Enumeration for a Large Number of Sources Based on a Two-Step Difference Operation of Linear Shrinkage Coefficients

A novel and computationally efficient source enumeration algorithm is proposed for large-scale arrays with a small number of samples, by employing a two-step difference operation of linear shrinkage (LS) coefficients of sample covariance matrix (SCM) in large-dimensional scenarios. It is firstly proved that the difference between noise LS coefficients tends to zero and there exists a clear gap between the last signal LS coefficient
${\hat{\alpha } ^{(d - 1)}}$
and the first noise LS coefficient
${\hat{\alpha } ^{(d)}}$
in relatively high signal-to-noise ratio (SNR) cases for
$m, n\to \infty$
and
$m/n\to c\in (0,\infty)$
, where
$m$
,
$n$
and
$d$
are the antenna number, sample number and source signal number, respectively. With this property, the first-step difference operation is designed to achieve initial source enumeration. Further considering relatively low or medium SNRs, the second step yields an improved estimation result and is capable of estimating a large number of sources. Furthermore, the applicability of the representative LS coefficients based
$\rm {SCD_{heur}}$
algorithm under various values of
$c$
is analyzed, and a more general condition for guaranteeing its effectiveness is provided. Simulation results are provided, which are consistent with the theoretical analysis.