CRB for 2-D DOA Estimation in MIMO Radar with UCA

Peng Zhenni（彭珍妮），Ben De（賁德），Zhang Gong（張弓），Cai Xiaobin（蔡曉斌）

1.UAV Research Institute，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；

2.Key Laboratory of Radar Imaging and Microwave Photonics，Ministry of Education，Nanjing 210016，P.R.China；

3.95835 PLA Troops，Malan 841700，P.R.China

Hence，the Eisher information matrix has the following expression

CRB for 2-D DOA Estimation in MIMO Radar with UCA

Peng Zhenni（彭珍妮）1*，Ben De（賁德）2，Zhang Gong（張弓）2，Cai Xiaobin（蔡曉斌）3

1.UAV Research Institute，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；

2.Key Laboratory of Radar Imaging and Microwave Photonics，Ministry of Education，Nanjing 210016，P.R.China；

3.95835 PLA Troops，Malan 841700，P.R.China

The Cramer-Rao bound（CRB）for two-dimensional（2-D）direction of arrival（DOA）estimation in multiple-input multiple-output（MIMO）radar with uniform circular array（UCA）is studied.Compared with the uniform linear array（ULA），UCA can obtain the similar performance with fewer antennas and can achieve DOA estimation in the range of 360°.This paper investigates the signal model of the MIMO radar with UCA and 2-D DOA estimation with the multiple signal classification（MUSIC）method.The CRB expressions are derived for DOA estimation and the relationship between the CRB and several parameters of the MIMO radar system is discussed.The simulation results show that more antennas and larger radius of the UCA leads to lower CRB and more accurate DOA estimation performance for the monostatic MIMO radar.Also the interference during the 2-D DOA estimation will be well restrained when the number of the transmitting antennas is different from that of the receiving antennas.

uniform circular array（UCA）；Cramer-Rao bound（CRB）；direction of arrival（DOA）estimation；multiple-input multiple-output（MIMO）radar

0 Introduction

The multiple-input multiple-out（MIMO）radar exploits the spatial diversity and the degree of freedom to improve resolution，clutter mitigation and classification performance［1］.Unlike a standard phased-array radar，which transmits scaled versions of a single waveform，MIMO radars emit orthogonal or noncoherent waveforms in each of the transmitting antennas and utilize a bank of matched filters to extract the waveforms at the receiver［2-3］.This paper focuses on array configuration problems of one class of MIMO radars which use closely spaced antennas to achieve coherent processing gain.Many literatures apply the uniform linear array（ULA）for array configuration and there have been various traditional algorithms for its direction of arrival（DOA）estimation.However，compared to ULA，uniform circular array（UCA）has its own advantages of DOA estimation in range of 360°and less antennas required than ULA for the same radar performance［4-5］.Here we will explore the property of UCA to design both the transmitting and receiving arrays and derive the CRB for DOA estimation of the MIMO radar system.The relationship between the parameters of the MIMO radar and the CRB will be investigated in detail.

1 Signal Model for MIMO Radar with UCA

Assume a MIMO radar system that utilizes a UCA with M transmitting antennas and N receiving antennas（Eig.1）.R1and R2denote the radii of the transmitting and receiving arrays，respec-tively.Each of the transmitting antennas emits M orthogonal signals sm∈CL×1，m=1，2，…，M where L is the number of samples of each pulse. Let S=［s1，s2，…，sM］.Assume there are also many（say，K）far field independent scattering point targets with the azimuth angleθand the elevation angleφ.Then the transmitting and receiving steering vectors can be described by the following expression respectively.

Eig.1 Signal model for MIMO radar with UCA

where

and i=0，1，…，M-1；j=0，1，…，N-1，and（·）Tdenotes the transpose of the matrix.

Then the received signal X∈CN×Lis

where E∈CN×Lis the interference plus Gaussian noise uncorrelated with X.

Thus，the output of the matched filters is［6］

where vec（·）denotes vectoring of the matrix. The DOA estimation method applied here is the multiple signal classification（MUSIC）algorithm［7-8］.

2 CRB for 2-D DOA Estimation

CRB is probably the best known lower bound on the mean square error（MSE）of unbiased estimators［9-10］.In this section we derive the CRB for 2-D DOA estimation of MIMO radar with UCA.

Assume that the parameters to be estimated corresponding to the k th target isΩ=［θ，φ，η，σ2］T，whereθis the azimuth angle，φthe elevation angle，ηthe scattering coefficients，andσ2the parameter of the noise.Besides，assume that during the q th pulse，the received signal through the l th sampling is x（l-1），l=1，…，L.Then a column vector is given by

The received signal without noise can be expressed as

Then we have

Eor each unknown parameter in vectorΩ=［θ，φ，η，σ2］T，its estimated variance has a lower bound，which is the CRB.It can be computed by the Eisher information matrix

where var（·）denotes the variance，Ωithe estimation of i th element ofΩ，andthe（ i，i） element of the inverse matrix of the Eisher information matrix F.The element of F can be expressed as

Hence，the Eisher information matrix has the following expression

Since F is a symmetric matrix，only one part of its elements needs to be calculated.Define a matrixthe following expression can be obtained from Eq.（12）

Considering that the transmitting array and receiving array of the MIMO radar satisfy the following expression

Similarly，the other elements of F can be calculated by

Thus，the Eisher information matrix can be simplified as

and its inverse matrix is

Einally，the CRB of azimuth angle can be obtained.

that is

And the CRB of the elevation angle is

Namely，

It can be inferred from Eqs.（27，29）that for the MIMO radar system with UCA，the more the number of antennas and samples is，the lower the CRB of the azimuth and elevation angle will become and the higher the estimation accuracy will be.Meanwhile，the CRB of the azimuth and elevation angle will be lower when the radius of the transmitting or receiving circular arrays and the scattering coefficient become larger.When the elevation angle is close to 0°（90°）and sinφ→0（cosφ→0），the CRB of azimuth angle（elevation angle）estimation tends to infinity and the estimation accuracy of azimuth angle（elevation angle）will become poor.

3 Simulation Results

The relationship between CRBs of the 2-DDOA estimation and the several parameters of a monostatic MIMO radar system is analyzed according to the simulation results.Assume that the signal-to-noise ratio（SNR）isυ=5 d B，the number of pulses is Q=512，and the number of samples during one pulse is L=256.The scattering coefficient of the target obeys the Rayleigh distribution with the parameter to be 1.All the simulations are conducted by the Matlab software.

Eirst，the number of the transmitting（receiving）antennas is set to be eight and the radius of the circular array be twice of the wavelength. The CRB of the 2-D DOA estimation is shown in Eig.2.The simulation results show that the CRB of the angle estimation relates to elevation angle. Eig.2（a）illustrates that the CRB for the azimuth angle estimation becomes lower and lower when the elevation angle increases from 0°to 90°.Contrarily，Eig.2（b）demonstrates that the CRB of the elevation angle increases as the elevation angle changes from 0°to 90°.This is in consistent with the theoretical analysis in the previous section.

Eig.2 Relationship between CRB of 2-D DOA estimation and elevation angle in MIMO radar with UCA

Then we test the relations between the CRB of the angle estimation and the number of the transmitting and receiving antennas.All the simulation parameters are the same as those in the above example except for the number of antennas.Set the number of antennas to be 5，8，10 and 15，respectively.The simulation results are shown in Eig.3.Both Eig.3（a）and Eig.3（b）indicate that the CRB of the angle estimation decrease when the number of antennas increase.It means that estimation accuracy will be improved when the number of the antennas increases. However，as the number of the antennas climbs，the complexity of the system and the computational load highly increases，which should be thought about seriously by radar designers.

Eig.3 Relationship between CRB of 2-D DOA estimation and UCA antenna number

Einally，the relations between the CRB of the angle estimation and the radius of the circular array are studied by tests.All the simulation parameters are the same as in the first example ex-cept for the radius of circular array.Set the radius of the circular array to be half of wavelength，one wavelength，two wavelengths and eight wavelengths，respectively.The simulation results are shown in Eig.4.It is obvious that both for the azimuth angle and the elevation angle，the CRB of angle estimation becomes lower with the larger radius of UCA.However，during the simulation the increasing of the radius brings in strong interference，which seriously affects the estimation accuracy.Moreover，when the radii of the transmitting array and the receiving array differ from each other，the interference problem will be restrained.The simulation results are shown in Eig.5.The 2-D DOA estimation performance of the MIMO radar shown in Eigs.5（b，c）is much better than that of Eig.5（a）.

Eig.4 Relationship between CRB of 2-D DOA estimation and radius of UCA in MIMO radar

4 Conclusions

Eig.5 2-D DOA estimation results of MIMO radar with different radii of UCA

A signal model is developed for 2-D DOA estimation in MIMO radar with UCA.The MUSIC method is applied to angle estimation.The CRB for DOA estimation with UCA is derived.The simulation results show that all the changes of the elevation angle，the number of the antennas，and the radius of the circular array have impacts on the CRB of the 2-D DOA estimation.Generally，the CRB of angle estimation is lower when the MIMO radar system has more antennas in a circular array and larger radius for the circular array. Besides，the simulation results also reveal that when the number of the transmitting and receiving antennas are different from each other，the interference problem during the DOA estimationwill be effectively restrained.The work has implications for MIMO radar design.

Acknowledgements

This work was supported by the National Natural Science Eoundation of China（Nos.61071163，61071164，61471191）and a project Eunded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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（Executive editor:Zhang Tong）

TN957 Document code:A Article ID:1005-1120（2015）01-0115-06

*Corresponding author:Peng Zhenni，Associate Researcher，E-mail:pengzhenni@nuaa.edu.cn.

How to cite this article:Peng Zhenni，Ben De，Zhang Gong，et al.CRB for 2-D DOA estimation in MIMO radar with UCA［J］.Trans.Nanjing U.Aero.Astro.，2015，32（1）:115-120.

http://dx.doi.org/10.16356/j.1005-1120.2015.01.115

（Received 20 November 2014；revised 7 January 2015；accepted 12 January 2015）