Incoherently Distributed Research Articles

Localization of mixed coherently and incoherently distributed sources based on generalized array manifold

In practice, signals received by an array may be a mixture of coherently distributed (CD) and incoherently distributed (ID) sources, due to some complicated multipath/scattering effects. To localize mixed CD and ID sources, a method based on the generalized array manifold (GAM) is developed in this work. Firstly, source enumeration and source type classification are achieved by jointly employing the minimum description length (MDL) criterion, the covariance matrix difference technique and the second-order statistics of associated eigenvalues. Secondly, with the source enumeration result, the nominal direction of arrival (DOA) of ID sources is estimated by the rank-reduction principle based one-dimensional (1-D) spectral search, where shift invariance of the GAM associated with ID sources is considered. Thirdly, the oblique projection technique is adopted to separate the CD sources from ID ones, and then the nominal DOA of CD sources is estimated via the sparsity-cognizant total least-squares (S-TLS) method. Finally, with aid of the nominal DOA estimates, the corresponding angular spreads are calculated. An analysis of the proposed method is discussed and the approximate Cramér-Rao bound (CRB) is derived. Simulations are provided to demonstrate the performance of the proposed method.

Vehicle Positioning With Deep-Learning-Based Direction-of-Arrival Estimation of Incoherently Distributed Sources

In this article, a novel vehicle positioning system architecture based on direction-of-arrival (DOA) estimation of incoherently distributed (ID) sources is proposed employing massive multiple-input–multiple-output (MIMO) arrays. Such an architecture with the associated signal model is more consistent with the actual array application and multipath transmission scenarios. First, an end-to-end two-dimensional (2-D) DOA estimation of ID sources utilizing a dual one-dimensional (1-D) convolutional neural network (D1D-CNN) under the deep learning (DL) framework is performed, where the normalized covariance matrix data is used for both offline training and online estimation. Then, the received SNR information is exploited to select a set of DOA estimates provided by multiple collaborative BSs for positioning. Moreover, transfer learning and an attention mechanism are employed to promote its generalization ability and achieve robustness against array perturbations. Simulation results are provided to show that the proposed method outperforms the state-of-the-art methods in terms of computational complexity, positioning accuracy, and robustness against array perturbations.

DOA estimation of incoherently distributed sources using importance sampling maximum likelihood

In this paper, an importance sampling maximum likelihood (ISML) estimator for direction-of-arrival (DOA) of incoherently distributed (ID) sources is proposed. Starting from the maximum likelihood estimation description of the uniform linear array (ULA), a decoupled concentrated likelihood function (CLF) is presented. A new objective function based on CLF which can obtain a closed-form solution of global maximum is constructed according to Pincus theorem. To obtain the optimal value of the objective function which is a complex high-dimensional integral, we propose an importance sampling approach based on Monte Carlo random calculation. Next, an importance function is derived, which can simplify the problem of generating random vector from a high-dimensional probability density function (PDF) to generate random variable from a one-dimensional PDF. Compared with the existing maximum likelihood (ML) algorithms for DOA estimation of ID sources, the proposed algorithm does not require initial estimates, and its performance is closer to Cramer-Rao lower bound (CRLB). The proposed algorithm performs better than the existing methods when the interval between sources to be estimated is small and in low signal to noise ratio (SNR) scenarios.

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.

Localization of Mixed Far-field and Near-field Incoherently Distributed Sources Using Two-stage RARE Estimator

In this paper, a mixed source localization method utilizing a two-stage rank-reduction (RARE) estimator is investigated. Different from the existing methods, the proposed one is built on the incoherently distributed (ID) source model, which is more appropriate for multipath and fast time-varying channels. Firstly, a general array manifold (GAM) model is established, where nominal DOAs and nominal ranges are extracted from the initial array manifold. By exploiting the shift invariance property of the far-field (FF) GAM and combining virtual source enumeration result, nominal FF DOA estimation is achieved by a one-dimensional (1-D) RARE spectral search. Secondly, the oblique projection operation is adopted to separate near-field (NF) sources, and the nominal DOA and range parameters of NF sources are subsequently obtained by jointly utilizing the manifold separation technique (MST) and another two 1-D spectral searches. With the estimated nominal DOA and range parameters, the angular spread and range spread are then successfully estimated. Moreover, the Cramér-Rao bound (CRB) for the considered case is also derived. Simulation results are presented to validate the effectiveness of the proposed method.

Noncircularity-based generalized shift invariance for estimation of angular parameters of incoherently distributed sources

In this paper, a reduced-rank angular parameters estimation algorithm is proposed for incoherently distributed (ID) noncircular sources based on a uniform linear array (ULA), which addresses the problems of central direction-of-arrival (DOA) estimation and angular spread estimation. Firstly, the noncircularity property of the signals is utilized to establish an extended generalized array manifold (GAM) model based on the first-order Taylor series approximation. Then, the central DOAs of source signals are obtained based on the generalized shift invariance property of the array manifold and the reduced-rank principle. Next, the angular spreads are estimated from the central moments of the angular distribution. Compared with the existing algorithm without exploiting the noncircularity information, the proposed one can achieve a higher accuracy and handle more sources. In addition, it can deal with a general scenario where different sources have different angular distribution shapes. Furthermore, the approximate stochastic Cramer-Rao bound (CRB) of the concerned problem is derived. Simulation results are provided to demonstrate the performance of the proposed algorithm.

Two-Dimensional Angular Parameter Estimation for Noncircular Incoherently Distributed Sources Based on an L-Shaped Array

In this paper, a two-stage reduced-rank estimator is proposed for two-dimensional (2D) direction estimation of incoherently distributed (ID) noncircular sources, including their center directions of arrival (DOAs) and angular spreads, based on an L-shaped array. Firstly, based on the first-order Taylor series approximation, a noncircularity-based extended generalized array manifold (GAM) model is established. Then, the 2D center DOAs of incident ID signals are obtained separately with the noncircularity-based generalized shift-invariance property of the array manifold and the reduced-rank principle. The pairing of the two center DOAs is completed by searching for the minimum value of a cost function. Secondly, the 2D angular spreads can be obtained in closed-form solution from the central moments of the angular distribution. The proposed estimator achieves higher accuracy in angle estimation that manages more sources and shows promising results in the general scenario, where different sources possess different angular distributions. Furthermore, the approximate noncircular stochastic Cramer-Rao bound (CRB) of the concerned problem is derived as a benchmark. Numerical analysis proves that the proposed algorithm achieves better estimation performance in both 2D center DOAs and 2D angular spreads than an existing estimator.

Parameter Estimation for Two-Dimensional Incoherently Distributed Source with Double Cross Arrays.

A direction of arrival (DOA) estimator for two-dimensional (2D) incoherently distributed (ID) sources is presented under proposed double cross arrays, satisfying both the small interval of parallel linear arrays and the aperture equalization in the elevation and azimuth dimensions. First, by virtue of a first-order Taylor expansion for array manifold vectors of parallel linear arrays, the received signal of arrays can be reconstructed by the products of generalized manifold matrices and extended signal vectors. Then, the rotating invariant relations concerning the nominal elevation and azimuth are derived. According to the rotating invariant relationships, the rotating operators are obtained through the subspace of the covariance matrix of the received vectors. Last, the angle matching approach and angular spreads are explored based on the Capon principle. The proposed method for estimating the DOA of 2D ID sources does not require a spectral search and prior knowledge of the angular power density function. The proposed DOA estimation has a significant advantage in terms of computational cost. Investigating the influence of experimental conditions and angular spreads on estimation, numerical simulations are carried out to validate the effectiveness of the proposed method. The experimental results show that the algorithm proposed in this paper has advantages in terms of estimation accuracy, with a similar number of sensors and the same experimental conditions when compared with existing methods, and that it shows a robustness in cases of model mismatch.

Spatial Spectrum Estimation of Incoherently Distributed Sources Based on Low-Rank Matrix Recovery

This article considers the problem of two-dimensional (2-D) angle estimation of incoherently distributed (ID) sources in massive MIMO systems. We first show that the matrix representing the 2-D spatial spectrum of ID sources is of low-rank rather than sparse. Then, by exploiting the low-rank property, we formulate a rank minimization problem to estimate the spatial spectrum, which is solved by an efficient iterative re-weighted procedure. Finally, the key parameters of the spatial spectrum are obtained via an off-grid estimator. The performance analysis of the proposed method, and the Cramer-Rao bound (CRB) of the key parameters with unknown angular distributions are also presented. Contrary to the existing methods which presume a specific type of angular distribution, the proposed method is suitable for general angular distribution as long as it is of low-rank. Moreover, the proposed method is computationally more efficient than the conventional methods that require a multi-dimensional search. Extensive simulations are provided to verify the performance 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.

Angular Parameters Estimation of Multiple Incoherently Distributed Sources Generating Noncircular Signals

In this paper, we introduce for the first time a new method for the estimation of the angular parameters [i.e., central directions of arrival (DOAs) and angular spreads] of multiple non-circular and incoherently-distributed (ID) sources. The new method is derived by going through three different stages to finally obtain a robust solution that allows decoupling the estimation of each central DOA from its associated angular spread by means of two consecutive 1-D searches. By doing so, we reduce significantly the complexity of the proposed technique as compared to the brute-force 2D grid search solution. This new technique is also oblivious to the sources' angular distribution or any mismatch thereof. By analyzing its performance, we show that our new estimator outperforms most well-known state-of-the-art techniques in terms of estimation accuracy and robustness, especially for small DOA separations and/or low SNR levels. We also derive an explicit expression for the stochastic Cramér-Rao lower bounds (CRLBs) of the underlying estimation problem. These CRLBs are compared to those for circular ID signals. It will be shown that the noncircularity of the signals is more informative about the angular parameters when the sources have different angular distributions and/or when the angular spreads increase. Besides, the noncircular CRLBs depend on the noncircularity rate, the noncircularity phase separation, and the DOA separation.

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.

An efficient method for angular parameter estimation of incoherently distributed sources via beamspace shift invariance

In this paper, we propose a beamspace-based method for nominal direction-of-arrival (DOA) and angular spread estimation of incoherently distributed (ID) sources using a uniform linear array (ULA). Firstly, with generalized array manifold of the ULA, we obtain the beamspace array manifold by performing beamspace transformation on the received vector of two overlapping subarrays, and further derive the beamspace shift invariance structure via designing appropriate beamforming matrix. Next, the total least squares approach is used to estimate the nominal DOAs of ID sources. Finally, with the DOA estimates, the corresponding angular spreads are obtained by means of the central moments of the angular distribution. The proposed method does not involve any spectral search and reduces the dimension of matrix operations, thus it is obviously more efficient than the traditional algorithms. Simulation results indicate that our proposed algorithm is comparable to the existing algorithms when the number of sensors is large.

Application of Manifold Separation to Parametric Localization for Incoherently Distributed Sources

By using the manifold separation technique (MST), we develop a computationally efficient yet accurate estimator for localization of multiple incoherently distributed (ID) sources. In this paper, we have made the following main contributions: first, we use the MST to derive a closed-form expression for the ID signal covariance matrix that is applicable to the case with arbitrary array geometries or large angular spreads; second, we find that the two-dimensional spatial spectrum can be computed efficiently by using the discrete Fourier transform algorithms coupled with the unweighted (or Gaussian-weighted) moving average for the uniformly (or Gaussian) distributed ID sources; finally, we employ the first-order Taylor expansion to formulate a weighted least-squares approach that can improve the estimation performance significantly. Numerical results demonstrate that with less complexity, the proposed estimator offers better estimation performance compared with several classical estimators.

Decoupled two-dimensional direction of arrival estimation of single distributed source by vectoring differential phases

In practical applications such as radar, sonar, and mobile communications, transmitted signals are often affected by the scattering and reflection phenomena, which causes the signal energy received by the antenna array to be distributed into a certain space. In this case, a distributed source model will be more applicable. In general, the distributed sources have been classified as coherently distributed (CD) source and incoherently distributed (ID) source, which prove to be suitable for the cases of slowly time-varying and rapidly time-varying channels, respectively.In this paper, we consider the two-dimensional direction of arrival (DOA) estimation of distributed sources (including CD source or ID source). Specifically, uniform circular array (UCA) is widely used because of its ability to measure full azimuth angle and high resolution. However, the existing estimation algorithms all require spectral peak searching and the eigenvalue decomposition, which can bring a large computational complexity. To solve this problem, a decoupled rapid two-dimensional DOA estimation algorithm is proposed based on vectoring differential phases considering the two cases of single CD source and ID source. Firstly, based on spatial frequency approximation model, it is proved that none of differential phases between the received signals of different sensors in the UCA is affected by angle spread parameters when there is only a single distributed source. Under the premise of such a property, the central DOAs can be decoupled through obtaining the differential phases. Next, we can obtain the phase angles of strictly upper triangular elements in the sample covariance matrix, which correspond to differential phases between different sensors. Finally, by vectoring these differential phases, the central azimuth and elevation DOAs are estimated in the closed form from a least-squared problem, where the spectral peak searching and eigenvalue decomposition can be avoided, hence the computational complexity is reduced greatly. Theoretical analysis and simulation results show that the proposed algorithm has higher estimation accuracy and does not require prior information about the distribution of angular signals. With both low computational complexity and low hardware complexity, the proposed algorithm is beneficial to the engineering practice of array direction finding in complex environment.

Computational Efficient Two-Dimension DOA Estimation for Incoherently Distributed Noncircular Sources With Automatic Pairing

In this paper, the problem of two-dimensional (2-D) direction-of-arrival (DOA) estimation for incoherently distributed (ID) noncircular sources is addressed. A low-complexity estimator with automatic pairing in the three-parallel uniform linear arrays (ULAs) is proposed. First, the signal non-circularity is applied to establish an extended generalized array manifold (GAM) model based on one-order Taylor series approximation. Resorting to such model, the generalized rotational invariance relationships are identified among the extended GAM matrices of the three ULAs. Thereafter, a modified propagator method is utilized to estimate the central elevation and azimuth DOAs. Without any spectrum searching, estimation or eigenvalue decomposition of the sample covariance matrix, the proposed approach is capable of considerably reducing the calculation cost in comparison with the existing methods. In addition, it can automatically pair the estimated central azimuth and central elevation DOAs. We also derive the Cramer-Rao lower bound regarding the 2-D DOA estimation of the ID noncircular source and conduct the computational complexity analysis. Numerical results demonstrate that the proposed method achieves better estimation performance than the existing methods. Furthermore, it can be applied in the multi-source scenario where different sources may have different angular distribution shapes.

An Angular Parameter Estimation Method for Incoherently Distributed Sources Via Generalized Shift Invariance

This paper proposes a new algorithm to estimate the nominal direction-of-arrival (DOA) and the angular spread of multiple incoherently distributed (ID) sources. With the general array manifold (GAM) model, the nominal DOAs are first separated from the original array manifold. Then, a generalized shift invariance property inside the array manifold is identified, based on which the nominal DOAs are obtained when the rank of a trickily formulated matrix drops. Next, the angular spreads are estimated from the central moments of the angular distribution. We also derive a polynomial rooting based search-free method for nominal DOA estimation. This method can greatly reduce the computational complexity. As compared with the popular ESPRIT-ID algorithm, the proposed algorithm can achieve higher accuracy, can handle more sources, and applies to a more general array structure. Extensive simulations are provided to demonstrate the superior performance of the proposed algorithm over the existing works.

Wearable Sensor Localization Considering Mixed Distributed Sources in Health Monitoring Systems.

In health monitoring systems, the base station (BS) and the wearable sensors communicate with each other to construct a virtual multiple input and multiple output (VMIMO) system. In real applications, the signal that the BS received is a distributed source because of the scattering, reflection, diffraction and refraction in the propagation path. In this paper, a 2D direction-of-arrival (DOA) estimation algorithm for incoherently-distributed (ID) and coherently-distributed (CD) sources is proposed based on multiple VMIMO systems. ID and CD sources are separated through the second-order blind identification (SOBI) algorithm. The traditional estimating signal parameters via the rotational invariance technique (ESPRIT)-based algorithm is valid only for one-dimensional (1D) DOA estimation for the ID source. By constructing the signal subspace, two rotational invariant relationships are constructed. Then, we extend the ESPRIT to estimate 2D DOAs for ID sources. For DOA estimation of CD sources, two rational invariance relationships are constructed based on the application of generalized steering vectors (GSVs). Then, the ESPRIT-based algorithm is used for estimating the eigenvalues of two rational invariance matrices, which contain the angular parameters. The expressions of azimuth and elevation for ID and CD sources have closed forms, which means that the spectrum peak searching is avoided. Therefore, compared to the traditional 2D DOA estimation algorithms, the proposed algorithm imposes significantly low computational complexity. The intersecting point of two rays, which come from two different directions measured by two uniform rectangle arrays (URA), can be regarded as the location of the biosensor (wearable sensor). Three BSs adopting the smart antenna (SA) technique cooperate with each other to locate the wearable sensors using the angulation positioning method. Simulation results demonstrate the effectiveness of the proposed algorithm.

An efficient central DOA tracking algorithm for multiple incoherently distributed sources

In this paper, we develop a new tracking method for the direction of arrival (DOA) parameters assuming multiple incoherently distributed (ID) sources. The new approach is based on a simple covariance fitting optimization technique exploiting the central and noncentral moments of the source angular power densities to estimate the central DOAs. The current estimates are treated as measurements provided to the Kalman filter that model the dynamic property of directional changes for the moving sources. Then, the covariance-fitting-based algorithm and the Kalman filtering theory are combined to formulate an adaptive tracking algorithm. Our algorithm is compared to the fast approximated power iteration-total least square-estimation of signal parameters via rotational invariance technique (FAPI-TLS-ESPRIT) algorithm using the TLS-ESPRIT method and the subspace updating via FAPI-algorithm. It will be shown that the proposed algorithm offers an excellent DOA tracking performance and outperforms the FAPI-TLS-ESPRIT method especially at low signal-to-noise ratio (SNR) values. Moreover, the performances of the two methods increase as the SNR values increase. This increase is more prominent with the FAPI-TLS-ESPRIT method. However, their performances degrade when the number of sources increases. It will be also proved that our method depends on the form of the angular distribution function when tracking the central DOAs. Finally, it will be shown that the more the sources are spaced, the more the proposed method can exactly track the DOAs.