Partial Relaxation Approach: An Eigenvalue-Based DOA Estimator Framework

Abstract

In this paper, the partial relaxation approach is introduced and applied to the DOA estimation problem using spectral search. Unlike existing spectral-based methods like conventional beamformer, Capon beamformer or MUSIC which can be considered as single source approximation of multi-source estimation criteria, the proposed approach accounts for the existence of multiple sources. At each considered direction, the manifold structure of the remaining interfering signals impinging on the sensor array is relaxed, which results in closed form estimates for the "interference" parameters. Thanks to this relaxation, the conventional multi-source optimization problem reduces to a simple spectral search. Following this principle, we propose estimators based on the Deterministic Maximum Likelihood, Weighted Subspace Fitting and covariance fitting methods. To calculate the null-spectra efficiently, an iterative rooting scheme based on the rational function approximation is applied to the partial relaxation methods. Simulation results show that, irrespectively of any specific structure of the sensor array, the performance of the proposed estimators is superior to the conventional methods, especially in the case of low Signal-to-Noise-Ratio and low number of snapshots, while maintaining a computational cost which is comparable to MUSIC.

Full Covariance Fitting DoA Estimation Using Partial Relaxation Framework

“…which still retains some geometric structure of the sensor array [1]. In the minimization procedure in (6), the objective function f ([a(θ), B]) is minimized with respect to B.…”

“…The sample covariance matrixR is assumed to be positive definite for T ≥ M , and the signals from other sources are assumed to be uncorrelated with the signals from the direction of interest a. By applying the decoupling approach to the optimization problem in (7) we obtain the PR-CF method [1] θ PR-CF = N arg min…”

“…The optimal σ 2 s is computed by the Capon spectrum in (11) [1] and the pseudo-spectrum is therefore given by…”

“…For an efficient computation of the eigenvalues λ k (θ), the secular function in (20) and the interlacing property in (21) of Theorem 1 are used [1], where z =Û H a, ρ = σ 2 s and D =Λ. In order to compute the minimizer σ 2 s of (19) numerically, the first and second order derivatives of the objective function in (19) w.r.t.…”

“…The Partial Relaxation (PR) framework was introduced to overcome the aforementioned disadvantages of the conventional spectral-based DoA methods [1], [2]. Instead of ignoring the presence of multiple sources, the PR approach considers both the signal impinging from the current direction of interest as well as the interfering ones.…”