Detection of the Number of Signals in Uniform Arrays by Invariant-Signal-Subspace Matching
M. Wax, Amir Adler
Abstract
We present a novel and computationally simple solution to the problem of detecting the number of signals in the case of uniform linear arrays (ULA) and uniform rectangular arrays (URA), which is applicable to white and colored noise, and to a very small number of samples. The solution is based on novel and non-asymptotic goodness-of-fit metric, referred invariant signal subspace matching (ISSM), which is aimed at matching the signal subspaces of two subarrays which are translation invariant. We form a pair of projection matrices on the signal subspaces – one for each subarray – which are parameterized by the number of signals <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> and constructed from the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> leading eigenvectors of the sample-covariance matrices of the subarrays. The value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> which minimizes this metric is selected as the number of signals. We prove the consistency of the ISSM criterion for the high signal-to-noise-ratio (SNR) limit, and also for the large-sample limit, conditioned on the noise being white. The evaluation of the ISSM criterion involves only the computation of eigenvectors of the sample-covariance matrix of the the array. Simulation results, demonstrating the improved performance of the ISSM criterion over existing criteria, especially for colored noise, are included.