Detection of the Number of Signals by Signal Subspace Matching

We present a novel and computationally simple solution to the problem of detecting the number of signals, which is applicable to both white and colored noise, and to a very small number of samples. The solution is based on a novel and non-asymptotic goodness-of-fit metric, referred to as signal subspace matching (SSM), which is aimed at matching a model-based signal subspace to its sampled-data-based counterpart. We form a set of hypothesized signal subspace models, with the $k$-th model being a projection matrix composed of the $k$ leading eigenvectors of the sample-covariance matrix. This set of hypothesized models is compared to their sampled-data-based counterpart – a projection matrix constructed from the sampled data – via the SSM metric, and the model minimizing this metric is selected. We show that this solution involves the principal angles between the column span of the model and the column span of the model. We prove the consistency of this solution for the high signal-to-noise-ratio limit and for the large-sample limit. The large-sample consistency is shown to be conditioned on the signal-to-noise ratio (SNR) being higher than a a certain threshold. Simulation results, demonstrating the performance of the solution for both colored and white noise, are included.