A search-free near-field source localization method with exact signal model

PAN Jingjing ,SINGH Parth Raj ,and MEN Shaoyang

1.Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China;2.Department of Electronic Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 211106,China;3.International Electronics Engineering S.A.,Bissen L-7795,Luxembourg;4.School of Medical Information Engineering,Guangzhou University of Chinese Medicine,Guangzhou 510006,China

Abstract:Most of the near-field source localization methods are developed with the approximated signal model,because the phases of the received near-field signal are highly non-linear.Nevertheless,the approximated signal model based methods suffer from model mismatch and performance degradation while the exact signal model based estimation methods usually involve parameter searching or multiple decomposition procedures.In this paper,a search-free near-field source localization method is proposed with the exact signal model.Firstly,the approximative estimates of the direction of arrival (DOA) and range are obtained by using the approximated signal model based method through parameter separation and polynomial rooting operations.Then,the approximative estimates are corrected with the exact signal model according to the exact expressions of phase difference in near-field observations.The proposed method avoids spectral searching and parameter pairing and has enhanced estimation performance.Numerical simulations are provided to demonstrate the effectiveness of the proposed method.

Keywords:near-field,source localization,polynomial rooting,approximation error,exact signal model.

1.Introduction

Source localization is of practical importance in many aspects of array signal processing,for example,radar,sonar,speech,and seismology [1−5].When a source is located in the Fresnel region of an array,the near-field,its wave front is spherical and the phase difference among sensors is non-linear.Therefore,both direction of arrival(DOA) and the range are necessary for the localization of near-field sources.

Most of the near-field source localization methods are based on an approximated signal model for simplification [6−8].The phase difference is usually approximated as a linear term which contains only the DOA,like in the far-field,and a quadratic non-linear term which contains both the DOA and the range.Several source localization methods aim at the elimination of the quadratic non-linear term to reduce the computational complexity,which only needs several one-dimensional (1-D) search.This type of 1-D spectral methods includes the covariance approximation method [9],the weighted linear prediction method [10],the rank reduced (RARE) method [11],the second order statistics (SOS) method [12],the reduceddimension method [13],and the high order statistics based methods [2,14−16].

Unfortunately,the approximated model based methods suffer from model mismatch,resulting in systematic estimation bias in practical environment [17−19].In the literature,there are two main approaches dealing with the exact signal model in near-field source localization.One is to directly estimate the DOA and the range with the exact model,including the maximum likelihood estimator(MLE) method [20],the Fourier transform method [21],the parallel factor (PARAFAC) decomposition method[22],and the cumulant based method [17].However,direct estimation methods generally require multi-dimensional or 1-D search,or multiple decomposition,which brings much computational load.The other approach performs an initial estimate of DOA and range by using the approximated signal model based methods,followed by error correction procedures [22,23].Although the family of the approximated model based methods [7,9−16] perform reduced-dimension estimation,they still need several 1-D search or parameter pairing operations,which have an influence on the estimation accuracy and efficiency.

In this paper,a search-free near-field source localization method is proposed with the exact signal model by using the correction method.The contributions of this work are as follows:

(i) Reduce the computational complexity of the approximated model based near-filed source localization methods.Unlike [23],the proposed method obtains the estimates of DOA and range with the approximated signal model through parameter separation and the polynomial rooting,which is free of searching and pairing.

(ii) Improve the parameter estimation accuracy in nearfield source localization. Compared with the spectral methods,rooting procedures are more effective in parameter estimation problems [24].Besides,the implementation of error correction also contributes to the reduction of systematic estimation error with the approximated signal model.

The rest of this paper is organized as follows.Section 2 presents the signal model.Section 3 describes the proposed method.Section 4 gives the theoretical analysis.Section 5 shows its performance with numerical data.Conclusionsare drawn in Section 6.

2.Signal model

ConsiderKindependent narrow band signals impinging on a uniform linear array (ULA) with elements,as shown inFig.1.

Fig.1 ULA configuration in near-field source localization

Taking the 0th element as reference,the received signal at themth sensor can be expressed as

wherenm(t) is the additive white Gaussian noise at themth sensor with zero mean and variance σ2;sk(t) is the signal emitted from thekth source and received by themth sensor at timet,t∈[1,L];Ldenotes the number of snapshots; δmkdenotes the phase difference of thekth signal between sensorsmand 0:

where θkdenotes the DOA of thekth source,θk∈rkis the range of thekth source,which is within the Fresnel regionrF∈[0.62(D3/λ)1/2,2D2/λ];Dis the array aperture; λ is the wavelength of incoming signals;andddenotes the distance between two adjacent elements of ULA.

The phase difference in (2) is usually approximated by using the second-order Taylor expansion [25,26]:

The vector of the received signal can be written as

wherex(t)=[x−M(t),x−M+1(t),···,xM(t)]T,s(t)=[s1(t),s2(t),···,sK(t)]T,n(t)=[n−M(t),n−M+1(t),···,nM(t)]T,A=[a(r1,θ1),a(r2,θ2),···,a(rK,θK)] is the steering matrix;a(rk,θk)=where δ−Mkcan either be the exact or the approximated phase differences.

The covariance matrix of the received signals is

whereRs=Eis the covariance matrix of the source signals,E[·] denotes the ensemble average,andIis the identity matrix.

The eigenvalue decomposition (EVD) ofRis

where Λ is a diagonal matrix whose elements are the eigenvalues ofRin descending order,λ1≥···≥λK>λK+1=···=λN=σ2andU=[u1,u2,···,uN] contains the c orresponding eigenvectors.

3.Methodology

For the source localization with the exact signal model,we propose to firstly apply an approximated signal model based method to obtain the initial approximative estimates of DOA and range,and then perform error correction with respect to the exact phase difference and signal model.The approximated estimation of DOA and range is achieved by the polynomial rooting procedure with the approximated signal model.

3.1 Polynomial rooting with approximated signal model

In approximated signal model based methods,the steering vector can be represented by using parameter separation in the following split form [13]:

where Γ(ω) only contains the information of DOA whileb(ϕ) contains both the information of DOA and range.

With the split of DOA and range,the searching spectrum of multiple signal classification (MUSIC) with the approximated signal model becomes

whereUn=[uK+1,uK+2,···,uN].

According to the Rayleigh quotient theorem [27],the minimum of the quadratic function in (8) over ω and ϕ is equal to the minimum eigenvalue ofQ(ω).SinceQ(ω) is with the quadratic form,its eigenvalues are always nonnegative [28].When ω corresponds to the true DOA of a near-field source,the minimum eigenvalue is equal to zero and the determinant ofQ(ω) reaches zero as well.

Therefore,the angle information of the near-field sources can be obtained by finding the values ofzsatisfying

where det[·] denotes the determinant.

Since Γ(ω) is a symmetric matrix,it is easy to find that the expression of detis an even function.According to the basic property of the even function,(10)can be represented as the following polynomial:

where γ1is a constant.

In theory,the roots of (11) are not only conjugate reciprocal but also rotational symmetric with respect to the origin.Therefore,there will be 2Kroots lying inside and close to the unit circle.However,the phase ofis limited tosincein the approximated signal model based methods.In this context,theKroots corresponding to the true DOAs can be selected for the estimation

where ∠ represents the phase of a complex variable.

Substituting each estimated DOA into (8),we can formulate another polynomial:

where γ2is a constant.

With the approximative estimate,there is only one root lying inside and close to the unit circle,vk,which can be used for the range estimation:

It should be noted that the proposed method is searchfree and computational efficient.The range parameter is calculated with each estimated DOA,which does not require an additional pairing procedure.

3.2 Error correction with exact signal model

As mentioned in [23],the parameters can be corrected with the approximative estimates by using (2):

Sincem∈[−M,M],we can form the following system of linear equations for thekth source:

Finally,the corrected estimates of each source can be obtained by

where Ψ=D+c.

The proposed method is summarized as follows:

(i) Estimate the data covariance matrixRand apply EVD on it;

(ii) Build polynom ials (11),(14) and obtain the approximated DOA and range;

(iv) Construct the linear system of equations for each source and obtain the corrected estimates of DOA and range.

The computational complexity of the proposed method isO{N2L+N3+(N2−1)/4+K((N2−1)/4+N3+N)},while that of [23] isO{N2L+N3+Knθ(N−K)(N+1)+nr(N−K)(N+1)(3N+1)/4+K(N3+N)},wherenθandnrdenote the searching number in the DOA and the range domain,respectively.

4.Theoretical analysis

4.1 Asym ptotic properties of the rooting procedure with approximated signal model

The asymptotic performance of the proposed method is analyzed with a sufficiently large number of snapshots.In the presence of noise,the errors in the eigenvectors lead to the perturbations in the signal zeros,both in the estimation of DOA and range.Here,we only show the estimation error of the rooting procedure with the approximated signal model.

Letzkandvkbe the signal zero in the estimation of DOA and range for the source respectively,that is,|zk|=1 and |vk|=1.Assume ∆zkand ∆vkas the estimation error.Therefore,the mean square error of the signal zerosandcan be given as follows.

Firstly,for the determ ination ofthe polynomial(11) with estimation errors can be expressed as

with higher order terms neglected.

Taking expectations of (20),we have

According to the L’ Hospital’s rule and the general rules of the derivative of a determinant [29],part of the expression in (21) can be w ritten as

where Γ′(ωk) denotes the derivative of Γ(ωk).

Let ηkbe the bias of thekth eigenvectoru k.The expression ofcan be w ritten as

As in [19],the eigenvectors and eigenvalues ofRin a complex Gaussian process have the following properties:

and

where ξk jdenotes the Kronecker delta.

Substituting the above expressions into (23) and then doing some mathematical operations yield [16]:

Through the calculations,we have the expression ofas follows:

Secondly,for the mean square errorwe have similar derivations by using polynomial (15):

Taking expectations of (28),we have

which can be partially represented as follows:

where R(·) denotes the real part andb′(ϕk) denotes the derivative ofb(ϕk) and

Therefore,we have the final expression of

4.2 Cramer-Rao lower bound (CRLB)

The closed-form expression of the CRLB in near-field with the exact signal model is similar to that in [30]:

The derivations offθ(rk,θk) andfr(rk,θk) are realized with the exact signal model (2),without approximation.

5.Simulation

In this section,several numerical examples are presented to evaluate the performance of the proposed method.Consider a ULA with nine isotropic sensors,M=4.The interval between two adjacent sensors is set asd=Therefore,the Fresnel region isrF∈[1.75λ,8λ].Assume two sources locate at (4.0λ,−20◦) and (6.0λ,20◦),respectively.

In the first simulation,the number of snapshots isL=500,and SNR =15 dB.Fig.2andFig.3show the roots lying inside and close to the unit circle,in the approximated parameter estimation.In the estimation of DOA,the roots are symmetrically distributed because of the even polynomial. The roots corresponding to the DOAs of the sources can be selected according to the phase rangeWith the approximated estimation of DOA,we can get the approximated range of each source,seeFig.3.Table 1gives the estimation results of the proposed method before and after correction.It is clear that the estimations after correction are closer to the true values.

Fig.2 Roots in the approximated estimation of DOA

Fig.3 Roots in the approximated estimation of range,the first source

Table 1 Near-field source localization by using the proposed method

In the second simulation,we analyze the statistical estimation performance of the proposed method in terms of SNR. For comparison,the estimation results of the searching method in [23] are also recorded,in which the searching step is 0.001.SNR varies from 0 to 30 dB.The other settings are the same as the first simulation.The root mean square error (RMSE) is defined as

Fig.4andFig.5illustrate the RMSEs of the proposed method and that in [23] at different SNRs,before and after correction.The number of Monte Carlo trials is 200.It is obvious from the figures that the RMSEs of both methods are decreasing with the increasing of SNR,not only for the estimation of DOA but also for range.The estimations after correction are less biased,compared with those before correction.The proposed method after correction has the lowest estimation error.

Fig.4 RMSE in the estimation of DOA versus SNR

Fig.5 RMSE in the estimation of range versus SNR

In the third simulation,we evaluate the statistical estimation performance of the proposed method with respect to the number of snapshots with 200 Monte Carlo trials,as shown inFig.6andFig.7.The other settings are the same as the first simulation while SNR=15 dB and the number of snapshots is within [100,1 000].The RMSEs of the proposed method decrease as the number of snapshots increases,like inFig.4andFig.5.The proposed method after correction outperforms the other methods at each number of snapshots,especially for the estimation of DOA.

Fig.6 RMSE in the estimation of DOA versus the number of snapshots

Fig.7 RMSE in the estimation of range versus the number of snapshots

In the fourth simulation,the execution time is recorded over 200 Monte Carlo trials withL=500 and SNR=15 dB,apart from the theoretical analysis of the complexity.The average execution time of one trail using the proposed method and that in [23] are 0.072 4 s and 0.587 4 s,respectively,with a computer equipped with a CPU of 2.2 GHz and 8 GB of RAM.In view of the execution operation time,the proposed method is much more effective compared with that in [23],since it is developed with polynomial rooting procedures.

6.Conclusions

In this paper,we propose to locate near-field sources with the exact signal model by using a search-free method.The proposed method firstly exploits the parameter separation procedure by splitting the steering vector with the approximated signal model.Then,the principle of polynomial rooting is adapted to the approximated estimation of DOA and range.The final estimates of the parameters are corrected according to the expression of the exact signal model.The proposed method requires neither searching nor pairing,which reduces the computational complexity.In addition,the estimation accuracy is improved thanks to the polynomial rooting and error correction operations.Its effectiveness is validated with the numerical simulation.