Elsevier

Signal Processing

Volume 83, Issue 8, August 2003, Pages 1827-1831
Signal Processing

Fast communication
A fast algorithm for 2-D direction-of-arrival estimation

https://doi.org/10.1016/S0165-1684(03)00118-XGet rights and content

Abstract

A computationally efficient method for two-dimensional direction-of-arrival estimation of multiple narrowband sources impinging on the far field of a planar array is presented. The key idea is to apply the propagator method which only requires linear operations but does not involve any eigendecomposition or singular value decomposition as in common subspace techniques such as MUSIC and ESPRIT. Comparing with a fast ESPRIT-based algorithm, it has a lower computational complexity particularly when the ratio of array size to the number of sources is large, at the expense of negligible performance loss. Simulation results are included to demonstrate the performance of the proposed technique.

Introduction

The problem of estimating the two-dimensional (2-D) directions-of-arrival (DOAs), namely, the azimuth and elevation angles, of multiple sources has received considerable attention in the field of array processing [1], [2], [3], [5], [7], [8], [9]. Although the maximum likelihood estimator [2] provides optimum parameter estimation, its computational complexity is extremely demanding. Simpler but suboptimal solutions can be achieved by the subspace-based approach, which relies on the decomposition of the observation space into signal subspace and noise subspace. However, conventional subspace techniques for 2-D DOA estimation such as MUSIC [1] and ESPRIT [3], [5], [7], [8], [9] necessitate eigendecomposition of the sample covariance matrix (SCM) or the singular value decomposition (SVD) of the data matrix (DM) to estimate the signal and noise subspaces, and huge computation will be involved particularly when the dimensions of the underlying matrices are large, for example, in the case of large towed arrays in sonar. Furthermore, many of the above methods [1], [2], [5], [7], [8], [9] require reasonably accurate initial DOA estimates, 2-D search and/or complex pair matching of the azimuth and elevation angles, although the ESPRIT-based technique proposed in [3] is direct, that is, free of these operations.

The aim of this paper is to develop a computationally simple 2-D DOA estimation algorithm by utilizing the propagator method (PM) [6], which is also a subspace scheme but does not involve eigendecomposition or SVD. In fact, Li et al. [4] have proposed a PM-based DOA estimation method but unfortunately, a 2-D peak search is needed. Our derived algorithm is direct and has a lower computational complexity comparing with [3], particularly when the ratio of the number of sensors to the number of sources is large, at the expense of negligible performance loss.

The rest of the paper is organized as follows. The data model is presented in Section 2. Our 2-D DOA estimation algorithm is developed in Section 3. In Section 4, simulation results are included to demonstrate the effectiveness of the proposed method, while in Section 5, conclusions are drawn.

Section snippets

Data model

Consider two parallel uniform linear arrays (ULAs) with interelement spacing equals d, such that one ULA consists of (N+1) sensors while another has N sensors, which is depicted in Fig. 1. From this configuration, we form three sub-ULAs of N sensors, namely, X, Y and Z, which have coordinates (d·i,0), (d·i+d,0) and (d·i,d), for 0⩽iN−1, respectively. As in common ESPRIT-based approach [5], it is necessary to have three subarrays for 2-D DOA estimation, while one-dimensional DOA estimation

Proposed method

We first give the definition of the propagator [6]. Let B=[AT(AΦy)T(AΦz)T]T. The matrix A is partitioned into A=[A1TAT2]T, where A1 and A2 contain the first to pth rows and the (p+1)th to Nth rows of the matrix A, respectively. Similarly, we partition B as B=[A1TB2T]T where B2 comprises the last (p+1)th to 3Nth rows of B. Under the hypothesis that A1 is nonsingular, the propagator P is defined as a unique linear operator of the formPHA1=B2,where H denotes the Hermitian transpose.

Grouping the

Simulation results

Computer simulations have been conducted to evaluate the 2-D DOA estimation performance of the proposed method. Comparison with the direct 2-D ESPRIT algorithm [3] is also made. The sensor displacement d is taken to be half the wavelength of the signal waves. We consider two uncorrelated signal sources with identical powers while the additive noises are white Gaussian processes. The number of snapshots at each sensor is M=200. We use root mean square error (RMSE), which is defined as E{(θ̂i−θi)2

Conclusions

A fast algorithm for 2-D DOA estimation is proposed based on the propagator method. The main motivation of using the propagator is that it only requires linear operations but does not involve any eigendecomposition or SVD as in common subspace methods. It is shown that the proposed method is more computationally efficient than a direct ESPRIT algorithm when the ratio of array size to source number is large, at the expense of negligible loss in estimation performance.

Acknowledgements

The work described in this paper was supported by a grant from City University of Hong Kong (Project No. 7001203).

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