carrier then if I am not mistaken , we will have bars and we won't know what is whose. I think I was mistaken to draw it this way since I took the best case, but basically X(t)*H(t) should swallow s(t)*g(t) and give a big mess of a signal Y(t). $\endgroup$ – Haydie Jul 2 '14 at 18:34
  • $\begingroup$ Perhaps some more concrete examples/illustrations would help. $\endgroup$ – Jason R Jul 2 '14 at 19:03
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If you want to separate two sources you need some kind of diversity. Your know almost nothing about these signals so the only diversity you can exploit is frequency one. So first both spectrum should not be overlapped. And the second: as you know not all transmitted energy of the strong source (that you called $X(t)$) is allocated in the band you draw. Some out of band emission exists and its power depends on shape filter and power amplifier. So if even $X(t)$ and $s(t)$ are not overlaped at all, $s(t)$ may be totally shaded by out of band emission of $X(t)$. And you can't filter it out. You may refer to the problem of multiple access interference (MAI) problem of multi-user systems for details.

Now suppose frequency diversity is big enough and no out of band emission of $X(t)$ interferes with $s(t)$. As you were told FFT can be used to allocate both of the signals and further we can do bandpass filtering to completely distinguish them. But you probably need a large amount of data to achieve high resolution with FFT approach. Though you can improve FFT performance by windowning data set. The main advantage of FFT in this case is its ability to allocate signal even at very bad SNR (0 or less). You only need to increase its size.

The alternative is using some correlation properties and second order statistics. As I understand Blind source separation rely on it (PCA, ICA, but I can mistake). You may estimate input signal autocorrelation matrix as $R = E[xx^H]$. Further if you make $svd(R)$ you'll get singular eigen values of the signal mixture. Using these methods you can separate sources only if their eigenvalues are distinguishable from each other and noise ones (noise subspace). If the diversity is low or signal is too weak its eigen value decreases and it becomes "invisible" to the algorithm. The main advantage is much less amount of data is needed to allocate sources in comparison with FFT approach.

It is also important how large band of search is. It restricts complexity of the computations required. Furthermore $svd$ based methods (e.g. spectrum estimation) are good for narrowband sources (as I've seen, can mistake) while FFT is well enough for both narrow- and broadband cases.

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  • $\begingroup$ Hello! Thanks for your precious input! I'm currently reading a paper about the method you've suggested. But do you think that we could use it when we have ONE mixture of the source signals which is Y(t)? I thought about using the same signal with a delay to create another mixture or by parsing the signal. What do you think? $\endgroup$ – Haydie Jul 4 '14 at 8:00
  • $\begingroup$ Are you talking about $svd$ or $fft$ method? Howbeit you need some amount of data, of cause (no single scalar input). In $fft$ approach you need to feed fft core with data vector of the same length as fft. In subspace based method you should obtain data matrix to estimate $R$ further (ref. Haykin - Adaptive filter theory, chapter 11 for further details). You also can pick signals blocks with overlap to guarantee no losses if the signals have weak stationarity. $\endgroup$ – Serj Jul 4 '14 at 9:21
  • $\begingroup$ I still some small questions but thank you for showing me the way. $\endgroup$ – Haydie Jul 8 '14 at 7:32

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