# How to distinguish signal and noise eigenvalues of autocorrelation matrix for MUSIC algorithm?

MUSIC (multiple signal classification) algorithm is introduced in Schmidt, Ralph. "Multiple emitter location and signal parameter estimation." IEEE transactions on antennas and propagation 34.3 (1986): 276-280.

An important part of this algorithm is to distinguish eigenvalues froma sample autocorrelation matrix ($R_{N}=\frac{1}{N}\sum{XX^{H}}$) into two groups: noise eigenvalues and signal eigenvalues.

Eigenvalues are plotted as below. A few nontrivial signal eigenvalues and lots of trivial (zero or close to zero) noise eigenvalues: The figure above is in a paper Quinlan, Angela et al. “Model Order Selection for Short Data: An Exponential Fitting Test (EFT).” EURASIP J. Adv. Sig. Proc. 2007 (2007): n. pag.. The paper is telling how to distinguish signal eigenvalues and noise eigenvalues. The authors use exponential fitting test (EFT), which is not that simple, but seems effective.

On the other hand, I've found two implementations of MUSIC algorithm in Python and MATLAB. Interestingly, they've used very simple approach: calculate a difference (i.e., a slope) between adjacent eigenvalues and find an index of the most largest difference. Then, noise eigenvalues are with indices larger than the index of the largest difference.

Intuitively, EFT can coincide with the largest difference approach. However, it is just an intuition, and does not include an analysis or reasoning.

Is it okay to use the largest difference approach to distinguish signal eigenvalues and noise eigenvalues? If so or if not, what is a good explanation of it?