Nominal Azimuth Research Articles

SIS-BAM Algorithm for Angular Parameter Estimation of 2-D Incoherently Distributed Sources

For two-dimensional (2-D) incoherently distributed sources, this paper presents an effective angular parameter estimation method based on shift invariant structure (SIS) of the beamspace array manifold (BAM), named as SIS-BAM algorithm. In the proposed method, a shift invariance structure of the observed vectors is established utilizing a generalized array manifold of a uniform linear orthogonal array. Then, based on Fourier basis vectors and the SIS, a beamspace transformation matrix can be obtained. It projects received signals into the corresponding beamspace, so as to carry out dimension reduction of observed signals in beamspace domain. Finally, according to the SIS of beamspace observed vectors, the closed form solutions of the nominal azimuth and elevation are derived. Compared with the previous works, the presented SIS-BAM method provides better estimation performance, for example: 1) the computational complexity is reduced due to dealing with low-dimension beamspace signals and avoiding spectral search; 2) it can not only improve the angular parameter estimation accuracy but also have excellent robustness to the change of signal-to-noise ratio (SNR) and snapshot number. The theoretical analysis and simulation results confirm the effectiveness of the proposed method.

2-D DOA Estimation of Incoherently Distributed Sources Considering Gain-Phase Perturbations in Massive MIMO Systems

In massive multiple-input multiple-output (MIMO) systems, accurate direction-of-arrival (DOA) estimation is important for the base station (BS) to perform effective downlink beamforming. So far, there have been few reports on DOA estimation considering gain-phase perturbations in massive MIMO systems. However, gain-phase perturbations indeed exist in practical applications and cannot be ignored. In this paper, an efficient method for two-dimensional (2-D) DOA estimation of incoherently distributed (ID) sources considering array gain-phase perturbations is proposed for massive MIMO systems. Firstly, a shift invariance structure is established in the subspace framework, and a constrained optimization problem is formulated to estimate the nominal azimuth and elevation DOAs as well as gain-phase perturbations with closed-form expressions, under the assumption that some of the BS antennas are well calibrated; secondly, the corresponding angular spreads are obtained with the aid of the estimated gain-phase perturbations. Theoretical analysis and an approximate Cramér-Rao bound are also provided. An improved estimation performance is achieved by the proposed method as demonstrated by numerical simulations.

Two-Dimensional Localization: Low-Rank Matrix Completion With Random Sampling in Massive MIMO System

In this paper, random sampling is considered for direction-of-arrival (DOA) estimation with reduced hardware complexity in massive multiple-input-multiple-output (MIMO) systems. The resulting problem is that the accuracy of the existing DOA estimators will significantly degrade with the availability of only a small subset of data entries as the low date rate is employed to reduce the system power consumption. To address that, an efficient approach based on the variant of matrix factorization is devised to complete the underlying data matrix. As a result, the nominal azimuth and elevation DOAs with the corresponding angular spreads are estimated from the underlying data matrix. Numerical results demonstrate that even in the case of missing entries, the proposed method is superior to the existing approaches with full data.

DOA Estimation of Two-Dimensional Coherently Distributed Sources Based on Spatial Smoothing of Uniform Rectangular Arrays

Aiming at the direction-of-arrival (DOA) estimation of two-dimensional (2D) coherently distributed (CD) sources which are coherent with each other, we explore the propagator method based on spatial smoothing of a uniform rectangular array (URA). The rotational invariance relationships with respect to the nominal azimuth and nominal elevation are obtained under the small angular spreads assumption. A propagator operator is constructed through spatial smoothing of sample covariance matrices firstly. Then, combination of propagator and identical matrix is divided according to rotational operators, and the nominal angles can be obtained through eigendecomposition lastly. Realizing angle matching automatically, the proposed method can estimate multiple DOAs of 2D coherent CD sources without spectral peak searching and prior knowledge of deterministic angular signal distribution function. Simulations are conducted to verify the effectiveness of the proposed method.

A 2D Nested Array Based DOA Estimator for Incoherently Distributed Sources via Sparse Representation Utilizing L1-Norm

Nested arrays are sparse arrays composed of subarrays with nonuniform sensor spacing. Compared with traditional uniform arrays, nested arrays have more degree of freedoms (DOFs) and larger apertures. In this paper, a nested array has been proposed as well as a direction-of-arrival (DOA) estimation method for two-dimensional (2D) incoherently distributed (ID) sources. A virtual array is firstly obtained through vectorization of the cross-correlation matrix of subarrays. Sensor positions of the virtual array and the optimal configuration of the nested array are derived next. Then rotational invariance relationship for generalized steering matrix of the virtual array with respect to nominal azimuth is deduced. According to the rotational invariance relationship, sparse representation model under l1-norm constraint is established, which is resolved by transferring the objective function to second-order cone constraints and combining a estimation residual error constraint for receive vector of the virtual array. Simulations are conducted to investigate the effectiveness of the proposed method in underdetermined situation and examine different experiment factors including SNR, snapshots, and angular spreads as well as sensor number of subarrays. Results show that the proposed method has better performance than uniform parallel arrays with the same number of sensors.

Estimation of Two-Dimensional Non-Symmetric Incoherently Distributed Source with L-Shape Arrays.

In the field of array signal processing, distributed sources can be regarded as an assembly of point sources within a spatial distribution. In this study, a two-dimensional (2D) non-symmetric incoherently distributed (ID) source model is proposed; we explore the estimation of a 2D non-symmetric ID source using L-shape arrays. The 2D non-symmetric ID source is established by modeling the angular power density function (APDF) as a Gaussian mixture model. Estimation of the non-symmetric distributed source is proposed based on the expectation maximization (EM) framework. The proposed EM iterative framework contains three steps in the process of each circle. Firstly, the nominal azimuth and nominal elevation of each Gaussian component are obtained from the phase parts of elements in sample covariance matrices. Then the angular spreads can be solved through a one-dimensional (1D) search by the original generalized Capon estimator. Finally, weights of each Gaussian component are obtained by solving the least-squares estimator. Simulations are conducted to verify the effectiveness of the estimation technique.

Efficient Beamspace-Based Algorithm for Two-Dimensional DOA Estimation of Incoherently Distributed Sources in Massive MIMO Systems

In massive multiple-input multiple-output (MIMO) systems, accurate direction-of-arrival (DOA) estimation is critical for the transmitters to conduct downlink precoding. In this paper, we focus on the problem of two-dimensional DOA estimation of incoherently distributed sources in massive MIMO systems using uniform rectangular arrays (URAs). First, with the generalized array manifold, we obtain the beamspace array manifold by performing beamspace transformation on the observed vector of the URA, and establish one beamspace shift invariance structure via appropriate beamforming matrix design and array manifold selection. Second, from the beamspace array manifold, we extract two array manifolds, and derive the other beamspace shift invariance structure. Final, the total least squares approach is used to estimate the nominal azimuth and elevation DOAs. With the DOA estimates, the corresponding angular spreads are given in closed-form solutions. Our approach avoids spectral search and reduces the dimensions in matrix operations, thus, it is more computationally attractive than the existing techniques. Numerical results show that the proposed algorithm achieves performance comparable to that of the conventional estimators in massive MIMO systems.

Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays.

Aiming at the two-dimensional (2D) incoherently distributed (ID) sources, we explore a direction-of-arrival (DOA) estimation algorithm based on uniform rectangular arrays (URA). By means of Taylor series expansion of steering vector, rotational invariance relations with regard to nominal azimuth and nominal elevation between subarrays are deduced under the assumption of small angular spreads and small sensors distance firstly; then received signal vectors can be described by generalized steering matrices and generalized signal vectors; thus, an estimation of signal parameters via rotational invariance techniques (ESPRIT) like algorithm is proposed to estimate nominal elevation and nominal azimuth respectively using covariance matrices of constructed subarrays. Angle matching method is proposed by virtue of Capon principle lastly. The proposed method can estimate multiple 2D ID sources without spectral searching and without information of angular power distribution function of sources. Investigating different SNR, sources with different angular power density functions, sources in boundary region, distance between sensors and number of sources, simulations are conducted to investigate the effectiveness of the proposed method.

Estimation for Two-Dimensional Nonsymmetric Coherently Distributed Source in L-Shaped Arrays

We explore the estimation of a two-dimensional (2D) nonsymmetric coherently distributed (CD) source using L-shaped arrays. Compared with a symmetric source, the modeling and estimation of a nonsymmetric source are more practical. A nonsymmetric CD source is established through modeling the deterministic angular signal distribution function as a summation of Gaussian probability density functions. Parameter estimation of the nonsymmetric distributed source is proposed under an expectation maximization (EM) framework. The proposed EM iterative calculation contains three steps in each cycle. Firstly, the nominal azimuth angles and nominal elevation angles of Gaussian components in the nonsymmetric source are obtained from the relationship of rotational invariance matrices. Then, angular spreads can be solved through one-dimensional (1D) searching based on nominal angles. Finally, the powers of Gaussian components are obtained by solving least-squares estimators. Simulations are conducted to verify the effectiveness of the nonsymmetric CD model and estimation technique.

A beamspace approach for 2-D localization of incoherently distributed sources in massive MIMO systems

In this paper, a generalized low-complexity beamspace approach is proposed for two-dimensional localization of incoherently distributed sources with a uniform cylindrical array (UCyA) in large scale/massive multiple-input multiple-output (MIMO) systems. The received signal vectors in the antenna-element space are transformed into the beamspace by employing beamforming vectors. As a beneficial result, the total dimensions of the received signal vectors are significantly reduced. In addition, it is shown that the error introduced by the transformation decreases as the number of UCyA antennas increases. The UCyA is composed of multiple uniform circular arrays (UCAs), and the beamspace array response matrices of adjacent UCAs are linearly related. Then, the linear relation is exploited to estimate the nominal elevation direction-of-arrivals (DOAs) directly and the nominal azimuth DOAs based on a low-complexity search algorithm. In contrast, the linear relation in the traditional approach is based on approximations and the associated search algorithm is more complicated. Numerical results demonstrate that the proposed approach outperforms the existing approach in terms of both performance and complexity in the context of massive MIMO systems.

An ESPRIT-Based Approach for 2-D Localization of Incoherently Distributed Sources in Massive MIMO Systems

In this paper, an approach of estimating signal parameters via rotational invariance technique (ESPRIT) is proposed for two-dimensional (2-D) localization of incoherently distributed (ID) sources in large-scale/massive multiple-input multiple-output (MIMO) systems. The traditional ESPRIT-based methods are valid only for one-dimensional (1-D) localization of the ID sources. By contrast, in the proposed approach the signal subspace is constructed for estimating the nominal azimuth and elevation direction-of-arrivals and the angular spreads. The proposed estimator enjoys closed-form expressions and hence it bypasses the searching over the entire feasible field. Therefore, it imposes significantly lower computational complexity than the conventional 2-D estimation approaches. Our analysis shows that the estimation performance of the proposed approach improves when the large-scale/massive MIMO systems are employed. The approximate Cram\'{e}r-Rao bound of the proposed estimator for the 2-D localization is also derived. Numerical results demonstrate that albeit the proposed estimation method is comparable with the traditional 2-D estimators in terms of performance, it benefits from a remarkably lower computational complexity.

Decoupled Nominal 2-D Direction-of -Arrival Estimation Algorithm for Coherently Distributed Source

A computationally efficient method for nominal 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation of coherently distributed source impinging on the far field is presented. Since the coherently distributed source is characterized by four parameters, the nominal azimuth DOA, angular spread of the nominal azimuth DOA, the nominal elevation DOA, and angular spread of the nominal elevation DOA, the computational complexity of the parameter estimation is normally high demanding. So a low complexity estimation algorithm is proposed in this paper, the key idea of which is to apply a subspace-based method without eigendecomposition in beamspace and a proposed second-order statistics for estimating the nominal elevation and azimuth DOAs. The proposed decoupled estimation algorithm does not involve any searching. It has a lower computational complexity particularly when the radio of array size to the number of source is large, at the expense of negligible performance loss. Simulation results are included to demonstrate the performance of the proposed technique.

A Simplified Estimator for Tridimensional Localization of Single Incoherently Distributed Source

In this paper, a simplified parametric estimator for tridimensional localization of single incoherently distributed (ID) source with small angular spread is proposed. The proposed estimator firstly obtains two sample covariance matrices using the observation data of a L-shape array. And then the secondary diagonal elements of the sample covariance matrices are used to estimate the nominal azimuth and elevation of single ID source. Our technique does not involve any spectrum searching and the eigen-decomposition of the sample covariance matrix, and thus the computational burden has been significantly alleviated. Moreover, it is also a blind estimator which doesn't require any prior knowledge about the angular power density of the ID source. Numerical examples illustrate the performance of the proposed estimator.

A Fast Decoupled Nominal 2-D Direction-of-Arrival Estimation for Coherently Distributed Source

In this paper, we consider the problem of the nominal 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation for coherently distributed source. This new approach is based on the rotation matrices of three parallel uniform linear arrays as deduced, which has decoupled the nominal 2-D DOA from those of angular spreads. The estimator makes use of the eigenvalue decomposition to beamspace data to estimate the nominal elevation DOA. And then using a new cross-correlation matrix, the nominal azimuth DOA estimates are decoupled from the elevation estimates and can be obtained with no searching. The proposed algorithm has lower computational complexity particularly when the radio of array size to the number of source is large, at the expense of negligible performance loss. Simulation results verify the effectiveness of the proposed method.

Low-Complexity Estimation of the Nominal Azimuth and Elevation for Incoherently Distributed Sources

This paper develops a new technique for estimating the two-dimensional direction-of-arrivals (DOAs) of incoherently distributed (ID) sources, which can estimate effectively the nominal azimuth and nominal elevation of multiple ID sources at the cost of less computational complexity. Using a pair of parallel uniform linear arrays (ULAs), a new approach for 2D DOA estimation of multiple ID sources is proposed. The proposed method firstly estimates the nominal elevation by the modified TLS-ESPRIT method, which is based on the approximate rotational invariance property with respect to the nominal elevation between two closely parallel ULAs. And then with the help of the nominal elevation estimates, the nominal azimuth is estimated by one-dimensional searching. Without multi-dimensional searching, the proposed method has significantly reduced the computational cost compared with the existing methods. Simulation results indicate that the proposed method can exhibit a good performance and be applied to the multisource scenario where different sources may have different angular distribution shapes.

Simplified Estimation of 2D DOA for Coherently Distributed Sources

In mobile communications, local scattering in the vicinity of the mobile results in angular spreading as seen from a base station antenna array. In this paper, we consider the problem of estimating the two-dimensional (azimuth and elevation) direction-of-arrival (DOA) parameters of spatially distributed sources. Based on double parallel uniform linear arrays (ULAs), a simplified method without spectrum-peak searching is proposed for the 2D DOA estimation of multiple coherently distributed (CD) sources. The proposed method firstly obtains two approximate rotational invariance relations with respect to the nominal DOAs of CD sources by using one-order Taylor approximation to the generalized steering vectors (GSVs) of two pairs of shifted subarrays. And then a new ESPRIT-based method is utilized to estimate the nominal azimuth DOA and nominal elevation DOA. In addition, a simple parameter matching approach is also given. Compared with the conventional methods, our method has significantly reduced the computational cost and can sustain the estimation performance within a tolerable level. Moreover, our method is a blind estimator without any prior knowledge about angular distribution shape. Numerical examples illustrate the performance of the method.

Low-complexity estimation of 2D DOA for coherently distributed sources

We consider the estimation of two-dimensional (azimuth and elevation) direction-of-arrival (DOA) using a pair of uniform circular arrays under a coherently distributed source model. Since the coherently distributed source is characterized by four parameters, the nominal azimuth DOA, angular extension of the azimuth DOA, nominal elevation DOA, and angular extension of the elevation DOA, the computational complexity of the parameter estimation is normally highly demanding. We propose a low-complexity estimation algorithm, called the sequential one-dimensional searching algorithm by concentrating only on the estimation of DOAs. The SOS algorithm has a basis on the eigenstructure between the steering matrix and signal subspace, and utilizes preliminary estimates obtained at a pre-processing stage. The SOS algorithm estimates the DOAs, but not the angular extensions: although the SOS algorithm does not provide estimates of angular extensions, it is useful when the angular extensions are small. Specifically, it is shown from simulation results that the SOS algorithm exhibits as good an estimation performance as the maximum likelihoood method for coherently distributed sources.