Peng Zhenni,Ben De,Zhang Gong,Gu Haiguang

(1.College of Electronic and Information Engineering,NUAA,29 Yudao Street,Nanjing,210016,P.R.China;2.Nanjing Research Institution of Electrical Technology,Nanjing,210039,P.R.China)

INTRODUCTION

Multiple-input multiple-output(MI MO)radar has draw n considerable attention recently because of a number of advantages[1-5],including high sensitivity of detecting slow moving targets,excellent interference rejection capability,good parameter identifiability,and enhanced flexibility for transmitted beam pattern design.Unlike a standard phased-array radar which transmits scaled versions of a single waveform,a MI MO radar system emits orthogonal waveforms in each of the transmit antennas and utilizes a bank of matched filters to extract the waveforms at the receiver.As a new radar system,MI MO radar has many problems to be dealt with among which non-uniform spacing can achieve higher performance for parameter identifiability,that is,the maximum number of targets that can be uniquely identified by the radar[1].It is known that the non-uniform linear array(N LA)of phased-array radar can be used to develop thinner beam patterns to improve the system performance.In this paper N LA is extended to the case of M IMO radars to obtain more distinct virtual array elements with fewer physical antennas,which is important to improve the parameter identifiability and also reduce the design cost and complexity of the radar system.A class of N LA called minimum redundancy linear array is employed for MI MO radar array configuration.A new method to generate large low redundancy arrays from small ones is also described.

1 SIGNAL MODEL OF MI MO RADAR

Assume a MI MO radar system that utilizes an array of M t for transmit antennas and M r for receive antennas,and many far field independent scattering point targets.Let x m(n)denote the discrete-time baseband signal transmitted by the m th transmit antenna,and ym(n)the signal received by the m th receive antenna.

where N is the number of samples of each signal pulses.

Letθdenote the direction-of-arrival(DOA)of a generic target.Then,under the assumption that the transmitted probing signals are narrow band and the propagation is non-dispersive,the transmitted and received steering vectors can be described by the following expression respectively

where f m(θ)is the time delay via the m th transmit antenna to the target located atθand f′m(θ)via the target to the m th receive antenna,and k 0 the carrier frequency.Then assume the number of the far field point targets is Q,the received signal y(n)is[1]

where X(n) denotes the interference noise uncorrelated with x(n),T q the complex amplitudes proportional to the radar cross-sections(RCSs)of those targets,(? )Hthe conjugate transpose,and(?)cthe complex conjugate. Then a new virtual array steering vector G is formed,and it is the Kronecker product of the transmitted and the received array steering vectors of MIMO radar

The concerned problem is the maximum number of targets that can be distinguished by a certain M IMO radar system.Consider the case that the transmitting array is also the receiving array(Fig.1),for most radar systems are active ones and the array is used for both transmitting and receiving.Assume that the array is a uniform linear array,i.e.Mt=Mr=M,so its steering vector is

where d denotes the distance of the adjacent antennas andλthe carrier wavelength.So G is

where G is supposed to have M×M distinct elements which represent the M×M distinct signal channels formed at the receiver for each of the transmitted waveforms.As there are overlaps in the results of the convolution products,only 2M-1 distinct elements can be obtained when the array is uniform linearly designed,however,for a N LA the number may reach M(M+1)/2[1,6].

Fig.1 ULA MI MO radar scenario

2 MINIMUM REDUNDANCY LINEAR ARRAY

The non-uniform linear array applied in this paper is the minimum redundancy linear array(MR LA)[7-8],which is to minimize the number of the antennas by reducing the redundancy of the spacing.The nomenclature used to denote MR LA of M antennas is a bracketed list of M numbers{uk}indicating the normalized antennas locations.For example,it is{0,1,4,6}as shown in Fig.2.This is a 4-antenna array whose aperture is equal to a 7-antenna ULA.Its steering vector is a(θ)=[1 e-jke-j4ke-j6k]T,where k=2πd sinθ/λ.Then its distinct elements in G=a(θ)? a(θ)are

However,for a 4-antenna ULA,a′(θ)=[1 e-jke-j2ke-j3k]T,its distinct elements in G′are(1,e-jk,e-j2k,e-j3k,e-j4k,e-j5k,e-j6k).

Note that the number of the distinct virtual array elements obtained by the 4-antenna N LA is up to M(M+1)/2=10,while it is only 2M-1=7 by ULA[1,6,7]. As the distinct elements represent the effective signal channels,it can be inferred that the MI MO radar parameter identifiability mostly depends on the number of the distinct elements in the Kronecker product G[1,9].More distinct elements lead to a higher identification performance under the same experimental conditions.

Fig.2 MR LA of 4 antennas

The idea of MR LA was first proposed by Moffet[6].It suggests that one should minimize the number of the physical antennas as long as the spacings between pairs of array antennas include all the integers between 1 and L,where L is the desired normalized aperture of N LA.The spacing are defined as{uk-uk′}.The optimization solution is

where|?|denotes the cardinality of the set.For a smaller M,the optimization solution can be found by an exhaustive search algorithm.However,when M becomes larger,it requires an extremely long time for the exhaustive searching,which is a problem not easy to solve.

In order to avoid the exhaustive search for a larger M,a new method is proposed to grow small MR LAs into large ones by inserting a seed repeatedly. Redundancy R is quantitatively defined by the ratio of the number of pairs of antennas to the desired aperture length L[7]

MR LAs are designed to make the redundancy R as small as possible.A bracketed list of M-1 numbers(m1,m2,…,m M-1)indicating the spacing between adjacent antennas is used to denote MR LA.Firstly,split the parent array(m1,m2,…,m M-1)into two parts.When M is odd,the number of elements in the bracketed list is even and it is split at the midpoint.When M is even,the list can be split at either the(M-1)/2 or the M/2 position.Then the new array is constructed by inserting a number repeatedly at the position where the list has been split.This number is equal to the number of antennas(M)in the parent configuration.The number inserted at the midpoint of the list must appear at least twice in order to ensure that the array is restricted.For example,when M=13,the array configuration can be generated by the following sequence with the redundancy R=1.34

In this way,the redundancy R of the large MR LAs can be constrained within R≤1.60 when M≤37[6-8],which is an acceptable redundancy.

3 NUMERICAL RESULTS

Several numerical examples are presented to compare the parameter identifiability of the N LA MI MO radar with its ULA counterpart.The applied transmitted waveforms are quadrature phase shift keyed(QP SK)sequences which are orthogonal to each other[1].

Firstly,consider a scenario where Q targets are located with Δθ=10°to adjacent ones.The number of the snapshot is N=256.Assume the received signal is mixed with a Gaussian noise with mean zero and variance 0.01.An MI MO radar system with M=7 antennas is used for testing.Let the array as MR LA{0,1,4,6,13,21,31}and all the distances between antennas are times of half-wavelength.Fig.3 shows the simple least-squares(LS)spatial spectrum Z LS(θ),as a function ofθ,when Q=12 and the targets are located from-50°to+60°.Note that all the 12 targets can be identified by the peak of the LS spatial spectrum.However,compared with an ULA MI MO radar[1],at least 10 antennas are needed to get 12 targets separated under the same simulation conditions.

Fig.3 LS spatial spectrum(M=7,Q=12)

Then consider a N LA MI MO radar system with M=6.Its array configuration is{0,1,4,5,11,13}. The corresponding 6-antenna ULA MI MO radar system is also tested where the adjacent antennas are half-wavelength spaced.All the simulation parameters are the same as tho se in the above example except that Q=10.It can be observed from Fig.4 that the N LA system can distinguish the 10 targets clearly,while it is very hard for the ULA system to gain the similar performance with the same number of antennas.The numerical results are also provided on the Cramer-Rao bound(CRB)ofθ,which is probably the best known lower bound on the MSE of unbiased estimators[5,10].CRB has the following form in the simulation

Fig.4 Comparison of LS spatial spectrum between N LA and ULA MI MO radars(M=6,Q=10)

Fig.5 shows CRB ofθas a function of Q and a comparison of the two systems.Note that CRB of N LA MI MO radar is always lower than its ULA counterpart as Q increases from 1 to 10.

Fig.5 CRB ofθf(wàn)or N LA and ULA MI MO radars

4 CONCLUSION

In this paper,a non-uniform linear array configuration method is presented for MI MO radars. As demonstration of the potential advantages that a N LA MI MO radar can offer,the LS and the Cramer-Rao bound are evaluated for parameter estimation.The numerical results show that compared with ULA M IMO radars,N LA MI MO radars can achieve the same parameter identifiability with fewer physical antennas and obtain more distinct virtual array elements and lower Cramer-Rao bound with the same number of the antennas. The N LA configuration method can reduce the cost and complexity of the array design in MI MO radars.How to design the optimum non-uniform linear arrays for a MI MO radar system is valuable in deep research.

[1] Li Jian,Stoica P,Xu Luzhou,et al.On parameter identifiability of MI MO radar[J]. IEEE Signal Processing Letters,2007,14(12):968-971.

[2] Friedlander B.On the relationship between MI MO and SI MO radars[J].IEEE Transactions on Signal Processing,2009,57(1):394-398.

[3] Fei Yuanyuan,Fan Yijia,Lau B K,et al.Optimal single-port matching impedance for capacity maximization in compact M IMO arrays[J].IEEE Transactions on Antennas and Propagation,2008,56(11):3566-3575.

[4] Chapursky V V,Ivashov S I,Vasiliev I A,et al.Short range radar with MI MO antenna system and?multifreq?uency sounding signal[J].PIERS Online,2009,5(8):716-720.

[5] Fishler E,Haimovich A,Blum R,et al.M IMO radar:an idea whose time has come[C]//Radar Conference,Proceedings of the IEEE.Philadelphia:IEEE,2004:71-78.

[6] Chen Chun-Yang,Vaidyana than P P.Minimum redundancy MI MO radars[C]//IEEE International Symposium on Circuits and Systems.Seattle:IEEE,2008:45-48.

[7] Camps A,Cardama A,In fantes D.Synthesis of large low-redundancy linear arrays[J]. IEEE Transactions on Antennas and Propagation,2002,49(12):1881-1883.

[8] Zhang Zhaoyang,Zhao Yongbo,Huang Jing fang.Array optimization for MI MO radar by genetic algorithms[C]//The 2nd International Congress on Image and Signal Processing.Tianjin,China:IEEE,2009:1-4.

[9] Li Jian,Stoica P.MI MO radar signal processing[M].New York:Wiley-IEEE Press,2008:5-11.

[10]Dogandzic A,Nehorai A.Cramer-Rao bounds for estimating range,velocity,and direction with an active array[J]. IEEE Transaction on Signal Processing,2001,49(6):1122-1137.