On the Limiting Distribution of the Likelihood Ratio Statistic in Expected Likelihood AMF Detector under Large Dimensional Regime

The expected likelihood adaptive matched filter (ELAMF) detector proposed by Abramovich et al. exhibits excellent performance in insufficient sample scenario. However, deriving an explicit expression for the median of the likelihood ratio (LR) statistic in EL-AMF remains a significant challenge due to the complicated expression of the LR's probability density function. In this paper, by using the random matrix theory, we derive the limiting distribution of the LR statistic under the large dimensional regime where both the sample size K and dimension N are assumed to tend to infinity whereas their quotient N/K converges to a constant c ∊ (0, 1). Our results reveal that under this regime, the LR converges in distribution to a Log-Normal distribution LN(μγ˜, σ²_γ˜) where μγ˜ = N[c−1/c log(1−c) − 1] and σ ²_γ˜ = − log(1−c) −c. Based on this limiting distribution, we derive an asymptotic yet analytical expression for the median statistic of the LR, given by e^−N (1−c)^−(K−N). Using this expression, we propose a modified EL-AMF detector. Numerical examples validate the theoretical results presented in this paper