Asymptotic Performance of Low Rank Adaptive Normalized Matched Filter Test Under Large Dimensional Regime

This paper investigates the performance of the low rank adaptive normalized matched filter (LR-ANMF) test under a new asymptotic regime where both the dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> and sample size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> tend to infinity while their ratio <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N / K$</tex> converges to a constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$c \in(0, \infty)$</tex>. To derive the asymptotic distribution of the test statistic of LR-ANMF under this regime, we establish theoretical results on the asymptotic behavior of random bilinear forms. Leveraging these results, we obtain analytical expressions for the asymptotic false alarm probability and detection probability of LR-ANMF. The theoretical expressions are validated through Monte Carlo simulations. The results indicate that the performance of LR-ANMF approaches optimality as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> increase at the same rate.