Music and Model-Order Selection for Spherically Invariant Random Vectors

Under Gaussian assumptions, the eigen decomposition of the sample covariance matrix (SCM) is the basis for MUSIC and Information Criterion methods. When signals are modeled by Spherically Invariant Random Vectors (SIRV), a natural extension of the SCM is the Normalized Sample Co- variance Matrix (NSCM). We show that the NSCM preserves the eigen subspaces of the covariance matrix of a signal plus white noise model. Moreover, the ratio of the arithmetic mean to the geometric mean of the NSCM lowest eigenvalues is asymptotically proportional to a chi²-distributed random variable. This allows one to estimate the number of signals and then to use MUSIC, as we show in simulations.