Abstract
The statistical resolution limit (SRL), which is defined as the minimal separation between parameters to allow a correct resolvability, is an important statistical tool to quantify the ultimate performance for parametric estimation problems. In this article, we generalize the concept of the SRL to the multidimensional SRL (MSRL) applied to the multidimensional harmonic retrieval model. In this article, we derive the SRL for the so-called multidimensional harmonic retrieval model using a generalization of the previously introduced SRL concepts that we call multidimensional SRL (MSRL). We first derive the MSRL using an hypothesis test approach. This statistical test is shown to be asymptotically an uniformly most powerful test which is the strongest optimality statement that one could expect to obtain. Second, we link the proposed asymptotic MSRL based on the hypothesis test approach to a new extension of the SRL based on the Cramér-Rao Bound approach. Thus, a closed-form expression of the asymptotic MSRL is given and analyzed in the framework of the multidimensional harmonic retrieval model. Particularly, it is proved that the optimal MSRL is obtained for equi-powered sources and/or an equi-distributed number of sensors on each multi-way array.
Highlights
The multidimensional harmonic retrieval problem is an important topic which arises in several applications [1]
We show that the multidimensional statistical resolution limit (SRL) (MSRL) based on the Cramér-Rao Bound (CRB) approach is equivalent to the MSRL based on the hypothesis test approach for a fixed couple (Pfa, Pd), and (iii) the hypothesis test is shown to be asymptotically an uniformly most powerful test which is the strongest statement of optimality that one could expect to obtain [28]
Let the MSRL denotes the l1 normc between two sets containing the parameters of interest of each source
Summary
The multidimensional harmonic retrieval problem is an important topic which arises in several applications [1]. We conduct a hypothesis test formulation on the observation model (5) to derive our MSRL expression in the case of two sources. Let the MSRL denotes the l1 normc between two sets containing the parameters of interest of each source (which is the naturally used norm, since in the monoparameter frequency case that we extend here, the SRL is defined as δ = f1 - f2 [13,14,34]). Assuming that CRB(δ) exists (under H0 and H1), is well defined (see section “MSRL closed-form expression” for the necessarye and sufficient conditions) and is a tight bound (i.e., achievable under quite general/weak conditions [36,37]), the noncentral parameter ’(Pfa, Pd) is given by [[35], p.
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