Low complexity and high accuracy angle of arrival estimation using eigenvalue decomposition with extension to 2D AOA and power estimation
© Al-Jazzar et al; licensee Springer. 2011
Received: 9 July 2011
Accepted: 7 October 2011
Published: 7 October 2011
In this paper, an angle of arrival (AOA) estimator is presented. Accurate AOA estimation is very crucial for many applications such as wireless positioning and signal enhancement using space processing techniques. The proposed estimator is based upon applying the eigenvalue decomposition (EVD) method on the crosscorrelation matrix of the received signals at two sides of the antenna array doublets. The proposed method is named the eigenvalue-decomposition-based AOA (EDBA) estimator. In comparison with the ESPRIT algorithm, the EDBA has less complexity because the decomposition in the EDBA method is performed only once and on a smaller matrix dimension than that in the ESPRIT algorithm where the decomposition is performed twice. The other advantage is that the EDBA method has better performance than the ESPRIT algorithm. The EDBA is also extended to two-dimensional (2D) AOA estimation with automatic pairing in two ways. The first one performs the 2D AOA estimation by considering the eigenvalues of the crosscorrelation matrix to estimate the azimuth angles and their corresponding eigenvectors to estimate their corresponding elevation angles. Thus, the 2D AOA estimation is performed with automatic pairing and without the need for any pairing or searching techniques. The first 2D extension of the EDBA is named the EDBA-2D estimator. Another 2D AOA estimator that is based upon the EDBA method is also presented and named the Two-EDBA estimator. This second 2D estimator performs the pairing between the azimuth and elevation angles using the alignment of the eigenvalues' magnitudes. An additional advantage for the EDBA estimator is the fact that it provides an estimate for the received signal power paired automatically with its corresponding AOA estimate. Simulations of the proposed EDBA method and its 2D extensions with the signals' power estimation are shown to assess their performance.
Many applications require accurate estimation of the angle of arrival (AOA) of the received signal. Examples of these applications are signal reception enhancement [1, 2] and accurate wireless positioning [3, 4]. Many effects cause errors in the AOA estimation. An error in the AOA estimate can cause huge error when locating wireless devices or an increase in the bit error rate when detecting the received signal. Thus, it is of interest to develop techniques to estimate the AOA accurately.
In some cases, it is required to estimate two-dimensional (2D) angles, i.e., the azimuth and elevation angles. Besides the accuracy requirement for estimating both the azimuth and elevation angles, it is of interest to pair them for each of their corresponding source. The complexity of the estimator is another issue that requires the designers of the estimator to take care of. Also, in 2D AOA estimation, the antenna array geometry plays an important role in the estimation procedure. In this paper, we will assume two antenna array geometries and we will provide a 2D estimator, based upon the proposed estimator, for each of these geometries. The first one is the rectangular antenna array shape formulated from two-parallel uniform linear arrays (ULAs). The second antenna array geometry considered in this paper is the L-shaped antenna array formulated from two perpendicular ULAs.
ESPRIT or ESPRIT-based algorithms are proposed in the literature to perform the 2D AOA estimation, such as in [5–8]. Also, another method proposed to perform the pairing procedure is the modified propagator method (PM) [9, 10]. Although these methods do not require complex searching techniques as in [11–14], they still require pairing techniques in matching the angles estimated. The 2D AOA estimator proposed in  uses two-parallel ULAs, whereas the method in  is based upon matrix enhancement and the matrix pencil algorithm. The proposed method in  constructs a second-order statistic based upon the Schur-Hadamard product steering vector to perform the 2D AOA estimation. In , the authors propose an estimator that requires two ULAs to pair the azimuth and elevation angles. In [19–22], the authors propose 2D AOA techniques that apply singular value decomposition (SVD) on specially constructed covariance matrices.
In this paper, we propose an AOA estimator that enhances the AOA estimation with low computational complexity. The proposed AOA estimator utilizes the eigenvalue decomposition (EVD) method and is named the eigenvalue decomposition-based AOA (EDBA) estimator. The EDBA applies the EVD on a crosscorrelation matrix constructed from both sides of the antenna array doublets.
The first advantage of the proposed EDBA method is its low complexity. In fact, it has lower computational complexity than that of the well-known ESPRIT algorithm due to the fact that the decomposition used in the EDBA method is applied only once and on a smaller matrix dimension than that in the ESPRIT algorithm where the decomposition is applied twice . The second advantage of the proposed methods is its better performance in estimating the AOA than that of the ESPRIT algorithm. Another benefit for using the EDBA method is its ability to estimate the received signals' power with automatic pairing with their corresponding AOA estimates.
The EDBA is also extended for 2D AOA estimation in two ways. The first one is named the EDBA-2D method. In this extension, the azimuth and elevation angles are estimated and paired automatically. The EDBA-2D method starts by formulating a crosscorrelation matrix from the received signals at both sides of two-parallel ULAs. Then, the azimuth angles are estimated from the eigenvalues of the crosscorrelation matrix, and the elevation angles are estimated from the corresponding eigenvectors of the same crosscorrelation matrix. Thus, automatic pairing between the estimated azimuth and elevation angles is provided by the correspondence between the eigenvalues and the eigenvectors of the crosscorrelation matrix. The second way presented to extend the EDBA method for 2D AOA estimation uses the EDBA method twice and separately at the two ULAs of the L-shaped antenna array to estimate the azimuth and elevation angles. Then, we pair them with their corresponding sources using the alignment of the eigenvalues' magnitudes. This method is named the Two-EDBA estimator. The EDBA-2D and Two-EDBA methods have the advantage of not requiring any searching process as well as possessing the advantages of the EDBA method over the ESPRIT algorithm as discussed above.
The paper is organized as follows: Section 2 introduces the system model that forms the foundation for the EDBA estimator. The proposed EDBA estimator is explained in Section 3. Section 4 presents the 2D AOA system models. The proposed 2D extensions for the EDBA method are presented in Section 5. Estimating the received signals' power is presented in Section 6. The simulated performance of the EDBA estimator and its 2D extensions with the signals' power estimation are presented in Section 7. Finally, conclusions are shown in Section 8.
2 System model for the EDBA estimator
where n is an integer so that s k (nT) = α k (n).
with λ is the signal wavelength, d d is the distance between consecutive antenna doublets, and θ k is the AOA for the k th signal. In addition, n1(i) is the noise vector added to the received signal at the antenna elements located at the upper part of the antenna doublets, which is additive white Gaussian noise (AWGN) and has a covariance matrix of σ2 I M × M , where I M × M is the M × M identity matrix. Finally, (.) T represents the transpose operation.
From now on, we will drop the term (i) from all terms for simplicity.
and n2 is an AWGN at the lower side of the antenna array doublets.
Now we showed the received signal model at the antenna arrays, and we will next propose the EDBA estimator.
3 Proposed eigenvalue-decomposition-based AOA (EDBA) estimator
where p k is the power of the k th received signal.
We use here the assumption that n1 and n2 are independent realizations of AWGN.
where the largest K values of the diagonal elements of ΓZ correspond to the K sources which we will call γ1→K≡ γ1 → γ K . Each element of γ1→Kcorresponds to one of the K sources.
where T is the appropriate matrix to change the basis vectors.
where Ω = ϒ-1 and is a diagonal matrix (since ϒ is a diagonal matrix). Thus, each column of is a rotated scaled version of a corresponding column in A(θ).
where u k , μ is the μ th element of the vector u k .
The performance of the proposed method will be shown in the simulation section (Section 7). In comparison with the ESPRIT method, the EDBA uses the EVD on an M × M matrix as shown in (10), whereas in the ESPRIT method, the EVD will be applied on a 2M × 2M for the same system model assumed . Also, the EVD in the EDBA is applied only once, whereas in the ESPRIT method, the EVD is applied twice for estimating the AOA as shown in . Next, we will present the system models that will be used for the extension of the EDBA method for estimating the 2D AOA with automatic pairing.
4 System model for 2D AOA estimation
4.1 Parallel ULA
with λ is the signal wavelength and d d is the distance between consecutive antenna elements at each ULA. In addition, n A is the noise vector added to the received signal at the antenna elements (from m = 1 → M) located along the right-hand side of each antenna doublet, which is AWGN and has a covariance matrix of σ2I M × M .
and n B is an AWGN vector at the left-hand side of the antenna array doublets.
The next subsection will provide some insight into the L-shaped antenna array.
4.2 The L-shaped antenna array
where n z is the AWGN noise vector with a covariance matrix of σ2I M × M .
Likewise, n x is the AWGN noise vector, and it also has a covariance matrix of σ2I M × M .
Thus, the received signal models for the different 2D antenna arrays structures are presented.
5 Extension of the EDBA for 2D AOA estimation
In this section, we will propose two methods for which the EDBA can be used to estimate the azimuth and elevation angles with automatic pairing. The first method is named the EDBA-2D method that is based upon the parallel antenna array structure illustrated in Figure 2. The second method is named the Two-EDBA method that is based upon the L-shaped antenna structure in Figure 3.
5.1 Proposed EDBA-2D estimator
where the largest K values of the diagonal elements of correspond to the K sources which we will call . Each element of corresponds to one of the K sources. Also, is the eigenvector matrix for R BA .
where is the appropriate matrix to change the basis vectors.
For the elevation angle (θ k ), it is estimated from the eigenvectors of R BA in a similar way as discussed in the EDBA estimator explained in Section 3. We will not repeat the discussion and derivation again in this section but rather we will give the final procedure on how to estimate the elevation angle θ k .
where is the μ th element of the vector .
Because of eigenvalue-eigenvector correspondence, estimation of the azimuth angle (ϕ k ) for the k th signal from the k th eigenvalue and its corresponding elevation angle (θ k ) from the corresponding k th eigenvector is performed with automatic pairing and without the need of any pairing procedure.
5.2 Proposed two-EDBA estimator
The Two-EDBA method is based upon the L-shaped antenna structure in Figure 3. The Two-EDBA starts by estimating the azimuth and elevation angles separately. This is a straightforward procedure using the EDBA method on each of the received signals on both ULAs at each axis. Then, after estimating the azimuth (ϕ k ) and elevation (θ k ) angles, the pairing is achieved by the alignment of the eigenvalues' magnitudes. From Section 3, it is shown that the EDBA method estimates the angles from the eigenvectors of the crosscorrelation matrix R21. Since the eigenvalues' magnitudes of the crosscorrelation matrix R21 depend on the received signals' power (which are assumed to be the same on both ULAs on both axis), then the pairing between the estimated azimuth and elevation angles is achieved by aligning the estimated azimuth and elevation angles depending on their corresponding eigenvalues' magnitudes.
6 Estimating the received signals' power
This section shows how the received signals' power can be estimated after estimating the received signals' AOAs. This is achieved with an automatic pairing between the received signals' power with their corresponding AOAs. Pairing can be performed for the EDBA estimator or its 2D estimator extensions. For sake of simplicity, we will show how the received signals' power is estimated for the simple EDBA estimator explained in Section 3, and the extension for the 2D EDBA estimators can be easily deduced.
Similarly, the matrix Z depends on θ k in its formulation as shown (6) and (7). Thus, from , we can reformulate the matrix Z, and we will give it the notation .
where (.)+ is the pseudo-inverse operator.
Thus, the received signal power can be deduced from the diagonal elements of , and we will give it the notation . The pairing between and is automatically provided from the fact that is estimated from the eigenvectors in U and is estimated from their corresponding eigenvalues in ΓZ. Since each eigenvector in U corresponds automatically to a specific eigenvalue in ΓZ, then is automatically paired with its corresponding . Thus, the estimation of the received signal power is estimated with automatic pairing with its corresponding received signal AOA.
Similarly, the estimation of the received signal power can be easily applied to the 2D AOA estimation extensions for the EDBA explained in Section 5, resulting in estimating the azimuth/elevation angles jointly with the received signals' power with automatic pairing.
The performance of the proposed methods will be shown in the simulation section (Section 7).
7 Simulation results
Simulations of the proposed EDBA estimator with its extensions were completed to assess their performance. The elements of each antenna array were separated by a half-wavelength (i.e., ). Unless mentioned otherwise, the number of antenna doublets was set to M = 10. The number of snap shots was set to W = 256 and W = 1,000 (where W is the number of snap shots over which the correlation matrices were estimated). The number of sources was set to 2. The near far ratio (NFR) was set to -10 dB, where the NFR is defined as . The proposed EDBA and its extensions were compared with the ESPRIT method.
Thus, the proposed EDBA-2D and Two-EDBA methods have high capability of pairing the estimated azimuth and elevation angles with good accuracy and in an automatic procedure.
In this paper, we propose an AOA estimator that we named the EDBA method. The EDBA method is applied by taking the EVD of the received signal crosscorrelation matrix. The AOA of the received signals is contained in the eigenvectors of the crosscorrelation matrix. So, the AOA of the received signals is deduced from these eigenvectors. Two extension methods for 2D estimation are introduced. The first extension method estimates the elevation angles from the eigenvectors of the crosscorrelation matrix, and the corresponding azimuth angles are estimated from the corresponding eigenvalues. This method is named the EDBA-2D. The second extension method is named the Two-EDBA estimator and uses the alignment of the eigenvalues' magnitudes to pair the azimuth and elevation angles. Also, the EDBA method is extended to estimate the received signals' power with automatic pairing with their corresponding AOAs. Numerical simulation indicated that the EDBA method and its extensions outperformed the ESPRIT AOA estimator and other pairing methods. Also, the EDBA method and its extensions are low complex and do not require any searching or pairing procedure to perform the estimation.
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