Zheng LI ,Jinqing SHEN ,Xiaofei ZHANG??

1College of Electronic and Information Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China

2Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space,Ministry of Industry and Information Technology,Nanjing 211106,China

Abstract:The three-dimensional localization problem for noncircular sources in near-field with a centro-symmetric cross array is rarely studied.In this paper,we propose an algorithm with improved estimation performance.We decompose the multiple parameters of the steering vector in a specific order so that it can be converted into the products of several matrices,and each of the matrices includes only one parameter.On this basis,each parameter to be resolved can be estimated by performing a one-dimensional spatial spectral search.Although the computational complexity of the proposed algorithm is several times that of our previous algorithm,the estimation performance,including its error and resolution,with respect to the direction of arrival,is improved,and the range estimation performance can be maintained.The superiority of the proposed algorithm is verified by simulation results.

Key words:Localization;Centro-symmetric cross array;Noncircular sources;Near-field;Steering vector decomposition

1 Introduction

Unlike the wavefronts of far-field (FF) sources,those of near-field (NF) sources are assumed to be spherical rather than planar (Swindlehurst and Kai‐lath,1988).Consequently,the direction of arrival(DOA) algorithms for FF sources are not applicable to NF sources.Besides DOA information,the range information for NF sources also needs to be obtained;i.e.,it is a problem not only for direction finding,but also for localization (Swindlehurst and Kailath,1988;Van Trees,2002).In DOA estimation,signals with noncircularity,e.g.,amplitude modulated (AM) or binary phase shift keying (BPSK) modulated signals(Chargé et al.,2001;Haardt and Romer,2004),have been widely studied because the noncircularity prop‐erty can be exploited to enhance estimation perfor‐mance (Chargé et al.,2001;Abeida and Delmas,2006;Steinwandt et al.,2014,2016;Zheng et al.,2017).

Some researchers have considered the localiza‐tion problem for NF and noncircular (NC) sources(Xie et al.,2015a,2015b;Chen et al.,2018;Shu et al.,2021).However,because the array configurations used in these papers are linear,the problem considered is actually two-dimensional localization.Studies of the three-dimensional (3D) localization problem for NF and NC sources with a centro-symmetric cross array(CSCA) have been reported very rarely.Although some studies used the same configuration,the source type was restricted to non-Gaussian sources (Challa and Shamsunder,1996;Wu and Yan,2020).What is more,the source in our study is NC,which means that there is one more parameter,i.e.,the NC phase.Consequently,the existing methods are not directly applicable.Recently,we proposed an algorithm us‐ing connection-matrices (CMs) for this problem (Li et al.,2021).This algorithm is able to obtain satisfac‐tory estimation performance while avoiding the multi-dimensional spatial spectral search.

In this study,to further improve the estimation performance for the same problem,we propose an algorithm that decomposes the multiple parameters of the steering vector in a specific order so that it can be converted into the products of several matrices,and each of the matrices includes only one parame‐ter.On this basis,each parameter to be resolved can be estimated by performing a one-dimensional (1D)spatial spectral search.Although the computational complexity of the proposed algorithm is several times that of our previous algorithm using CMs,simulation results have confirmed that it can obtain better angle estimation performance,including error and resolu‐tion,while maintaining the range estimation perfor‐mance.Note that the proposed algorithm does not need additional angle-pairing.

Notations:Bold upper-and lower-case letters denote matrices and vectors,respectively;(·)Tand (·)Hdenote the transpose and conjugate transpose opera‐tors,respectively;(·)+and (·)?1denote the pseudoinverse and inverse operators,respectively;?denotes the Kronecker product;det(·) and diag(·) denote the determinant calculation and diagonal operator,respec‐tively;Imdenotes them×midentity matrix.

2 Data model

Fig.1 shows the CSCA configuration.CSCA con‐sists of two symmetric uniform linear arrays (ULAs)that are orthogonal and intersectant.Both ULAs,in‐cluding 2M+1 sensors,are assumed to be located on thex-andy-axis,and the distance between adjacent sensors is withinλ/4 (Grosicki et al.,2005),whereλis the wavelength.The intersection of CSCA is the central sensor of the two ULAs,and is regarded as the reference sensor.All the sources are considered to be in the Fresnel region,i.e.,0.62(D3/λ)0.5

Fig.1 The centro-symmetric cross array configuration

Assume that CSCA receivesKsignals located in the Fresnel region with zero-mean.All the signals are uncorrelated,narrow-band,and strictly NC.Kis known andK≤2M.Under these assumptions,the sig‐nals received from thex-andy-axis are (Gan et al.,2008;Steinwandt et al.,2014)

wheres(t)=fs0(t) denotes the signal vector ands0(t)∈RK,?=diagandφk(k=1,2,…,K)denotes the NC phase of thekthsource.Ax=[ax(v1,r1),ax(v2,r2),…,ax(vK,rK)] andAy=[ay(u1,r1),ay(u2,r2),…,ay(uK,rK)] are the steering matrices withuk=sinθksin?kandvk=sinθkcos?k.nx(t) andny(t) are the additive white Gaussian noises.The steering vectorsax(vk,rk),ay(uk,rk)ofAxandAyare expressed as (Challa and Shamsunder,1996)

3 The proposed algorithm

3.1 Steering vector decomposition

We firstly stackx(t),y(t),and their conjugations to construct the extended received signal as (Li et al.,2021)

denote the extended steering matrix and steering vector,respectively.Note that we decomposeandin a special order as

According to Eqs.(8)–(11),we can rewriteb(αk,βk,rk,φk)as

Eqs.(16) and (21) show that the multiple parame‐ters in the steering vector can be decomposed into differ‐ent matrices so that each matrix has only one parameter.

3.2 Initial estimation for αk

We obtain an estimation of the covariance matrix ofz(t)from

whereJdenotes the number of snapshots.Then we perform eigenvalue decomposition(EVD)onas

to obtain the noise subspacecomposed of the 8M+4?Kminimum eigenvalues ofand have

due to the multiple signal classification principle(Sch?midt,1986).According to Eq.(16),Eq.(29) can be rewritten as

Consequently,the rank-reduction (RARE) tech‐nique (Pesavento et al.,2001) can be combined to yield

3.3 Estimation of rk and βk

Then we find the maximum peak of

3.4 Final estimation of αk

According to Eq.(21),Eq.(29)can be rewritten as

whereΔdenotes the partial search scope.After per‐forming Eq.(39),are obtained.

3.5 Procedure of the algorithm

We summarize the procedure of the proposed algorithm as follows:

Step 1:Calculate the covariance matrixRofz(t)in Eq.(5) and perform EVD forRto yield the noise subspace matrixas Eq.(28).

Step 2:Perform 1D global spatial spectral search once to findcorresponding to theKmaximum peaks off1(α)in Eq.(32).

4 Complexity analysis

Fig.2 shows a comparison of the computational complexity between different algorithms versusM,whereJ=300,K=3,ds=0.01°,anddr=0.01λ.dsanddrare the search steps for angles and range,respective‐ly.The complexity of the proposed algorithm is sev‐eral times that of the previous algorithm,but the pro‐posed algorithm has better performance in terms of angle estimation precision and resolution.This supe‐riority is demonstrated in the next section.

Fig.2 The computational complexity versus M

5 Simulation results

The Cramer-Rao bound (CRB)in this section was provided by Li et al.(2021).The root mean square error(RMSE)is defined as

whereCis the number of Monte-Carlo trials,,c,,c,and,cdenote the estimates forαk,βk,andrkin thecthtrial,respectively.In this section,we setC=300 andd=0.25λ.

Simulation 1Figs.3 and 4 show the scatter plots of angle–angle and angle–range.We supposed thatM=5,K=2,(α1,β1,r1,φ1)=(70°,75°,2.5λ,10°),(α2,β2,r2,φ2)=(80°,85°,3.5λ,20°),ds=0.01°,dr=0.01λ,andJ=200.The signal-to-noise ratio (SNR) was set to 10 and 15 dB.Figs.3 and 4 clearly show that the proposed algorithm was effective regarding both angle–angle and angle–range in all 300 trials.Note that the algorithm using CMs needs angle-pairing,while the proposed algorithm does not.Consequently,theoretically the proposed algorithm does not suffer from the pairing error.

Fig.3 Scatter plot of angle–angle

Fig.4 Scatter plot of angle–range

Simulation 2In this simulation,we compared the RMSE performance of the initial and final estima‐tions of the proposed algorithm forαkversus SNR,whereM=5,K=2,(α1,β1,r1,φ1)=(80°,80°,3λ,10°),(α2,β2,r2,φ2)=(100°,100°,5λ,20°),Δ=2°,ds=0.01°,dr=0.01λ,andJ=400 in Fig.5 and SNR=10 dB in Fig.6.From these simulations,we found that the final estimation can yield more precise results than the ini‐tial estimation,which reveals the necessity of the final estimation.A possible explanation is that the initial estimation is performed without any information aboutβkandrk,while before performing the final es‐timation,have been obtained,and this infor‐mation is exploited in the final estimation.

Fig.5 The accuracy of the initial and final estimations of the proposed algorithm for αk versus SNR

Fig.6 The accuracy of the initial and final estimations of the proposed algorithm for αk versus the number of snapshots

Simulation 3In this simulation,we compared the RMSE performance of different algorithms versus SNR.All parameters were identical to those in Fig.5.Figs.7 and 8 show that the proposed algorithm out‐performed the algorithm using CMs in terms of the DOA estimation error,while the two algorithms had very similar range estimation errors.This indicates that the range estimation performance of the pro‐posed algorithm can be maintained.

Fig.7 The RMSE results regarding DOA estimation versus SNR

Fig.8 The RMSE results regarding range estimation versus SNR

Simulation 4In this simulation,we compared the RMSE performance of different algorithms versus the number of snapshots,where SNR=10 dB.All parameters were identical to those in Fig.6.Similar to Figs.7 and 8,Figs.9 and 10 show the improvement in performance of the proposed algorithm for DOA estimation,and the similar performance for range esti‐mation compared with the algorithm using CMs.

Fig.9 The RMSE results regarding DOA estimation versus the number of snapshots

Fig.10 The RMSE results regarding range estimation versus the number of snapshots

Simulation 5Figs.11 and 12 illustrate the DOA resolution performance of different algorithms versus angle and range separation for two sources with close angle and close range,respectively.In Fig.11,M=5,SNR=20 dB,J=500,ds=0.01°,dr=0.001λ,(α1,β1,r1,φ1)=(80°,80°,3λ,10°),(α2,β2,r2,φ2)=(80°+Δα,80°+Δα,3.5λ,10°+Δα),and in Fig.12,M=6,(α1,β1,r1,φ1)=(80°,80°,3λ,10°),(α2,β2,r2,φ2)=(81°,81°,3λ+Δr,11°).Other parameters were identical to those in Fig.11,whereΔαandΔrrespectively denote angle and range separation.The two sources were consid‐ered to be successfully resolved ifwherek=1,2.Figs.11 and 12 show that the proba‐bility of successful recognition for the proposed algo‐rithm was obviously higher than that of the algorithm using CMs.This indicates that the proposed algo‐rithm has better DOA resolution for sources with close angle or close range.

Fig.11 The DOA resolution versus angle separation

Fig.12 The DOA resolution versus range separation

6 Conclusions

The 3D localization algorithm proposed in this study is for NF and strictly second-order NC sources with CSCA.The algorithm decomposes the multiple parameters of the steering vector in a specific order so that it can be converted into the products of several matrices,and each of the matrices includes only one parameter.On this basis,each parameter to be resolved can be estimated by performing a 1D spatial spectral search.Although the computational complexity of the proposed algorithm is several times that of our previ‐ous algorithm using CMs,its estimation performance,including estimation error and resolution,with respect to DOA,is better than that of the algorithm using CMs,while the range estimation performances of the two algorithms are very close.

Contributors

Zheng LI designed the research and processed the data.Zheng LI and Jinqing SHEN drafted the paper.Xiaofei ZHANG helped organize the paper.Zheng LI,Jinqing SHEN,and Xiaofei ZHANG revised and finalized the paper.

Compliance with ethics guidelines

Zheng LI,Jinqing SHEN,and Xiaofei ZHANG declare that they have no conflict of interest.