Source enumeration in reduced-rank reference techniques for source extraction

In source extraction, source enumeration is essential when the true number of sources is unknown. This is the case in reduced-rank reference (coherence) techniques, where a number of references higher than the number of sources is used. The estimation of the number of sources is critical for accurate source extraction. However, this ill-posed inverse problem has not been sufficiently addressed in the literature within the framework of reference techniques. In this paper, after providing a unified formalism for all reference techniques in the literature, three alternative source enumeration methods applicable to all reference techniques are presented: a direct likelihood ratio test (LRT) against the saturated model, a parametric bootstrap technique, and a cross-validation approach. A comparative study is conducted among the three methods based on simulated numerical data, real sound experimental data, and real industrial data from an electric motor. The results reveal two important findings. First, the number of snapshots (spectral windows) used in spectral analysis significantly affects the performance of the three methods, and they behave differently for the same number of snapshots. Second, parametric bootstrapping proves to be the best method in terms of both estimation accuracy and robustness concerning the number of snapshots used.