Sensor Array Calibration in Presence of Mutual Coupling and Gain/Phase Errors by Combining the Spatial-Domain and Time-Domain Waveform Information of the Calibration Sources

2018

Sensors

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This paper focuses on the localization methods for multiple sources received by widely separated arrays. The conventional two-step methods extract measurement parameters and then estimate the positions from them. In the contrast to the conventional two-step methods, direct position determination (DPD) localizes transmitters directly from original sensor outputs without estimating intermediate parameters, resulting in higher location accuracy and avoiding the data association. Existing subspace data fusion (SDF)-based DPD developed in the frequency domain is computationally attractive in the presence of multiple transmitters, whereas it does not use special properties of signals. This paper proposes an improved SDF-based DPD algorithm for strictly noncircular sources. We first derive the property of strictly noncircular signals in the frequency domain. On this basis, the observed frequency-domain vectors at all arrays are concatenated and extended by exploiting the noncircular property, producing extended noise subspaces. Fusing the extended noise subspaces of all frequency components and then performing a unitary transformation, we obtain a cost function for each source location, which is formulated as the smallest eigenvalue of a real-valued matrix. To avoid the exhaustive grid search and solve this nonlinear function efficiently, we devise a Newton-type iterative method using matrix Eigen-perturbation theory. Simulation results demonstrate that the proposed DPD using Newton-type iteration substantially reduces the running time, and its performance is superior to other localization methods for both near-field and far-field noncircular sources.

“…It can be seen from (60) that

and

are the gradient and Hessian matrix of

, respectively. Therefore, we can get the estimation of

following the Newton-type iterative procedure [ 28 ]:

where the superscript

denotes the iteration number and

is the step factor deciding the iteration step size. In this way, each transmitter can be located when an initial position is given.…”

Section: Methodsmentioning

confidence: 99%

An Efficient Direct Position Determination Method for Multiple Strictly Noncircular Sources

Yin

2018

Sensors

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“…Because in the present case the full parameter set contains both the deterministic parameters

, and

and the stochastic parameter

, the CRB derivation should follow the Bayesian theory frame [ 68 , 69 , 70 ]. It is noteworthy that the CRB derivation can also be used for stochastic parameters, as processed in [ 68 , 69 , 70 ]. To this end, a novel parameter vector that comprises all the deterministic and stochastic unknowns is introduced

where

…”

Section: Cramér-rao Bound On Covariance Matrix Of Localization Errmentioning

confidence: 99%

Performance Analysis of the Direct Position Determination Method in the Presence of Array Model Errors

et al. 2017

Sensors

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The direct position determination approach was recently presented as a promising technique for the localization of a transmitting source with accuracy higher than that of the conventional two-step localization method. In this paper, the theoretical performance of a direct position determination estimator proposed by Weiss is examined for situations in which the array model errors are present. Our study starts from a matrix eigen-perturbation result, which expresses the perturbation of eigenvalues as a function of the disturbance added to the Hermitian matrix. The first-order asymptotic expression of the positioning errors is presented, from which an analytical expression for the mean square error of the direct localization is available. Additionally, explicit formulas for computing the probabilities of a successful localization are deduced. Finally, Cramér–Rao bound expressions for the position estimation are derived for two cases: (1) array model errors are absent and (2) array model errors are present. The obtained Cramér-Rao bounds provide insights into the effects of the array model errors on the localization accuracy. Simulation results support and corroborate the theoretical developments made in this paper.

“…In this paper, the calibration method especially tailored to constant modulus auxiliary sources is investigated, and a remarkable improvement can be obtained by making use of this property. On the other hand, the azimuths of auxiliary sources may deviate slightly from the nominal values in practice [23,24,27], which would seriously deteriorate calibration precision. Consequently, the novel calibration algorithm is extended to the case where the azimuth deviations exist with a prior known Gaussian distribution.…”

Section: Research Articlementioning

confidence: 99%

“…The performance of Algorithm 1 is compared with that of the algorithms in [15][16][17]26], along with the corresponding CRB presented in Section 5.1. In addition, note that the method in [27] requires a variety of time-disjoint auxiliary sources to jointly estimate the sensor gain/phase errors and mutual coupling effects, its calibration issue is not the same as the one studied here. Thus, the algorithm in [27] is not suitable as reference.…”

Section: Comparing the Calibration Accuracy In Absence Of Azimuth Devmentioning

confidence: 99%

“…The majority of the research views the error calibration as a parameter estimation problem. The parametric error calibration methods can be generally grouped into auto-calibration [7][8][9][10][11][12][13][14] and active calibration [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Specifically, the auto-calibration method tries to correct for array errors and simultaneously estimates the unknown DOAs of incident sources according to some optimisation criterions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Calibration algorithm for multiplicative modelling errors using constant modulus auxiliary signals

2015

IET signal process.

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In this paper, the active calibration method for multiplicative modelling errors using phase-modulated (also called constant modulus) auxiliary sources is presented. Compared with some existing calibration methods, the proposed approach can exploit the constant modulus characteristic of the sources and has significantly better performance. For the purpose of incorporating the constant modulus information into the procedure for finding array error parameters, the maximum likelihood criterion is chosen as the optimisation function and a concentrated alternating iteration algorithm is developed, which has rapid convergence rate. In addition, to reduce the effects of azimuth deviations of the auxiliary sources, the study proceeds to extend the novel algorithm to the scenario where the true azimuths of the sources deviate slightly from the nominal values with a prior known Gaussian distribution. The Cramér-Rao bound (CRB) expressions for the unknowns are derived for the case when it is known that the sources are phase-modulated. Simulation results show that the performance of the proposed algorithms are considerably better than that of subspace-based calibration methods and closely follows the CRB for array error estimation. NomenclatureIn this paper, lowercase and uppercase boldface letters are used to denote vectors and matrices, respectively. In addition, the following conventions are used throughout this paper.Hadamard product of X 1 and X 2 vecd [X] column vector with diagonal elements of X as its elements diag [x] diagonal matrix with the elements of x as diagonal elements dim [x] dimension of x O m×n m × n matrix of zeros I n n × n identity matrix i (m) n the mth column of I n [X] † Moore-Penrose inverse of X Π[X] and Π ⊥ [X] orthogonal projection matrix onto the range of X and the null space of X H , respectively 〈x〉 n the nth entry of x 〈X〉 nm the (n, m)th entry of X