Sensor-array calibration using a maximum-likelihood approach

Ideal array responses are often desirable to a multiple‐input multiple‐output (MIMO) system. Unfortunately, it may not be guaranteed in practice as the mutual coupling (MC) effects always exist. Current works concerning MC in the MIMO system only account for the uniform array geometry scenario. In this paper, we generalize the issue of angle estimation and MC self‐calibration in a bistatic MIMO system in the case of arbitrary sensor geometry. The MC effects corresponding to the transmit array and the receive array are modeled by two MC matrices with several distinct entities. Angle estimation is then recast to a linear constrained quadratic problem. Inspired by the MC transformation property, a multiple signal classification‐ (MUSIC‐) like strategy is proposed, which can estimate the direction‐of‐departure (DOD) and direction‐of‐arrival (DOA) via two individual spectrum searches. Thereafter, the MC coefficients are obtained by exploiting the orthogonality between the signal subspace and the noise subspace. The proposed method is suitable for arbitrary sensor geometry. Detailed analyses with respect to computational complexity, identifiability, and Cramer‐Rao bounds (CRBs) are provided. Simulation results validate the effectiveness of the proposed method.

An auxiliary source-based near field source localization method with sensor position error

An expression for mutual reaction between two arbitrarily located slots on an infinitely long conducting cylinder is derived. This stationary expression is used to obtain a formula for mutual admittance and coupling coefficient. The numerical results are presented and discussed. >

Recently developed super nested array families have drawn much attention owing to their merits on keeping the benefits of the standard nested arrays while further mitigating coupling in dense subarray portions. In this communication, a new mutual coupling model for nested arrays is constructed. Analyzing the structure of the newly formed mutual coupling matrix, a transformation of the distorted steering vector to separate angular information from the mutual coupling coefficients is revealed. By this property, direction of arrival (DOA) estimates can be determined via a grid search for the minimum of a determinant function of DOA, which is induced by the rank reduction property. We also extend the robust DOA estimation method to accommodate the unknown mutual coupling and gain-phase mismatches in the nested array. Compared with the schemes of super nested array families on reducing the mutual coupling effects, the solutions presented in this paper has two advantages: (a) It is applicable to the standard nested arrays without rearranging the configuration to increase the inter-element spacing, alleviating the cross talk in dense uniform linear arrays (ULAs) as well as gain-phase errors in sparse ULA parts; (b) Perturbations in nested arrays are estimated in colored noise, which is significant but rarely discussed before. Simulations results corroborate the superiority of the proposed methods using fourth-order cumulants.

Due to the existence of dense subarray, nested array (NA) is susceptible to mutual coupling, which seriously degrades the parameter estimation performance. To deal with this problem, we propose an improved scheme to obtain direction of arrival (DOA) and mutual coupling estimation in this paper. Specifically, a new sparse subarray can be formed by moving specific sensors of the nested array, which enables well-performed estimation free from severe mutual coupling effect. Subsequently, the contaminated steering vector of the whole non-uniform sparse array is constructed and a quadratic optimization problem is established to simultaneously eliminate DOA estimation ambiguity and estimate mutual coupling coefficients (MCCs). Numerical simulations demonstrate the superiority of the proposed scheme in terms of estimation accuracy and computation complexity.

In this paper, single snapshot direction-of-arrival (DOA) estimation under mutual coupling (MC) is considered with arbitrary array structures. A compressed sensing approach is utilized and a joint-sparse recovery algorithm is proposed. In this respect, both spatial source directions and MC coefficients are embedded into a joint-sparse vector. A new dictionary matrix is defined using the symmetricity of MC matrix. The proposed approach does not depend on the structure of MC matrix and it is suitable for any type of array geometry. In the simulations, the proposed method is evaluated for a planar array where the antennas are placed randomly in 2-D space. It is shown that the proposed method effectively estimates the source parameters and performs significantly better than the alternative methods.

Calibrating the error from sensor position uncertainty in TDOA-AOA localization

The problem of Angle-of-Arrival estimation of multiple sources in the presence of mutual coupling is addressed. In this paper, we derive a Mean-Squared-Error (MSE) expression of a recently proposed algorithm that could estimate the Angles-of-Arrival of multiple sources in the presence of more mutual coupling parameters, compared to traditional methods. The MSE expression is compared with the MSE of MUSIC with known mutual coupling parameters and to the Cramer-Rao bound (CRB) of any unbiased estimator that estimates the Angles-of-Arrival in the presence of mutual coupling. It is shown that the proposed method is asymptotically unbiased. In addition, it is also shown that the method attains CRB for large number of antennas with fixed coupling parameters and uncorrelated sources. For high SNR, the CRB is not necessarily attained, however, we study the gap between the derived MSE and the CRB.

The limited space of a conformal array may lead to a serious mutual coupling effect, which will significantly affect the performance of direction of arrival (DOA) estimation algorithms. In this paper, an efficient 2-D direction finding method is developed in the presence of unknown mutual coupling for the uniform cylindrical conformal array (CCA). To avoid the time-consuming two-dimensional spectral peak searching, the 2-D DOA estimation is decoupled and divided into two 1-D DOA estimations. Elevation is first estimated based on a subarray estimation of signal parameters via rotation invariant technique (ESPRIT), and then azimuth is estimated based on the rank reduction (RARE) method by using the elevation estimation result. Consequently, the mutual coupling coefficients can be estimated after getting the DOA estimates. The proposed method can well calibrate the mutual coupling effect of a cylindrical array with a low computational complexity. The final simulation results corroborate our analysis.

Many direction-of-arrival (DOA) estimation algorithms have been proposed recently. However, the effect of mutual coupling among antenna elements has not been taken into consideration. In this paper, a novel DOA and mutual coupling coefficient estimation algorithm is proposed in intelligent transportation systems (ITS) via conformal array. By constructing the spectial mutual coupling matrix (MCM), the effect of mutual coupling can be eliminated via instrumental element method. Then the DOA of incident signals can be estimated based on parallel factor (PARAFAC) theory. The PARAFAC model is constructed in cumulant domain using covariance matrices. The mutual coupling coefficients are estimated based on the former DOA estimation and the matrix transformation between MCM and the steering vector. Finally, due to the drawback of the parameter pairing method in Wan et al., 2014, a novel method is given to improve the performance of parameter pairing. The computer simulation verifies the effectiveness of the proposed algorithm.

Active array sensor localization

This paper discusses the problem of Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation of Binary Phase Shift keying (BPSK) modulated signals using the Expectation Maximization (EM) Algorithm. In addition, the Cramer-Rao Lower Bounds (CRLB) for the estimation of Data Aided (DA) and Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation is derived. Multiple Input Single Output (MISO) channels with Space Time Block Codes (STBC) is used. The EM algorithm is a method that finds the Maximum Likelihood (ML) solution iteratively when there are unobserved (hidden or missing) data. Extension of the proposed approach to other types of linearly modulated signals in estimating SNR is straight forward. The performance of the estimator is assessed using the NDA CRLBs. Alamouti coding technique is used in this paper with two transmit antennas and one receive antenna. The authors’ assumption is that the received signal is corrupted by additive white Gaussian noise (AWGN) with unknown variance, and scaled by fixed unknown complex channel gain. Monte Carlo simulations are used to show that the proposed estimator offers a substantial improvement over the conventional Single Input Single Output (SISO) NDA SNR estimator due to the use of the statistical dependences in space and time. Moreover, the proposed NDA SNR estimator works close to the NDA SNR estimator over Single Input Multiple Output (SIMO) channels.

Processing conformal array signal is sensitive to antenna patterns on horizontal (H) and vertical (V) planes while mutual coupling (MC) impacts distinctly on them. Many existing MC calibration algorithms ignore that and thus lose their efficacy. In this study, MC is modelled as distortions of H and V patterns. The direction-dependent and slowly varied properties indicate that MC can be estimated via sampling in the interested spatial range and calibrated. A novel MC estimation and calibration algorithm for all kinds of conformal arrays is proposed based on it. The process requires the use of three signals with diverse sets of polarisation emitted by a known calibrated source in each sampling direction. The selection of diverse polarisations is analysed in terms of optimisation and practical application. Simulations exhibit the potentialities of the proposed algorithm regarding the precision of MC estimation and the efficiency of calibration. It shows that significant superiority influences greatly in the analysis of conformal arrays.

An improved element pattern reconstruction method is introduced to calibrate the mutual coupling effect of antenna arrays, and its applications in wide-angle direction-of-arrival estimation are presented. In this method, direction-of-arrival estimations and a calibration matrix are achieved by alternate iteration. The improved method can achieve a more accurate calibration matrix and direction-of-arrivals with less calculation time and iterations than the joint estimation methods for direction-of-arrivals and calibration matrix. Additionally, the angle ranges of direction-of-arrival estimations can be widened by this method for practical antenna arrays. The validity was verified by some numerical examples with a microstrip antenna array.

In this article, we focus on indoor direct positioning using a practical massive multiple-input and multiple-output (MIMO) array with imperfections. First, the imperfect array needs to be calibrated in an anechoic chamber from the view of array signal processing. This brings inconveniences due to the large size of massive MIMO systems. To solve this problem, we propose onsite calibration, where a banded calibration matrix, calibrating both phase-gain error and mutual coupling, can be estimated by performing least-squares on the measure near-field channels, i.e., the site survey data. The feasibility of this calibration method can be explained by that the non-line-of-sight (NLOS) channel is able to be modeled using Rayleigh fading; i.e., the sum of the multipath components can be treated as Gaussian noise. Then, to estimate the user position, we propose a new two-stage direct positioning scheme. In the first stage, the coarse position is obtained via global search-based conventional beamforming (CBF) using a global calibration matrix. In the second stage, fine positioning is achieved via local search-based multiple signal classification (MUSIC) using a local calibration matrix. The proposed method is computationally efficient, insensitive to model order selection, free of dense site survey, and robust to partial line-of-sight (LOS) blocking. Finally, the proposed method is verified by practical measurements from a distributed array (DA), uniform linear array (ULA), and uniform rectangular array (URA), and compared with a deep learning positioning-method. The results show that the DA with the proposed method performs the best, achieving sub-centimeter accuracy (typically below 5 mm).