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I've been looking at every tutorial and paper I can find, but I keep losing the intuition behind what's going on when they get into the matrix operations.

There is a step in which the signal vector is multiplied by its hermitian transpose and the correlation matrix of the product is computed. What do both of these steps accomplish from a high level?

Where is the 'e' term arising from in the steering vector?

I also don't understand the last part about how the orthogonality of the signal and noise subspaces is used to disentangle signals. We use a function in which we have 1 divided by the product of vectors representing the signal and noise subspaces. If they equal 0, wouldn't this function be undefined since it would be equal to 1/0?

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    $\begingroup$ Welcome to DSP.SE! Your question is a little vague: it would help if you wrote out the equations you are having problems with so we can see what you mean. $\endgroup$ – Peter K. Sep 26 '15 at 16:46
  • $\begingroup$ you can understand it, studying PCA (Principal Components Analysis) $\endgroup$ – Stefano Jan 28 '16 at 20:15
  • $\begingroup$ @Stefano, perhaps you should explain more, and submit an answer. I'd be interested to see how PCA is linked to MUSIC. $\endgroup$ – user2699 Apr 28 '16 at 2:53
  • $\begingroup$ FWIW dsp.stackexchange.com/questions/33652/… $\endgroup$ – Atul Ingle Aug 27 '16 at 2:45
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First of, if you can read German: I was author on a thing, back in the day.

There is a step in which the signal vector is multiplied by its hermitian transpose and the correlation matrix of the product is computed. What do both of these steps accomplish from a high level?

So, first of all, MUSIC can be used for a couple of things. Originally, it was used for direction finding (underwater microphones, if I remember correctly), and later was used for spectrum estimation. The idea is this:

You have two inputs. In the direction finding usage, those are two different microphones, in the frequency estimation case, it is but one signal.

You then estimate the covariance matrix – I'm going to focus on the frequency estimation part for now, so you're focussing on the autocovariance matrix – which basically says "If I shift my signal vector by N samples, this is how similar it looks" ("similarity" = dot product in each element of the matrix). That's your dyadic produce (ie. multiplication with conjugate transpose) – nothing but an estimator for the autocovariance matrix.

Then you go

"OK, let's abstract this and take it from a pure linear algebra perspective: these column vectors of this $M\times M$ matrix describe my signal, and knowing that they must be linearly independent (noise being uncorrelated, and present), they form a base of the receive signal space.

What about we decompose that space into one subspace with only noise vectors, and one containing only signal vectors?

If we do that, the base vectors of the signal subspace must be the periodicities contained in my signal.

So you go ahead and do that with an appropriate Eigenvalue decomposition, sort the eigenvectors by magnitude, draw an arbitrary line between noise and signal, and say "ok, the eigenvectors belonging to eigenvectors above that threshold are signal, the others noise representatives".

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  1. The music algorithm decomposes the spatial space into a signal and noise sub spaces. Thus given number of sources one can perform a one dimensional angle search to obtain the DOA.

  2. The denominator is never zero due to noise. Thus a simple peak detector is utilized in MUSIC.

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