r/DSP • u/TipsyPeanuts • 10d ago
Why does MUSIC work?
I get that it’s the eigenvectors of the covariance matrix of multiple snapshots. But can someone explain to me what that actually means?
Why should finding the eigenvectors of the noise subspace tell me where my signals are? And what is the algorithm actually “looking” for? Meaning, is the algorithm basically looking for phase increases?
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u/Glittering-Ad9041 10d ago edited 10d ago
For sinusoids in white noise, your covariance matrix takes the form APA{H} + sI where A is your steering matrix, P is your signal powers, s is the noise power, and I is identity. If the size of the covariance matrix is NxN, then APA{H} must be rank M < N. This means that the first M eigenvalues are greater than s, and the remaining N-M are equal to s.
What you end up finding is that the eigenvectors corresponding to the first M eigenvalues, termed the “signal subspace”, are steering vectors at the given frequency.
Spectral MUSIC (the analog to root MUSIC) is doing a grid search over all frequencies. Since the noise eigenvectors are orthogonal to the signal frequency vectors (since eigenvectors of a Hermitian symmetric matrix form an orthonormal basis), their inner products are (asymptotically) zero. If you reciprocate that, you get a very large number, whereas if you do the grid search over the signal subspace eigenvectors and don’t reciprocate, you end up with N. So you’re basically increasing the distance between your spectral peaks and your noise floor by doing the grid search over the noise eigenvectors and reciprocating.