We know that the MUSIC algorithm requires independent signal sources. Suppose the channel model is r=Ms+n, where r is the received signal vector, M is the matrix of steering vectors, s is the signal source vector and n is the noise vector. We have R=MSM*+I, and hence R-I=MSM*. Since MSM* is not full rank, there are vectors x, such that (R-I)x=MSM*x=0. Therefore x is an eigenvector of R. If S is full rank, MSM*x=0 means M*x=0, this is the basic idea of the MUSIC algorithm.
However, if S is rank deficient, a correct estimation of the steering vector still implies MSM*=0 (no more a necessary condition but still sufficient), which means the algorithm could still guess the correct steering vectors but at the same time offer some false ones. But the algorithm is still partially valid. Am I right?
Thanks for the answer :-) Do you mean, if we have n Rx antennas and the rank of S matrix is p, the solution set dimensions of the equation system MSM*x=0 is (n-p), which means the dimension of the noise subspace is (n-p) and there could only be p peaks. If S is rank deficient, there would be more than p signal sources, so the algorithm fails?
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