Elsevier

Signal Processing

Volume 90, Issue 3, March 2010, Pages 809-820
Signal Processing

Unitary-JAFE algorithm for joint angle–frequency estimation based on Frame–Newton method

https://doi.org/10.1016/j.sigpro.2009.08.013Get rights and content

Abstract

The joint angle–frequency estimation (JAFE) based on the multidimensional ESPRIT is recently developed high-resolution parameter estimation technique for determining the directions of arrival (DOAs) and center frequencies of a number of narrow-band sources in a band of interest impinging on the far field of a planar array. In this paper, we present a simple and efficient joint angle–frequency estimation method via the unitary transformation, named as Unitary-JAFE algorithm. In the final stage of Unitary-JAFE, the matrix required to eigendecomposition is a complex unsymmetric matrix. For the eigendecomposition of a complex unsymmetric matrix, we will develop an effective algorithm based on Frame method and Newton iteration method, referred as Frame–Newton algorithm. The proposed method offers a number of advantages over other recently proposed ESPRIT-based techniques by taking advantage of the temporal smoothing, spatial smoothing, and forward–backward averaging techniques. Firstly, except for the final eigenvalue decomposition of dimension equal to the number of sources, it is efficiently formulated in terms of real-valued computation throughout. Secondly, in the final stage of the proposed algorithm, the real and imaginary parts of the i th eigenvalue of a matrix are one-to-one related to the frequency and direction of arrival (DOA) of the i th source. That is, the pairing of the estimated frequency and DOA is automatically determined. Thirdly, the proposed method avoids the joint diagonalization processing, which reduces the computation complexity. Finally, simulation results are presented verifying the efficacy of the proposed algorithm.

Introduction

The recovery of signal parameters from noisy observations is a fundamental problem in array signal processing. Optimal techniques based on maximum likelihood [1], [2] are often applicable but might be computationally prohibitive. To reduce the complexity, Fleury et al. [3] proposed a space-alternating generalized expectation maximization (SAGE)-based algorithm, in which only a subset of the parameters is estimated iteratively while the other parameters remain fixed. Despite its effectiveness, it needs the signature of the transmit signal to achieve the signal decomposition in the expectation (E) step, and the computational load may still be too high due to the iteration process. Algebraic techniques based on a batch of data have an edge in terms of computational complexity. Such techniques make specific use of certain algebraic structures present in the data matrix. For example, a two-dimensional (2-D) multiple signal classification (MUSIC) [4]-based algorithm was considered in [5]. However, it estimates the parameters by carrying out high-dimensional eigendecompositions of the signal covariance matrices. In addition, it requires a 2-D search on the DOA-frequency plane, and thus still calls for an enormous amount of computations. To alleviate the computational overhead, several one-dimensional (1-D) subspace-based algorithms were reported. For example, Lin et al. [6] proposed FSF MUSIC which describes a tree-structured frequency-space-frequency (FSF) MUSIC-based algorithm for joint angle and frequency estimation. Although, with such a tree-structured estimation scheme, the estimated angles and frequencies are automatically paired without extra processing, this method employs three 1-D MUSIC-type algorithms, i.e. two F-MUSICs and one S-MUSIC. Thus, this method requires multiple 1-D search, which is computationally not very attractive, too. Another prime example of an algebraic technique is the ESPRIT algorithm [7]. Due to its simplicity and high-resolution capability, ESPRIT has been used for joint azimuth and elevation angle estimation (2-D DOA estimation) [8], [9], [10], [11], joint DOA and delay estimation [12], [13], [14], joint angle–frequency estimation (JAFE) [15], [16], [17], [18], [19], [20], etc. Their parameter estimates are obtained by exploiting the rotational invariance structure of the signal subspace, induced by the translational invariance structure of the associated sensor array. This can be achieved without computation or search of any spectral measure. In particular, Zoltowski et al. [15] discussed a similar problem of angle–frequency estimation using multiple scales in time and space. Because of ambitious goals, however, their solutions are very much directed by engineering considerations, which incur a certain sacrifice in elegance and clarity. Haardt et al. [16] discussed the problem in the context of mobile communications for space division multiple access (SDMA) applications. Their method is based on unitary-ESPRIT, which involves a certain Cayley transformation of the data to real-valued matrices. This provided a computationally efficient solution scheme but might lead to numerical inaccuracies, particularly when the eigenvalues are closed to ±π. Chen et al. [17] made use of two 1-D estimation of signal parameters via ESPRIT algorithm to estimate these two parameter pairing with a marked subspace scheme to overcome the parameter pairing problem. This method, however, cannot discriminate the rays with very close DOAs or very close frequencies. Another 1-D ESPRIT-based algorithm was also suggested in [18], which, nevertheless, requires a special array configuration. Yet another 1-D ESPRIT-based algorithm, termed C-JAFE, was addressed recently in [19], [20]. C-JAFE algorithm achieves more accurate results than previous ESPRIT-based techniques by taking advantage of the temporal smoothing, spatial smoothing, and forward–backward averaging techniques. When two or more signals have the same DOA, the array steering vectors corresponding to signals with the same DOAs are identical, and therefore, the data matrix is rank deficient. Under a certain condition, the temporal smoothing can restore the rank of the data matrix by stacking m temporally shifted versions of the original data matrix. This is because the temporal smoothing enriches the structure of the array steering matrix which is referred to as the extended array steering matrix, such that the extended array steering matrix has a double Vandermonde structure. Employing a similar technique in the spatial domain, coherent signals can be separated by the spatial smoothing. Despite its high-resolution capability, the C-JAFE still involves the joint diagonalization processing apart from the singular value decomposition (SVD) or eigenvalue decomposition (EVD). That is, the C-JAFE algorithm needs to find a nonsingular matrix T such that Eφ (which includes the information of the frequencies of incident signal sources) and Eθ (which includes the information of the DOAs of incident signal sources) are simultaneously diagonalizable by the same matrix T. It is well-known that the joint diagonalization processing is a complex nonlinear optimization procedure. If Eφ and Eθ are n×n real-valued matrices, the complexity of the joint diagonalization is O(n4), namely, the computational complexity of the joint diagonalization is extremely demanding. Another drawback of the C-JAFE algorithm based on the joint diagonalization processing is that it needs an extra pairing procedure to match the separately estimated DOAs and frequencies. The parameter matching makes the joint angle–frequency estimation more difficult to solve.

The objective of the paper is to give a more accuracy and high resolution algorithm for the joint angle–frequency estimation, referred as Unitary-JAFE algorithm. In the final stage of Unitary-JAFE, the i th eigenvalue of a matrix is the form tan(vi/2)+jtan(ui/2), where vi=2πfi/P and ui=(2πfi/c)dsinθi. fi, and θi denote the frequency and DOA of the i th source, respectively. c is the speed of propagation. d is the spacing between the sensors of a uniform linear array (ULA). P is the sample rate. The eigenvalue for each source is thus unique such that Unitary-JAFE provides the real and imaginary parts of the i th eigenvalue of a matrix are one-to-one related to ui and vi, respectively. That is, the paring of the estimated frequency and DOA is automatically determined, in the final stage of new algorithm. Unitary-JAFE is developed as an extension of the C-JAFE and unitary ESPRIT [21] for a ULA. It is similar to [22]. Except for the final EVD of dimension equal to the number of sources, it is efficiently formulated in terms of real-valued computation throughout. However, the data model is different, which in turn results in different temporal and spatial signal structures. In addition, since the parameter to be estimated is now angle and frequency instead of 2-D DOA, the array steering matrix is not the same either.

The outline of this paper is organized as follows. Section 2 briefly introduces the data model and C-JAFE algorithm for joint angle–frequency estimation. In Section 3, we firstly develop the Unitary-JAFE algorithm. Secondly, the Frame–Newton method is given for solving the EVD problem of the complex unsymmetric matrix in the final stage of the Unitary-JAFE algorithm. Finally, the complexity of the proposed algorithm is addressed as well. Section 4 presents several simulation results to verify the performance of the proposed approach. Section 5 provides a concluding remark to summarize the paper.

Section snippets

Data model and C-JAFE algorithm

In this section, we firstly introduce the data model for the joint angle–frequency estimation. Secondly, a generalized data model is derived by the temporal smoothing and spatial smoothing processing. Finally, we review a brief development of C-JAFE algorithm for the joint angle and frequency estimation.

The proposed Unitary-JAFE algorithm

In this section, we firstly discuss a lower computational complexity but higher performance algorithm to solve the joint angle–frequency estimation. Secondly, the Frame–Newton method is developed for solving the EVD problem of the complex unsymmetric matrix. Finally, the computational complexity of the proposed algorithm is addressed.

Simulation results

In this section, we construct several simulations to evaluate the proposed algorithm.

Conclusions

In this paper, an Unitary-JAFE algorithm based on Frame–Newton method is proposed for joint angle–frequency estimation. In this proposed method, the computational complexity is reduced significantly by the real-valued computation, except for the final eigenvalue decomposition. Unitary-JAFE represents a simple method to constrain the estimated signal parameters to the complex eigenvalues of the space–time factor matrix, yielding more accurate signal subspace estimates and avoiding the complex

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