Joint angle and frequency estimation for linear array:an extended DOA-matrix method

CHEN Luo ,DAI Xiangrui ,and ZHANG Xiaofei

1.College of Electronic and Information Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;2.The 28th Research Institute of China Electronics Technology Group Corporation,Nanjing 210007,China

Abstract: An approach for joint direction of arrival (DOA) angle and frequency estimation for a linear array is investigated in this paper.Specifically,we make the utmost of the autocorrelation and cross-correlation information to propose an extended DOAmatrix (EDOAM) method.Subsequently,we obtain the autopaired angle and frequency estimates by the eigenvalues and the corresponding eigenvectors of the novel DOA matrix.Furthermore,the proposed method surpasses the DOA-matrix method which partly ignores the autocorrelation and cross-correlation information.Finally,the proposed method works well for both uniform and non-uniform linear arrays.The simulation consequences indicate the superiority of our proposed approach.

Keywords: direction of arrival (DOA),frequency,linear array,time delay,extended DOA-matrix method.

1.Introduction

Array multi-parameter estimation is an essential problem in array signal processing.Joint estimation of frequency,Doppler shift,direction of arrival (DOA) and time/time difference of arrival (TDOA) is a standard application in multi-parameter estimation.Up to now,joint angle and frequency estimation is of great significance for various applications such as radar and wireless communication[1-5].

Over the decades,DOA estimation has become a hot topic.Many high accuracy algorithms have been presented,which include multi-dimensional multiple signal classification (MUSIC) algorithm [6],estimating signal parameters via rotational invariance techniques (ESPRIT)algorithm [7],Capon algorithm [8],Propagator method(PM) [9],maximum likelihood (ML) method [10],and DOA matrix (DOAM) method [11,12].Among these methods,the MUSIC algorithm and the ML method can achieve better estimation accuracy,but their computational complexity is higher.While the ESPRIT algorithm and the PM have elevated computational efficiency,their estimation performance has been satisfactorily evaluated.

Since the introduction of the least-squares (LS) method of the frequency-wave number,many conventional algorithms,such as linear prediction method for azimuth-frequency joint estimation [1],MUSIC algorithm [13-15],ESPRIT algorithm [16,17],PM algorithm [18],ML method [19] and DOAM method [20],have been applied into the estimation of joint angle and frequency [13-31].Moreover,Zoltowski et al.considered the problem of joint two-dimensional (2-D) DOA and frequency estimation in radar signal processing and application fields [2]while Haardt et al.discussed joint 2-D DOA and frequency estimation in the field of mobile communication[3].In order to reduce the amount of algorithm calculation,Zoltowski et al.designed a novel algorithm based on the unitary-ESPRIT algorithm which converts the data matrix into a real matrix by Cayley transform [21].

In [11],Yin et al.proposed the DOAM approach for 2-D DOA angle estimation with two parallel linear arrays.Specifically,the DOAM method structures a matrix through the auto-correlation matrix and the crosscorrelation matrix,and then the 2-D DOA angle estimates can then be calculated by executing eigenvalue decomposition (EVD) to the matrix.Since peak search is not required,the DOAM method has a lower computational complexity than the MUSIC algorithm.However,the DOAM method partly ignores the auto-correlation and cross-correlation information,which internally degenerates the estimation precision.To tackle this problem,Dai et al.proposed an extended DOAM (EDOAM) method,where an extended matrix that involves all the autocorrelation and cross-correlation information is constructed [22].

In this paper,we combine the EDOAM method with linear arrays and derive an efficient joint DOA and frequency estimation method.Specifically,we perform EVD on the EDOAM.Then,the estimation of the frequency as well as the DOA can be obtained with the resulting eigenvalues and eigenvectors.We have evaluated the angle and frequency estimation performance by extensive Monte Carlo simulations.The simulation consequences show that the proposed EDOAM approach exhibits higher angle and frequency estimation precision than the DOAM method.

The main contributions of our research are summarized as follows:

(i) To solve the problem of joint estimation of angle and frequency,we propose an EDOAM method.

(ii) We adequately utilize auto-correlation and crosscorrelation information to construct the EDOAM,while the DOAM method partly exploits the information.

(iii) The EDOAM method has a better angle and frequency estimation performance than the DOAM method.

2.Data model

Suppose there areKuncorrelated far-field narrowband sources impinging on anM-sensor linear array as shown in Fig.1.We define the first array element as the origin array element anddk(k=0,1,···,M-1) as the distance between themth element and the origin element,whered0=0.Each element is connected to a delay unit with delay of τ as shown in Fig.2.Assume thatθi(i=1,2,···,K) is the elevation angle andfi(i=1,2,···,K) is the carrier frequency of theith source.

Fig.1 Linear array

Remark 1In this paper,we assume that these sources are uncorrelated.If these sources are correlated,we can use spatial smoothing technology to complete decorrelation.

The received signal of themth(1,2,···,M) array element and the corresponding time delay unit [15] are

where c is the transmission speed of electromagnetic wave,andni(t) is the array output noise.Hence,after vectori-zation [15],we can obtain

3.Joint angle and frequency estimation via DOAM method

According to the received data of the signal receiving system,we can derive four auto-correlation and crosscorrelation matrices ofx(t) andy(t).However,the conventional DOAM method only uses the auto-correlation matrix ofx(t) and the cross-correlation matrix ofy(t) andx(t)[11] defined by

where Ψ=E[s(t)sH(t)] represents the covariance matrix of the signals.

To estimate the variance of noise,we perform EVD toRxxand obtain the eigenvalues ε1,ε2,···,εM,where ε1＞···＞εK＞···＞εM.Hence,the noise variance can be calculated as σˆ2=(εK+1+···+εM)/(M-K).Therefore,we can eliminate the impact of noise [11] as

Define the DOA matrixR∈CM×M[11] as

Hence,frequency estimation can be achieved by the eigenvalues ofR,and angle estimation can be achieved by the corresponding eigenvectors.The correspondence relation realizes the auto-pair of angle and frequency.

4.Joint angle and frequency estimation via EDOAM

The conventional DOAM method underutilizes the autocorrelation and cross-correlation information of received data leading to the degradation of the DOA and frequency estimation performance.In this paper,we propose an extended estimation method based on the conventional DOAM method,which makes the utmost use of the information in the four auto-correlation and cross-correlation matrices.The auto-correlation matrix and crosscorrelation matrix ofy(t) andx(t) can be denoted [22] as

According to the relation between the auto-correlation matrix and the cross-correlation matrix,we define two novel matricesR1,R2∈C2M×Mwhich contain all the autocorrelation and cross-correlation information,i.e.,

Therefore,similar to the traditional DOA matrix,an extended DOA matrix can be defined [22] as

Lemma 1IfAand Ψ are nonsingular,the signal frequency elements of Γ are equal to theKnon-zero unequal eigenvalues ofRE,and the corresponding eigenvectors are equal to the signal direction vectors,i.e.,

In summary,the major flow of the method proposed in this paper can be generalized as follows:

Step 1Compute the auto-correlation and cross-correlation matrices:

Remark 2In this paper,we assume that the source numberKis known.IfKis unknown,we can utilize the matrix decomposition method,smoothed rank method,Gerschgorin discs method or information theory method to estimateK[32-35].

5.Discussions

5.1 Complexity

5.2 Cramer-Rao bound (CRB)

In this subsection,we derive the CRB of the angle and frequency estimation for the received system.We define the data model of the system as

where

Hence,by utilizing (36),we can compute

5.3 Uniform linear arrays

In this subsection,we discuss the estimation performance of the proposed method for uniform linear arrays.Other things being equal,we define inter-element spacing asd.Hence,Acan be reformulated into

Replacing the matrixAwithA′in (3)-(25),we can obtain the frequency estimation (26).

For a uniform linear array,(27),(28),and (30) can be rewritten as

5.4 Advantages

The proposed method has the following advantages:

(i) The parameter estimation performance of the proposed method is superior to the conventional DOAM method.

(ii) The proposed method adequately utilizes the autocorrelation and cross-correlation information to construct the extended DOA matrix,while the conventional DOAM method partly exploits the information.

(iii) The proposed method suits both non-uniform and uniform arrays.

6.Simulation results

We consider the signal receiving system with single delay and supposeK=3 sources impinge on the linear array.These sources located at angles θ1=10◦,θ2=20◦,and θ3=30◦,withf1=6 MHz,f2=8 MHz,andf3=10 MHz.K,M,andNare the number of sources,array elements,and snapshots,respectively.In the following simulations,c=3×108m/s is the signal propagation velocity.The delay of system is τ=40 ns.We assess the angle and frequency estimation performance of the proposed method by the root mean square err or (RMSE) as

Fig.3 shows the estimation consequences of our proposed method for a uniform linear array while the distance between any two adjacent array elements isd=12 m.It depicts that our proposed method can accurately achieve joint estimation of DOA and frequency for uniform linear array.

Fig.3 Angle and frequency scatter with uniform linear array

Fig.4 and Fig.5 present the RMSE comparison of our proposed method and some other method for uniform linear array,whereM=12,K=3,andN=500.We find the proposed algorithm can also work well in the case of uniform linear arrays.Furthermore,the proposed method is more accurate than the conventional DOAM method[20] and the PM algorithm [18].From Fig.4 and Fig.5,it is shown that IM-ESPRIT performs slightly better than our proposed method.

Fig.4 RMSE of angle estimation with different SNRs for uniform linear array

Fig.5 RMSE of frequency estimation with different SNRs for uniform linear array

Fig.6 and Fig.7 show the DOA and frequency estimation precision of the proposed method under different snapshotsNfor uniform linear array,where S NR=10 dB,M=12,andK=3.Fig.6 and Fig.7 indicate that the estimation accuracy of both angle and frequency improves as the number of snapshotsNincreasing.Furthermore,these reconfirm that the proposed method has higher angle and frequency estimation precision than the conventional DOAM method [20] and the PM algorithm [18].

Fig.6 RMSE of angle estimation with different snapshots for uniform linear array

Fig.7 RMSE of frequency estimation with different snapshots for uniform linear array

Fig.8 presents the estimation results of our proposed method for a non-uniform linear array,where SNR=10 dB,N=500,M=12,K=3.Fig.8 is a nonuniform array design example where the elements positions ared={0 9.6 16.8 30 45.6 54 60 73.2 84 99.6 117 60 73.2 8499.6 117 140.4}.Fig.8 explicitly shows that our proposed method can accurately achieve joint estimation of DOA and frequency.Fig.9 and Fig.10 depict the angle and frequency estimation performance comparison of the proposed method and the traditional DOAM method [20] for a non-uniform linear array,whereN=500,M=12,andK=3.Fig.4 and Fig.5 show clearly that our proposed method has higher angle and frequency estimation precision compared to the conventional method since the conventional DOAM method [20] partly ignores the information of the autocorrelation and the cross-correlation matrices while the proposed method gets the utmost out of the information.IM-ESPRIT fails to work in a non-uniform linear array,and then we have not shown IM-ESPRIT in Fig.8 and Fig.12.

Fig.8 Angle and frequency scatter with non-uniform linear array

Fig.9 RMSE of angle estimation with different SNRs for non-uniform linear array

Fig.10 RMSE of frequency estimation with different SNRs for non-uniform linear array

Fig.11 and Fig.12 illustrate the DOA and frequency estimation accuracy of the proposed method under different snapshotsNfor a non-uniform linear array,where signal-to-noise ratio SNR=10 dB,M=12,andK=3.It can be observed that the estimation performance of both angle and frequency improves with the increase of snapshotsN.Furthermore,these reconfirm that the proposed method is more precise than the conventional DOAM method [20] with all the snapshots.

Fig.11 RMSE of angle estimation with different snapshots for nonuniform linear arrays

Fig.12 RMSE of frequency estimation with different snapshots for non-uniform linear arrays

7.Conclusions

In this paper,we propose the EDOAM method to jointly estimating the angle and frequency.The proposed method constructs EDOAM which makes the best of all crosscorrelation and autocorrelation information,while the conventional DOA-Matrix method ignores part of the information.We can obtain the auto-paired frequency and angle estimates by the eigenvalues and the corresponding eigenvectors of the EDOAM.The emulation consequences demonstrate that the proposed method estimation precision outperforms the conventional DOA-Matrix method.