Multiple-input–multiple-output radar multistage multiple-beam beamspace reduced-dimension...

Published in IET Radar, Sonar and NavigationReceived on 9th March 2012Revised on 16th July 2012Accepted on 19th July 2012doi: 10.1049/iet-rsn.2012.0078

ISSN 1751-8784

Multiple-input–multiple-output radar multistagemultiple-beam beamspace reduced-dimensionspace-time adaptive processingWei Zhang, Zishu He, Jun Li, Hongmin Liu

School of Electronic Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu,

People’s Republic of China

E-mail: [email protected]

Abstract: Multistage multiple-beam (MSMB) reduced-dimension (RD) space-time adaptive processing (STAP) algorithm formultiple-input–multiple-output radar has been proposed to reduce ground clutter in an airborne radar system. It isdemonstrated that the MSMB RD STAP can reduce the degrees of freedom much less than the rank of clutter or minimumamount of channels of other RD algorithms without significant loss, exhibiting rapid convergence to the performance boundset by the optimum filter. The proposed algorithm provides the upper-bound output signal-to-interference-noise for RD STAPalgorithm with fixed number of auxiliary channels, and it can use less auxiliary channels to achieve comparative performance.Consequently, the real time computation complexity of estimation and inversion of localised clutter covariance matrix isreduced, and the requirement of the sample support used to estimate the localised clutter covariance matrix is also reduced.

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1 Introduction

Multiple-input–multiple-output (MIMO) radar has been avery active area for several years. In recent literatures [1–6],MIMO radars have been shown to provide a number ofpotential benefits as compared with conventional radar,such as increased angle resolution, increased Dopplerresolution, sharper airborne radar clutter notches etc. Forsome applications, these results indicate significant potentialimprovements in clutter mitigation signal-to-interference-noise ratio (SINR) loss [7, 8]. The performance of a groundmoving target indication (GMTI) radar is strongly driven bythe length of the radar aperture, as long apertures enablelower minimum detectable velocity (MDV), that is, MDV issensitive to both the aperture size and integration interval.Both of these characteristics can be improved by usingMIMO radars. MIMO GMTI radars can be employed toimprove minimum detectable velocities [6].Space-time adaptive processing (STAP) has been a

well-established technique, whose basic concept and theoryare first put forward by Brennan and Reed [9]. However, itis difficult to be implemented in the practical systembecause of the computational complexity and the samplelimitation for estimating the clutter covariance matrix. ForMIMO radar, the challenges became more serious thanphased radar because of utilising multiple independentsignals. Following the landmark publication by Brennanet al., research on STAP evolved rapidly [10–31]. Up tonow, much work has been done to seek for this kind ofreduced-dimension (RD) [16–20] or reduced-rank (RR)[21–31] STAP algorithms. Some typical suboptimal

IET Radar Sonar Navig., 2013, Vol. 7, Iss. 3, pp. 295–303doi: 10.1049/iet-rsn.2012.0078

dimension-reduced STAP approaches have been proposed,such as auxiliary channel receiver [16], the factor approach(FA) and the extend FA (EFA) [17], the joint domainlocalised (JDL) approach [18]. Reduced rank STAPapproaches utilising data-dependent transformations, such asthe principal-components inverse algorithm [21], eigencanceller algorithm [22, 23], cross-spectral metric (CSM)[26] and multistage Wiener filter (MWF) [27]. Both theCSM and MWF all have high performances and lowsampling requirements. However, the detection performancedegrades rapidly as the number of eigenvectors retained isdecreased below the rank of clutter. The penalty forunderestimating the rank of the total noise contribution canbe great [28]. It is worth to point out that the adaptivedegrees of freedom (DoFs) of CSM approach and MWFapproach are still great if the system performance should bekept well especially for MIMO radar since the interferenceclutter rank is large [32].JDL utilises relative fixed auxiliary channels. It is thought

that the beams around the search beam are more important inJDL. However, it cannot provide the optimal or sub-optimalauxiliary beam channel selection method, that is, if wefixed the number of total auxiliary beams, which set ofbeams should be selected as auxiliary channels to cancelthe interference components in the main beam to maximisethe output SINR. Minimising the DoFs with littleperformance degradation is the target of RD STAP, forboth phased radar and MIMO radar. Maximising theconvergence rate is important for satisfactory performancein many practical situations [35], especially when theinterference and noise environment is not strictly wide

295& The Institution of Engineering and Technology 2013

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sense stationary. The issue at hand is then as follows, if thetotal number of auxiliary beam channels in RD beamspaceSTAP is fixed, how to make the selection optimally. A newRD STAP named multistage multiple-beam (MSMB) forMIMO radar will be proposed which will provide thenearly optimal performance when the total number ofauxiliary beam channels is fixed.The MIMO signal model and virtual transmitting–
receiving beamforming are reviewed in Section 2. The RDSTAP algorithms for MIMO radar is derived in Section3. The equivalent full-dimensional generalised sidelobecanceller (GSC) form for MIMO radar and themultiple-beam RD STAP equivalent GSC are derived inSection 4, also the proposed MSMB STAP algorithm isderived in this section, and a possible implementationscheme is considered. The simulation and performanceanalysis of the MSMB is presented in Section5. Conclusions are provided in Section 6.

2 Airborne MIMO radar receiving signalmodel and virtual transmitting–receivingbeamforming

2.1 Airborne MIMO radar receiving signal model

The assumed airborne MIMO radar system geometry isshown in Fig. 1.As illustrated in Fig. 1, the radar operates in a side-looking

geometry. The radar is monostatic and moves along ahorizontal path at a constant velocity. The long axis of theantenna is aligned with the velocity vector. The radarplatform travels at velocity v. There are NT transmittingelements with uniform space dT, NR receiving elementswith uniform space dR = λ/2, λ is the wavelength, thereceiving antennas are collocated with transmittingantennas. Generally, the array of MIMO radar can beconfigured as sparse array which can provide the longestcontinuum virtual aperture [6, 32, 33]. Thus, define γ = dT/dR, and let γ = NR. At each transmitter, a coherentprocessing interval (CPI) consists of M pulses. Assumingthe receivers and transmitters are close together relative totargets in the far field, the transmitting array and receivingarray have the same azimuth angle θ and elevation angle φ.

Fig. 1 MIMO radar geometry


In a MIMO radar system, one allows for the possibility ofmultiple distinct waveforms being transmitted. In each pulserepetition interval Tr, the system will transmit a waveform set

s(t) = s1(t), s2(t), . . . , sNT(t)

[ ]T(1)

where snt (t) (nt = 0, 1,…, NT− 1) is the transmitting signalfrom the ntht element, they are orthogonal to each other, thesuperscript T means matrix transpose. After matched filterin each receiving channel, and stacking the total space-timeclutter return into one column, we can obtain the NRNTMvector [32]

x = x0,0,0, x1,0,0, . . . , xNR−1,NT−1,M−1

[ ]T=∑NC

i=1

risD fd,i( )⊗ sT fs,i

( )⊗ sR fs,i( )+ n

=∑NC

i=1

ris fd,i, fs,i( )+ n

(2)

where s( fd,i, fs,i) denotes the size-NRNTM space-time steeringvector, n is the white noise,

sD fd,i( ) = 1, ej2pfd,i , . . . , ej2p(L−1)fd,i

[ ]Tis the Doppler

steering vector, sT fs,i( ) = 1, ej2pgfs,i , . . . , ej2pg NT−1( )fs,i[ ]T

is

the transmitting steering vector and

sR fs,i( ) = 1, ej2pfs,i , . . . , ej2p NR−1( )fs,i[ ]T

is receiving

steering vector, γ = dT/dR, dR = λ/2, the clutter ring at adistance of r is divided into NC discrete patches, fd,i =2vsinθicosj·Tr/λ is the normalised Doppler frequency ofclutter at θi, fs,i = dR sinθi cosj/λ is the spatial frequency ofclutter at θi, ρi is the reflected coefficient of clutter at θi.The goal of STAP is to find a linear combination of the

extracted signals so that the SINR can be maximised, it canbe expressed as the following optimisation problem [11]

min{w}

wHRw

subject to wHs fd,t, fs,t( ) = 1

⎧⎨⎩ (3)

where R = E[xxH] is the covariance matrix of clutter. Thecoefficients w of the STAP should be designed such thatminimises clutter power while maintaining a space-timetwo-dimensional (2D) mainbeam constraint.

2.2 Virtual transmitting–receiving beamforming

For each pulse, the space domain beamforming can be done.For simplicity, the subscript m will be omitted. The receivedclutter signal vector of the mth pulse is

x = x0,0, x1,0, . . . , xNR−1,NT−1

[ ]T(4)

For forming virtual antenna beams pointing to a broadsideangle of θ, the coefficients ws of the beamforming filtershould be designed such that the filter output in theintended beam direction reaches the maximum value. Thedigital filter coefficient vector to be designed for


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beamforming is assumed to be
ws = w0,0, w1,0, . . . , wNR−1,NT−1

[ ]T(5)

where wnr ,ntis the filter weighting coefficient for the output of

the matched filter at nthr element and matched to waveformtransmitted by ntht element, where nr = 0, 1,…, NR− 1, nt =0, 1,…, NT− 1. It is well known that, the coefficients ws ofthe optimal beamforming filter for target direction can bedetermined by maximising signal-to-noise ratio (SNR), theSNR reaches the maximum value if and only the followingequation holds

ws = ss fs,t( )

= sT fs,t( )⊗ sR fs,t

( )= wT fs,t

( )⊗ wR fs,t( ) (6)

where ss( fs,t) is the target signal from the expected directionθt, wT( fs,t) = sT( fs,t) and wR( fs,t) = sR( fs,t) are the weightingcoefficient for the virtual transmitting beamforming filterand receiving beamforming filter, respectively. The steeringvector sT( fs,t)⊗ sR( fs,t) is equivalent to the steering vectorthat would result from a spatial convolution of thetransmitting and receiving phase centres providing anaugmentation of the conventional array steering vectors.

3 RD STAP algorithm

The transformed signals are less correlated in theangular-Doppler domain [17], thus, for RD STAP, the datain the space-time domain is first transformed to theangle-Doppler domain. A linear transform T is applied tothe received echo data. This multidimensional transform Tshould be invertible to avoid any information loss. Forauxiliary channel processing (ACP) [16], (N +M ) beams areformed, where N is the number of receiving channels inreceiver, one of them being the search beam. The beamselection mode of ACP is illustrated in Fig. 2a. LocalisedSTAP algorithms, such as JDL, restrict adaptive processingin localised processing region (LPR) in angle-Dopplerdomain. Therefore JDL has different LPR, the size can be

Fig. 2 Angular-Doppler beam selection mode


3 × 3, 3 × 4 or other else. The beam selection modes of JDL3 × 3 is illustrated in Fig. 2b [18].The transform T can be written in a general form for RD

STAP algorithms. For MIMO radar, let fs,nt = nt − 1( )

/NTg(nt = 1, 2,…, NT), fs,nr = fs,nt+ nr − 1

( )/NR (nr = 1,

2,…, NR) and fd,i = (i− 1)/M (i = 1, 2,…, M ) represent thetransmitting spatial frequency, receiving spatial frequencyand Doppler frequency, respectively. Then the space-timesteering vector of a space-time 2D beam [18, 20] can beextended to MIMO, represented by

s fd,i, fs,nt , fs,nr

( )= sD fd,i

( )⊗ sT fs,nt

( )⊗ sR fs,nr

( )(7)

where sD( fd,i) is the length M temporal steering vectorassociated with the desired signal’s Doppler shift fd,i,sT( fs,nt ) and sR( fs,nr ) are the length NT, NR spatialtransmitting steering vector with the desired signal’ spatialfrequency fs,nt and receiving steering vector associated withthe desired signal’ spatial frequency fs,nr , respectively.Furthermore,⊗ is the Kronecker product [34, 36]. Then, Ttakes the form

[T]row = sH fd,i, fs,nt , fs,nr

( )[ ](8)

where i = 1, 2,…,M, nt = 1, 2,…, NT and nr = 1, 2,…, NR. Thesubscript row denotes the row of T. With T, all theangular-Doppler beams can be achieved. Each row of Tcorresponds to one 2D angular-Doppler beam. Thedifference among different RD STAP algorithms is onlythat they select different row subsets of T as the transformmatrix.

4 Multistage multiple-beam RD STAP

4.1 GSC-form processor for MIMO radar

The GSC can reformulate the STAP structure into the form ofa standard Wiener filter and can provide a deeper insight tothe rank-reduction and dimensionality reduction problems[26]. One angular-Doppler beam is considered asmain-beam and the rest as auxiliary beams. Main-beam is atraditional antenna beam used for detecting target signalsembedded in interference; auxiliary beams receive theinterference signals and are correlated to those in themainbeam. The transformation of the radar return x bythe operator T in (8) yields a vector xT which has the form

xT = Tx = sHxBx

[ ]= d

b

[ ](9)

where the scalar-valued beamformed output is denoted by d,b [ C

NTNRM−1( )×1 is termed the noise subspace data vector.The first row of transform matrix T is the expected signal s,all the other rows are orthogonal to s. Thus, after thetransformation of x by T, the first element is the projectionof x to the expected signal s, and b is the projections of xto other orthonormal basis.The partitioning of the matrix operator T leads naturally to

the form of the GSC array processor, which is illustrated inFig. 3.


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The transformed data vector xT has an associatedcovariance matrix RxT

which takes the form

RxT= TRTH = s2

d rHbdrbd Rb

[ ](10)

The (NTNRM− 1) × (NTNRM− 1) noise subspace covariancematrix Rb is expressed by

Rb = E bbH[ ] = BRBH (11)

The J × 1 cross-correlation vector between the noise subspacedata vector and the beamformer output is given by

rbd = E bd∗[ ] = BRs (12)

where * represents the complex conjugate operator.The optimal value of the vector w is provided by the

Wiener solution [26]

w = R−1b rbd (13)

The output of the arrays y is

y = sH − wHB( )

x (14)

It is worth to note that, the blocking matrix B is different fromRR-STAP algorithm such as CSM or Eigen-subspace inwhich B can be any matrix only needs to satisfy Bs = 0 andresults in an invertible T. Here, the signal blocking matrixB will be constructed by the 2D angular-Doppler channelsin the beamspace RD STAP. The transform matrix Tthat transforms the data from space-time domain toangle-Doppler domain can be expressed as

{T = sH

B

[ ]

= sD fd,i1

( )⊗ sT fs,j1

( )⊗ sR fs,k1

( )[ ]HB

[ ](15)

When one of the beams is selected as the expected signalchannel, just as sD( fd,i1) ⊗ sT( fs,j1) ⊗ sR( fs,k1) in (15), all theother space-time 2D beams will be selected to constructingthe signal blocking matrix B. Each row of B takes the form

[B]g = sD fd,i( )⊗ sT fs,nt

( )⊗ sR fs,nr

( )[ ]H,

i = i1 or nt = j1 or nr = k1( ) (16)

Fig. 3 Full-dimension GSC processor for MIMO radar


The subscript g of B means gth row of B. Total NRNTM 2Dangular-Doppler channels are formed by combinationdifferent Doppler frequency, transmitting angular frequencyand receiving angular frequency. One of them is the expectedsignal s or called the main channel, all the other channelsexcept the main channel are used to construct the blockmatrix B. So the signal blocking matrix B is a row subset ofT without the expected angular-Doppler beam. All the rowsof the signal blocking matrix B form the orthogonal beamchannels, and they are orthogonal to the expected signal

Bs= 0 (17)

As transmitting sparse array which can achieve the longestcontinuous virtual array [6, 32] is considered in the paper,that is, dT =NRdR. For this case, the MIMO radar can beviewed as phased array radar which is consisted of NRNT

elements in receiving array from the viewpoint of equivalencephase centres. Thus, all the NRNTM 2D angular-Dopplerchannels are also orthogonal to each other, the signalblocking matrix B which is constructed by all 2D channelsexcept the interested channel will always satisfy BBH = I andBs = 0. After the transformation by the B,J = (MNTNR− 1) angular-Doppler beams are available.

4.2 Multiple-beam RD equivalent GSC

All the beams have the possibility to be selected as theauxiliary beam channels. A beam selecting matrix V isintroduced. If all the beams are selected as auxiliary beamchannels, then V will be IJ × J. If only some beams arechosen as the auxiliary beam channels, that is, V is some Lcolumns of IJ × J. If L < J, the dimensionality reductionoccurs. For example, L will be 8 or 11 for JDL 3 × 3 andJDL 3 × 4, respectively. The RD GSC processor structure isillustrated in Fig. 4.A J × L transformation operator V is utilised, instead of

IJ × J which corresponding full-dimension, to form the LRD observation data vector

z=VHb (18)

The L-dimension vector z is termed the RD noise subspacedata vector. The associated L × L RD covariance matrix isgiven by

Rz = VHRbV (19)

The cross-correlation between the process z and thebeamformed signal d is given by

rzd = E zd∗[ ] = VHrbd (20)

Fig. 4 RD GSC processor in beamspace


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From Wiener filter theory this weight vector is expressed by
wL = R−1z rzd (21)

The data vector z is then processed by the RD weight vectorwL. If all the available beams are selected as the auxiliarychannels, that is, V = IJ × J, the output is given by

y = d − wHJ I

HJ×J

( )b (22)

When only some beams are selected as the auxiliarychannels depicted in Fig. 4, the output yr is given by

yr = d − wHL V

H( )b (23)

Compared (22) and (23), the reduced-dimensional processoroutput may be expressed by the relation

yr = 1, wHJ I

HJ×J − wH

L VH[ ] y

b

[ ](24)

Using yr to represent the output which only using some 2Dangular-Doppler channels instead of all the other NRNTM −1 channels to cancel the interference components in the mainchannel. Equation (39) denotes that yr equal to the sum of yand an extra output wH

J IHJ×J − wH

L VH( )

b. Just because of theextra output, the output SINR is degraded. Thus, how tominimise the extra output is the objective. Denote theweight error vector between the full-dimension weightvector and its RD version by

1 = I J×JwJ − VwL = wJ − VwL (25)

The output noise power for the RD processor is nowcomputed in this notation to be

Pr = s2d − rHbdR

−1b rbd + 1HRb1 ≥ P (26)

Note that the target signal power is not affected by reducingthe dimension of the problem; the reduction indimensionality is only a function of the noise fieldestimation. The SINR of the RD processor in Fig. 4 isexpressed as

jr =|a|2

s2d − rHbdR

−1b rHbd + 1HRb1

≤ j (27)

4.3 MSMB selection

The target is to minimise the scalar term εHRbε where ε isdefined in (25) to maximise ξr.

R1/2b 1 = R1/2

b R−1b rbd − VR−1

z rzd( )

= R−1/2b rbd − R1/2

b VR−1z rzd

(28)

The observation data covariance matrix Rb is expressed interms of its eigenvalues and eigenvectors as follows

Rb = ULUH (29)

Where U is a unitary J × J matrix, composed of the


eigenvectors ui{ }J

i=1and Λ is the diagonal matrix of

associated eigenvalues li{ }J

i=1. Then

R−1/2b rbd = UL−1/2UH( )

rbd

=∑Ji=1

1��li

√ uiuHi

( )rbd

=∑Ji=1

uHi rbd��li

√ ui

(30)

The R−1/2b rbd can be written as the linear combination of

ui{ }J

i=1with the weights uHi rbd/

��li

√( ){ }Ji=1

which are thecross-spectral energy projected along the ith eigenvector. And

R1/2b VR−1

z rzd = R1/2b V VHRbV

( )−1VHrbd (31)

In (31), for avoiding the inversion of VHRbV directly, wecan just select one best channel from the available beamsat every step, then remove the affect of the currentoptimal beam from the correlation between noise subspaceand expected signal, and select the next optimal channeland so on. Thus, if only one beam is selected, thechannels selecting matrix V will be a vector. For example,if the nth beam is selected as the auxiliary matrix, thenall the components will be zero except the nth component

of V, that is,V = 0, 0, . . . , 0,︸��︷︷��︸n−1

1, 0, . . . , 0

⎡⎣

⎤⎦T

, where T

denotes transpose operator. Then, Rz =VHRbV will be acomplex equal to the nth row and nth column componentof Rb,

Rz = VHRbV = Rb(n, n) (32)

So if the nth beam is selected, then

R1/2b VR−1

z rzd

( )n= R1/2

b V VHRbV( )−1

VHrbd

=∑Ji=1

��li

√uiu

Hi

( )V

1

Rb(n, n)VHrbd

=∑Ji=1

��li

√uHi VV

HrbdRb(n, n)

ui

=∑Ji=1

��li

√uHi,nrbd,n

Rb(n, n)ui

(33)

where (·)n represents the nth beam is selected, Rb(n, n)represents (n, n) component of Rb, uHi,n represents theconjugate transpose of nth component of ui, rbd,nrepresents the nth component of cross-correlation rbd.From (33), when only the nth beam is selected, we cansee that the term R1/2

b VR−1z rzd in (28) can also be written

as the linear combination of ui{ }J

i=1, but with weights��

li√

uHi,nrbd,nRb(n, n)

{ }J

i=1

. Then, for each beam, according to (30)


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and (33), R1/2b 1

( )ncan be written as the linear combination

of ui{ }J

i=1, with the weightsuHi rbd��

li√ −

��li

√uHi,nrbd,n

Rb(n, n)

( )J

i=1

R1/2b 1

( )n=∑Ji=1

uHi rbd��li

√ ui −∑Ji=1

��li

√uHi,nrbd,n

Rb(n, n)ui

=∑Ji=1

uHi rbd��li

√ −��li

√uHi,nrbd,n

Rb n, n( )

( )ui (34)

and

1HRb1( )

n= R1/2

b 1( )

n

( )HR1/2b 1

( )n

( )(35)

We select the beams with minimum (εHRbε)n as the currentoptimal beam. Before the selection of the next optimumbeam, the affect of the current optimal beam should beremoved through updating the cross-correlation betweennoise subspace and expected signal.

4.4 Updating cross-correlation in the recursiveprocess

If there are already k beams numbered lk have been selected asthe optimal beam in the former k steps, that is, the optimalbeam in step i is li, where i = 1, 2, …, k. Assumed that I isJ × J identity matrix, and can be represented as

I = e1, e2, . . . , eJ[ ]

(36)

Thus, after the former k steps, the beam selection matrix Vk is

V k = el1 , el2 , . . . , elk

[ ](37)

The corresponding RD covariance matrix is Rzk is

Rzk = VHk RbV k (38)

The cross-correlation between the reduced processorzk = VH

k b and the beamformed signal d is given by

rzdk = VHk rbd(k−1) (39)

where rbd(k−1) is the cross-correlation between b and residual ofd after cancelling by 2D angular-Doppler beams numbered l1,l2,…, l(k − 1) which have been acquired in the former (k− 1)thstep. The optimal value of the wLk is provided by the Wienersolution, can be calculated by

wLk = R−1zk rzdk (40)

If all the selected beams are utilised to cancel the interference inthe main beam in the (k + 1)th step, then the output aftercancelling by these beams will be

d̃ = d − wHLk bl1 , bl2 , . . . , blk

[ ]T(41)

where l1, l2, …, lm represent the number of the beams alreadyselected as the optimal beams in the former steps, bli is a thelthi component of b and b = [b1, b2, …, bJ]

T. When one more


beam is selected to cancel the interference component, theoutput after cancelling will decrease, so d̃ is monotonicallydecreasing. According to the approach described above, wecan find the optimal beams which can make d̃ achieve thesteepest descent.In the next step, another optimal beam will be selected, so

the cross-correlation between the noise subspace andexpected signal in (k + 1)th step should be updated as follows

rbd̃ = E b d −vHLk bl1 , bl2 , . . . , blk

[ ]T( )H[ ]

= E bd∗[ ]− E b b∗l1 , b

∗l2, . . . , b∗lk

[ ][ ]vLk

= rbd − Rb :, l1, l2, . . . , lk[ ]( )

R−1zk rzdk

( )(42)

where rbd is defined in (12), Rb(:,[l1, l2, ..., lk]) represents thelth1 , l

th2 ,…, lthm columns of Rb. The selected numbered l1, l2,…,

lm optimal beams have been used to cancel the interferencecomponents, it means that the corresponding interferencecomponents in the main channel have been cancelled. Thus,rbd̃ will be used to updating as the new cross-correlationbetween b and residual of d in the next (k + 1)th step,

rbdk = rbd̃ (43)

So, the rbd(k− 1) in (39) can be calculated in recursive style

rbd(k−1) = rbd − Rb :, l1, l2, . . . , lk−1

[ ]( )× R−1

z(k−1)rzd(k−1)

( )(44)

When the subscript (k− 1) is added for rbd, rzd and Rz to denoteit is the (k− 1)th step, using (42), the rbd(k− 1) can becalculated in recursive style. For the first step, where k = 0

rbd0 = rbd (45)

Then, calculate the R1/2b 1

( )nfor each beam channel

R1/2b 1

( )n=∑Ji=1

uHi rbdk��li

√ ui −∑Ji=1

��li

√uHi,nrbdk,n

Rb(n, n)ui

=∑Ji=1

uHi rbdk��li

√ −��li

√uHi,nrbdk,n

Rb(n, n)

( )ui

(46)

Using rbdk,n to replace rbd, n in (49) to denote the index ofcurrent step, then, calculate (R1/2

b 1)n for each beam channel.It should be noted that, the rbd in (34) has been replaced byrbdk. The beam with minimal (εHRbε)n will be selected incurrent step. Choosing auxiliary channels of JDL algorithmis relative fixed, such as JDL 3 × 3, the eight 2D channelsaround the interested channel are selected as the auxiliarychannels. The optimal auxiliary channels for each interestedchannel corresponding to certain Doppler frequency areachieved in the proposed approach through iteration andupdating to minimise the εHRbε.

4.5 Possible implementation scheme

In general, the RD STAP algorithms all use the relative fixedauxiliary channels selection approach. The proposedalgorithm can provide the optimal selection approach of


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auxiliary channels to cancel the interference components in themain beam to maximise the output SINR. For example, eightauxiliary channels at least will be chosen for JDL 3 × 3, 12for JDL 3 × 4, or 15 for JDL 3 × 5. However, the proposedalgorithm can use less auxiliary channels to achievecomparative performance. Owing to less auxiliary channelsare needed, the computation complexity of estimation andinversion of localised clutter covariance matrix RxT
which isrequired to be real time is reduced, and the requirement ofthe sample support used to estimate RxT

is also reducedbased on the RMB rule. For example, if only three auxiliarychannels will be used to cancel the interference componentsin the proposed algorithm, then, the computation complexityof inversion 4 × 4 localised clutter covariance matrix ratherthan the 9 × 9 for JDL 3 × 3.The superiority is acquired at the cost extra computation

burden resulting from the inversion of Rzk andeigendecomposition of Rb. If iteration and updating processare needed for each detecting, the algorithm will be highcomputation complexity. Fortunately, real time calculationis not needed. As the optimal auxiliary beam channels usedto cancel the interference components in the mainbeam isrelatively invariable for each normalised Doppler frequencyof target. Thus, the proposed algorithm can be used inhomogenous clutter or non-homogeneous clutter which isassumed to be slowly changing.The entire operating process can be divided into two parallel

parts, training process and detecting process. For example, eachCPI is considered as one period. We can obtain the optimalauxiliary channels which will be used in the next CPIthrough the time-consuming iteration in the training process.In the detecting process, the optimal auxiliary channelsselecting approach has been obtained in the training processfrom the last CPI. Under the assumption that the clutter ishomogeneous or slowly changing non-homogeneous, the

Fig. 5 Total operating process


statistical characteristics of clutter can be expected to be thesame between previous CPI and current CPI. The totaloperating process is illuminated in Fig. 5.It is worth to be noted that, only the auxiliary channels can

be pre-calculated and stored, rather than the inversion oflocalised clutter covariance matrix RxT

which need to becalculated in real-time to acquire the weight coefficients. Ifthe clutter is relative homogenous, the obtained optimalauxiliary channels are not needed to be changed and can besuitable all the time.

5 Simulations

The principle parameters in the simulations are assumedto be as follows: the radar platform travels at velocity v =125 m/s; there are NT = 4 transmitting elements with uniformspace dT = 4(λ/2) = 2λ, where λ is the wavelength; and NR = 4receiving elements with uniform space dR = λ/2;carrier frequency f0 = 1.2 GHz; pulse repletion frequencyfr = 2000 Hz; platform altitude H = 8000 m; clutter-to-noiseratio = 30 dB; M = 16. Assume the array is side looking. Insimulation experiments, the look angle θ0 is set to be zero,that is, then intended beam pointing at broadside angles of 0°.

Fig. 6 Output SINR of RD processors as function of Wiener filterorder for testing Doppler channel 2

Fig. 7 SINR loss of MSMB with different Wiener filter order


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Fig. 8 Comparison of MSMB, JDL with the same DoFs

a Number of auxiliary beams = 8b Number of auxiliary beams = 14

The performance of the proposed MSMB, cross-spectralsubspace and eigensubspace may be evaluated by plottingthe output SINR as a function of Wiener filter order. Forthe testing Doppler channel 3, that is, the normalisedDoppler frequency is 0.125, the full-dimension Wiener filteryields an output SINR of 8.81 dB. The SINR for thecross-spectral subspace, eigensubspace and the proposedMSMB are compared with the full-dimension SINR inFig. 6 as the dimension increases from 1 to 255.It can be seen from Fig. 6 that the eigensubspace processor

and cross-spectral subspace converge to the optimal SINR inan abrupt manner compared with the MSMB method. Bycontrast the MSMB approach converges to the optimal SINRfast. As depicted in Fig. 6, when the order of Wiener filter is3, that is, only three angular-Doppler beam channels areselected as auxiliary channel, the output SINR and SINR lossof MSMB approach are 7.912 and −0.898 dB, respectively.When eight angular-Doppler beams are selected, the SINRloss is only −0.338 dB. This cannot be achieved by CSM orJDL with the same DoFs. Only when the DoFs of CSM andeigensubspace reach 25 and 31, respectively, they can achievethe same performance as MSMB. Fewer DoFs, means fewerindependent identically distributed (i.i.d.) samples are neededto estimate the covariance matrix.The performances of MSMB for all Doppler channels with

different Wiener filter order are compared in Fig. 7.From Fig. 7, we can see that, when the number of auxiliary

beams increases, there is a slight performance improvement.The most important advantage of the proposed algorithm isthat, it can use less auxiliary channels to achievecomparative performance. Just as illustrated in Fig. 7, thereis little performance improvement for eight auxiliary beamscompared with three auxiliary beams. This indicates thatonly a few channels are needed to cancel the interferencecomponents in the main beam to achieve excellentperformance, the proposed algorithm can use less auxiliarychannels to achieve comparative performance. In practicalapplication, we can only use three auxiliary channels toachieve the satisfied performance.Although 8 or 14 auxiliary channels are not necessary in

the proposed algorithm, the number of auxiliary channels


will be set as 8 and 14 for illustrating that the proposedalgorithm can provide the upper-bound output SINR forRD STAP algorithm with fixed number of auxiliarychannels. The performances of MSMB, joint domainlocalised and space-time multiple beams are compared inFig. 8. NT = 4, NR = 16, M = 16, the number of auxiliarybeam channels is set to be same. If 8 angular-Dopplerbeams are selected as the auxiliary beam channels, theLPR is 3 × 3 for JDL. If 14 angular-Doppler beams areselected as the auxiliary beam channels, the LPR is 3 × 5or 5 × 3 for JDL.It can be seen from Fig. 8 that, the MSMB can provide an

upper-bound to the SINR performance achievable by JDLwith the same number of auxiliary beam channels.The superiority of proposed algorithm comes from the

characteristics of clutter which is high correlated. Theinterference components in the mainbeam are highcorrelated with some auxiliary beam. So, if the appropriatebeams are selected as the auxiliary beams, most theinterference components will be cancelled.

6 Conclusion

A MSMB RD STAP methodology for MIMO radar isintroduced in this paper. The optimal performance will beachieved utilising the proposed algorithm when the numberof total auxiliary channel is fixed. When three 2D beamchannels are chosen as the auxiliary channels to cancel theinterference components in the testing mainbeam, the SINRloss will be <2 dB.Due to the fact that the clutter angle-Doppler spectrum is

non-white, the clutter components on all angle-Dopplerbins are high correlated. By utilising the appropriateangular-Doppler 2D beam channel, major interferencecomponents in the mainbeam can be cancelled well.This algorithm is also suitable for the conventional array

radar. For the MIMO radar which utilising sparse array toobtain the longest continuous virtual array, that is, dT =NRdR,this is similar to phased array radar which consisting of NRNT

elements in receiving array from the viewpoint of equivalence


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phase centres. Thus, the phased array radar can be viewed as aspecial case which only transmitting one signal.
7 Acknowledgment

This work was supported by the National Science AssociatedFoundation (grant no. 11076006) and National ScienceFoundation of China (grant no. 61032010).

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