Two layers beamforming robust against direction-of-arrival mismatch

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Published in IET Signal ProcessingReceived on 21st January 2013Revised on 19th May 2013Accepted on 3rd June 2013doi: 10.1049/iet-spr.2013.0031

T Signal Process., 2014, Vol. 8, Iss. 1, pp. 49–58oi: 10.1049/iet-spr.2013.0031

ISSN 1751-9675

Two layers beamforming robust againstdirection-of-arrival mismatchMostafa Rahmani, Mohammad Hasan Bastani, Sarah Shahraini

Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran

E-mail: [email protected]

Abstract: The presence of the desired signal (DS) in the training snapshots makes the adaptive beamformer sensitive to anysteering vector mismatch and dramatically reduces the convergence rate. The objective of the present study is to propose anew adaptive beamformer which is robust against direction-of-arrival (DOA) mismatch and its convergence rate is notsensitive to the presence of the DS. This method is applicable to the arrays with specific structure such as the linear array.Our approach is based on the DS elimination from the training snapshots and the sub-array beamforming technique. Toaccomplish this goal, a blocking matrix which converts the primary data to the DS-free data is synthesized. The Synthesisprocess is based on the desired degree of freedom and the uncertainty of DOA of the DS. Using the signal-free data, thebeamforming vector is calculated through the presented algorithm. Owing to elimination of the DS from the trainingsnapshots and performing the adaptive operations in the sub-array level, our algorithm has high convergence rate andexcellent performance even in cases with the small sample size. Simulations show that the proposed beamformer can achievea much better performance in terms of output SINR compared to the existing ones.

1 Introduction

Beamforming has been used in many applications such asradar, sonar, seismology, speech processing and wirelesscommunication [1, 2]. In several applications such asradar and sonar, the DS-free data are available. In thesecases, adaptive beamforming is robust and has a highconvergence rate. Although in many applications such aspassive locating, wireless communication, passive sonarand speech processing, the DS-free data are unavailable.When the DS is present in the training data, twoproblems arise:

1. The presence of the DS in the training data dramaticallyreduces the convergence rates of the adaptive beamformingalgorithms as compared with the signal-free training datacase [3]. This may cause a substantial degradation of thebeamformer performance in situations of small trainingsample size.2. The presence of the desired signal (DS) in the trainingsnapshots makes the adaptive beamformer sensitive to anymismatch between the actual and presumed steering vectorsof the DS. Steering vector of the DS can be impreciseowing to the effect of various factors such as DOA errors,local scattering, near-far spatial signature mismatch,spatially distributed source, imperfect calibrated arrays anddistorted antenna shape [4–7]. In the presence of suchmismatches, the beamformer treats the DS as interferenceand the output signal-to-interference-plus-noise ratio (SINR)is decreased dramatically [3, 8].

There are several methods for robust beamforming problem.In [9, 10], linear constraints have been imposed whenminimising the output variance. The linear constraints canprevent the DS’s suppression. These beamformers are calledlinearly constrained minimum variance (LCMV)beamformers. In [11], to reduce the sensitivity of the DS,the noise variance has been increased artificially by thediagonal loading technique. The main shortcomingassociated with this approach is that it is not clear how toobtain the optimum value of the diagonal loading level. In[3], the eigenspace-based approach has been proposed. Inthis approach, the presumed steering vector is projected ontothe signal-plus-interference subspace to reduce the steeringvector mismatch. This method is essentially restricted in itsperformance at low signal-to-noise ratio (SNR) and whenthe dimension of the signal-plus-interference is large. In [12]Vorobyov et al. have used a non-convex constraint, whichforces the magnitude responses of the steering vectors in asphere set to exceed unity. This non-convex problem hasbeen converted to a second-order cone programming(SOCP) problem. In addition, it has been proved that thisbeamformer is equivalent to a diagonal loading beamformer,where its diagonal level is calculated inherently [13]. In [14,15], the real steering vector of the DS is estimated and theestimated steering vector is used instead of the presumedone. Therefore these methods remove the steering vectormismatch and are robust against any type of the steeringvector mismatch.In the present study, we propose a new beamformer

algorithm which is robust against the DOA mismatch andyields an excellent performance even in the small sample

49& The Institution of Engineering and Technology 2014

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size cases. In this method, the desired degree of freedom andthe range of the angular location of DS are primarily defined.Based on the desired degree of freedom, the array is dividedinto sub-arrays. Subsequently, the sub-arrays data arecombined with a weighting vector which is orthogonal tothe subspace of possible steering vectors of the DS. As aresult, the new data are free of the DS. It is shown that thisDS blocking process is equivalent to a simple lineartransformation. Sub-array covariance matrix is estimated bythis new data and then the sub-array beamforming vector iscomputed. At the end, the final beamforming vector iscomputed based on the sub-array beamforming vector. Theproposed method decreases the degree of freedom andproposes a high convergence rate. Based on the proposedsub-array technique, this algorithm is applicable to the arraywhere its structure can be synthesised by scrolling thesub-array.The rest of this paper is organised as follows: In Section
2, the background of adaptive beamforming and a numberof previous studies on robust beamforming are reviewed.In Section 3, we develop the theory and the algorithm forour new robust beamformer. Numerical examples arepresented in Section 4. Section 5 contains concludingremarks.

2 Background

Consider a uniform linear array of N omnidirectional sensorswith inter-element spacing d. The DS and the interferences

arrive from angles θs and{u ji

}Pi=1

, respectively (all signalsare narrowband). The baseband array output can beexpressed as

x(t) = s(t)v us( )+∑P

i=1

zi(t)v u ji

( )+ n(t) (1)

where s(t), v(θs), zi(t), v(θji) and n(t) are the DS, the basebandarray response to the DS, the ith interference, the basebandarray respond to ith interference and the noise vector,respectively. The array response is called the ‘steeringvector’ and can be expressed as

v(u) = 1 ej(2p/l)d cos (u) · · · ej(2p/l)(N−1)d cos (u)( )T(2)

in which λ and θ are the operating wavelength and the arrivalangle, respectively. The output of the beamformer can beexpressed as

y(t) = wHx(t) (3)

where w is the complex weighting vector (beamformingvector), and (·)H stands for the Hermitian transpose. Theoptimal weighting vector can be found from the maximumof the SINR

SINR = s2s wHv us

( )∣∣ ∣∣2wHRi+nw

(4)


where

Ri+n=E∑Pi=1

zi(t)v uji

( )+n(t)

( ) ∑Pi=1

zi(t)v uji

( )+n(t)

( )H{ }

(5)

is the interference-plus-noise covariance matrix and s2s is the

DS power. To find this optimal weighting vector, thefollowing problem can be considered [16]

minw

wHRi+nw subject to wHv us( )=1 (6)

From (6), the following well-known solution can be found forthe optimal weight vector

wopt=aR−1n+iv us

( )(7)

where a=(v us( )H

R−1n+iv us

( ))−1is the normalisation constant.

The solution (7) is commonly referred to as the minimumvariance distortionless response (MVDR) beamformer [16].However, in practice, the exact interference-plus-noisecovariance matrix is not available and the samplecovariance matrix

R_= 1

M

∑Mm=1

x(m)xH(m) (8)

should be used instead [17] (M is the number of trainingsnapshots). In this case, (6) should be rewritten as

minw

wHR_

w subject to wHv us( )=1 (9)

The solution to this problem is commonly known as samplematrix inversion (SMI) algorithm, whose weight vector isgiven by [17]

wSMI=R_−1v us

( )(10)

When the DS is present in the training snapshots, theconvergence rate of the SMI algorithm is considerablyreduced. This may cause a substantial degradation of theperformance of the adaptive beamforming techniques insituations of small training sample size. In addition, theprecise direction of the DS is unavailable, thus the realdirection should be replaced with the presumed one.Therefore the beamformer becomes

wSMI=R_−1v up

( )(11)

where θp is the presumed direction of the DS. This can beconsidered as a solution to the minimisation problem

minw

wHR_

w subject to wHv up

( )=1 (12)

In this circumstance, if the DS is present in the trainingsnapshots, the expected value of the objective function canbe indicated as

wHRw=wHRi+nw+s2s vH us

( )w

∣∣ ∣∣2 (13)

IET Signal Process., 2014, Vol. 8, Iss. 1, pp. 49–58doi: 10.1049/iet-spr.2013.0031

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where R = E{x(t)xH(t)}. Therefore since wHv(θs) = 1 is nolonger valid due to the mismatch, the DS magnituderesponse might be attenuated as a part of the objectivefunction. This suppression leads to severe degradation inSINR because the DS is treated as interference in this case.This phenomenon is called ‘signal cancellation’. A smallmismatch can lead to severe degradation in the SINR.Several robust approaches have been developed to improve
the performance of the SMI algorithm in the aforementionedcases [10–15, 18–25]. Table 1 presents a summary ofimportant approaches. In the first row (LSMI algorithm), λis the diagonal loading factor and I is the identity matrix. Inthe second row, (LCMV beamformer), C is the constrainsmatrix and f is the vector of appropriate constrains value. Inthe third row (eigenspace-based beamformer), the matrix Eis the dominant eigenvectors of the covariance matrix. Inthe fourth row (worst-case-based beamformer), S is asphere, which contains all the possible DS steering vectors,e models the difference between the real and presumedsteering vector and ε is the upper bound of the norm ofvector e. In the fifth row (robust beamforming based onsteering vector estimation), the real steering vector of theDS is estimated and vector a is the optimisation variablevector. In this problem, the first constrain guarantees thatthe estimated vector is a steering vector and the secondconstrain guarantees that the estimated steering vector doesnot belong to an interference. The matrix C is calculated by

Table 1 Different robust adaptive beamforming methods

Robust beamformers Algorithm’s idea

1 – loaded SMI (LSMI)algorithm [11]

increasing the noise varianceartificially in order to reduce thesensitivity to the DS

minw

wH

subject

wHv up

(

2 – linear constrainminimum variance(LCMV) beamformer[9]

adding linear constraints to theSMI beamformer in order toprevent the DS suppression

minw

wH

subject

CHw =

3 – eigenspace-basedbeamformer [3]

projection of the presumed DSsteering vector onto thesignal-plus-interference subspace

weig = R_

4 – worst-case-basedbeamformer [12]

constraining the magnituderesponse of beamformer toexceed unity in a set, containingall the possible steering vectors

minw

wH

subject

|wHa| ≥

where S

Containvector abetweensteering

5 – beamformingbased on the steeringvector estimation[14, 15]

the DS steering vector estimationin order to remove the steeringvector mismatch

mina_

a_

subject

a_

∥∥∥ ∥∥∥2= N


the following equation

C =∫Q

v(u)vH(u)du (14)

where Θ is the angular sector, in which the DS is located andQ is the complement of the sector Θ [15]. In the secondconstraint, the parameter t0 is determined in a way thatonly the steering vectors of the sector Θ can satisfy thesecond constrain.

3 Proposed robust beamformer

Our algorithm is based on the DS elimination from thetraining snapshots. Precise DOA of the DS is not known;however, a spatial region can be assumed for the possibleDOA. This region represents the range of the angularlocation of the DS. For instance, an imprecise pre-estimateof DOA of the DS can define this spatial region or asanother example in the radar applications; usually the 3 dBbeam-width of the transmitter’s beam-pattern is consideredas the expected spatial region of the DS. We call this spatialregion as ‘uncertainty region’. Therefore to assure the DSelimination from the training snapshots, any signal comingfrom the uncertainty region should be eliminated. Usingthis DS-free data, the weighting vector is calculated.

Method Weaknesses

R_

+lI( )

w

to)= 1

the optimum value of thediagonal loading factor cannot bedetermined

R_

w

to

f

poor performance in high SNRand small sample size cases

−1 EEHv up

( )( )poor performance at lowsignal-to-noise ratio (SNR) andwhen the dimension of thesignal-plus-interference is high

R_

w

to

1 for all a [ S

= v up

( )+ e| e‖ ‖ ≤ 1

{ }.

any possible DS steeringnd e models the differencethe real and presumedvector

poor performance facing a largeuncertainty on DOA, noclose-form solution

H R_ −1 a

_

to

, a_ HC a

_ ≤ t0

no close-form solution, poorperformance when theinterferences are close to the DS


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We primarily explain the proposed algorithm in few steps.
Consequently, a simple close-form formula for the proposedalgorithm is derived.

3.1 Theory of the proposed algorithm

For better understanding, the theory of the proposedalgorithm is explained in few steps.

3.1.1 First step (defining the desired degree offreedom): In the SMI beamformer, the degree of freedomis equal to the array length. In many arrays, this degree offreedom is more than needed. For example, in a 30-elementlinear array, we do not expect the beamformer to suppress29 interferences. Hence, we define L as the desired degreeof freedom. For instance, in the 30-element linear array, wecan choose 10 as the desired degree of freedom.An N-element linear array can be synthesised by scrolling an

L-element linear array on an (N− L + 1) element linear array.Accordingly, we call the L-element array as sub-array, the(N− L + 1) element array as synthesiser array and theN-element array as the main array. Evidently, the elementspacing in sub-arrays and synthesiser array is the same as themain array. Fig. 1 illustrates the sub-arrays, the synthesiserarray and the main array. For example, a 30-element lineararray has 21, 10-element sub-arrays. Hence, in this case, thesub-array is a 10-element linear array and the synthesiserarray is a 21-element linear array. In fact, the synthesiserarray and the sub-array can have any structure and an infinitenumber of array structures can be synthesised; however, inthis research, only the linear structure is studied.

3.1.2 Second step (DS elimination from the trainingsnapshots): Matrix Di is defined as the matrix of trainingsnapshots of the ith sub-array. Thus Di

{ }N−L+1

i=1are L ×M

matrices, in which M is the number of snapshots. Eachsub-array data matrix can be decomposed as

Di = Dsi +

∑Pm=1

D jmi + Dn

i (15)

where Dsi is the signal component, D jm

i is the interferencecomponent due to the mth interference and Dn

i is theadditive noise. Each sub-array is the shifted one from theprevious sub-array. Thus, the signal component can beexpressed as

Dsk = Ds

1 ej 2p/l( )(k−1)d cos (u)( )

(16)

The same formula can also be written for the interferencescomponent. According to (16), for every element of the

Fig. 1 Sub-arrays, synthesiser array and main array


signal component of the sub-array data matrices, we have

Ds1(i, j), D

s2(i, j), . . . , D

sN−L+1(i, j)

[ ]T= vz us

( )Ds

1(i, j)( ) (17)

where vz(θs) is the DS steering vector of the synthesiser arrayand Ds

k(i, j) represents the element at ith row and jth columnof the matrix Ds

k . Next, we define vector wo to combine thesub-array data matrices in the following form

D =∑N−L+1

k=1

w∗o(k)Dk (18)

where w∗o(k) is the complex conjugate of wo(k) and wo(k) is the

kth element of vector wo. According to (17), we have

D = wHo vz us

( )( )Ds

1

+∑pm=1

wHo vz u jm

( )( )D jm

1 +∑N−L+1

k=1

w∗o(k)D

ni (19)

We call D as new data matrix.We intend to generate the DS-free data. Thus, if wo has the

following properties:

wHo vz(u) = 0, for u [ Uncertainty region

wHo vz(u) = 1, for u [ Complement of the uncertainty region

(20)

then the new data matrix will be free of the DS and theinterference components remain undistorted. In reality, novector has exactly the above properties. Desirable wo isorthogonal to the subspace of the steering vectors of theuncertainty region and has relatively uniform response in thecomplement of the uncertainty region. In this investigation,we have generated wo through the following algorithm:

† Constructing wc which has nearly a uniform pattern overthe entire space;† Building the positive-definite matrix

S =∫u2u1

vz(u) vHz (u)du (21)

where [θ1, θ2] is the uncertainty region.† Constructing the column orthogonal matrix

U = u1, u2, . . . , uK[ ]

(22)

where uk{ }K

k=1 are K principle eigenvector of S.† Projecting wc onto the subspace, which is orthogonal to U

wo = I − UUH( )wc (23)

Hence, the generated wo is approximately orthogonal to anysteering vector in the uncertainty region because wo is
orthogonal to the subspace of the steering vectors of theuncertainty region. In fact, the vector wo is similar to afinite impulse response (FIR) filter and its rejection region isthe uncertainty region. According to (20), we need a sharptransition from the rejection region to the pass region.Similar to the FIR filter, for a sharp transition, wo must have

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adequate length. Consequently, the proposed method issuitable for the array which has its number of sensorssufficiently larger than the desired degree of freedom. Forexample, consider a linear array with 30 omnidirectionalsensors spaced half a wavelength apart. We define theuncertainty region as [88°, 92°], define L equal to 10 andconstruct U with the four dominant eigenvectors. In thisexample, wc is a vector which has all of its elements equal tozero except the 11th element which is equal to one. Fig. 2represents the spatial response of the generated wo. One canobserve that it has a relatively sharp transition. Evidently, ifthe number of array sensors (N) is decreased or the desireddegree of freedom (L) is increased, the length of vector wo isreduced, and consequently, this leads to a slower transition.In the proposed method, the length of the sub-array is fixed
and is equal to L, but the length of the synthesiser array can bechanged. For instance, if we make the sub-arrays by shiftingtwo elements, the length of the synthesiser array will behalved and clearly the element spacing of the synthesiserarray will be doubled. The increase in the element spacingof the synthesiser array can cause the spatial ambiguityproblem. Therefore in addition to the uncertainty region,signals from other directions may be eliminated. Fig. 3

Fig. 3 Spatial response of the vector wo (element spacing of thesynthesiser array is equal to the wavelength)

Fig. 2 Spatial response of the vector wo


presents the spatial response of the generated wo for theabove example; however, its sub-arrays are made byshifting two elements. It is clear that in addition to theuncertainty region, signal from other directions may beeliminated and if an interfering source is eliminated fromthe training snapshots, the adaptive beamformer cannotsuppress it. In this paper, the sub-arrays are made only byshifting one element.Next, we prove that the aforementioned DS elimination

process is equivalent to a linear transformation. If T isdefined as the matrix of the training snapshots of the mainarray, then we have

Dk =eTk

..

.

eTk+L−1

⎡⎢⎣

⎤⎥⎦T (24)

where eTk is the transpose of ek, that is, a N × 1 vector, whichhas its entire elements zero; except the kth element which isequal to one. Thus, we can expand (18) as

D =∑N−L+1

k=1

wo(k)

eTk

..

.

eTk+L−1

⎡⎢⎣

⎤⎥⎦T (25)

Thus, if we define

G =∑N−L+1

k=1

wo(k)

eTk

..

.

eTk+L−1

⎡⎢⎣

⎤⎥⎦ (26)

Then, the following equation can be written as

D = GT (27)

Therefore the DS elimination process is equivalent tomultiplying the matrix G to the input data. We call G asblocking matrix. Other kinds of blocking matrix havealready been used [26, 27]. The main shortcoming of theseapproaches is that they cannot guarantee the elimination ofall signals coming from the uncertainty region. On the otherhand, we need a blocking matrix that reduces the degree offreedom to a specific predetermined value. From the pointof view of both, the proposed method is a unique approachof the blocking matrix design.

3.1.3 Third step (sub-array beamforming): After thecombination of the sub-arrays data as described in step 2,we have the new data which is free of the DS. Therefore,the sub-array interference-plus-noise covariance matrix canbe estimated based on the new data matrix

R_

s =1

MDDH (28)

However, there is a mismatch between the estimated and idealsub-array interference-plus-noise covariance matrix. Thespatial distribution of the additive noise is changed duringthe DS elimination in (27). It is evident that the projectionof the noise component in the subspace of steering vectorsof the uncertainty region is eliminated. We assume theadditive noise as a zero-mean spatially and temporally


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white process. According to (27), the covariance matrix of thenoise component of the new data is expressed as
E DnDHn

{ } = E GTnTHnG

H{ } = s2GGH (29)

where Dn, Tn and σ2 are the noise component of D, noisecomponent of T and noise variance, respectively. Therefore,(29) is not the ideal noise covariance matrix. We cancompensate this mismatch by adding matrix C to theestimated covariance matrix, where C is given by

C = s2 I − GGH( )(30)

In addition, in practical applications, owing to the finitesample size, there is a mismatch between the ideal andestimated covariance matrix. Thus similar to the procedurein [25], we can write the sub-array beamforming problem as

minws

maxD‖ ‖≤b

wHs R

_

s + C + D( )

ws

subject towHs vs up

( )= 1

(31)

where the matrix Δ models the covariance matrix mismatch,D‖ ‖ is the Frobenius norm of Δ, β is the upper bound ofFrobenius norm of Δ, ws is the sub-array beamformingvector, and vs(θp) is the sub-array steering vector of thepresumed direction of the DS (we consider θp as the middleof the uncertainty region). Similar to [25], solution to (31)can be expressed in a closed-form and is given by

ws = a R_

s + C + bI( )−1

vs up

( )(32)

where α is the normalisation constant. Therefore, thesub-array beamformer is a LSMI beamformer.Next, using ws, we perform beamforming over all the

sub-arrays. Beamforming in the sub-array level can beperformed by matrix B, which is defined in the followingform

B =

wHs 0 0 . . . 0

0 wHs 0 . . . 0

..

. . .. ..

.

0 0 . . . wHs

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦

(N−L+1)×N

(33)

Each row performs beamforming over the appropriatesub-array.

3.1.4 Fourth step (second-stage beamforming):After performing the sub-array beamforming, the outputsignal is

r(t) = Bx(t) = s(t)Bv us( )+∑P

i=1

zi(t)Bv u ji

( )+ Bn(t) (34)

As a result of performing the adaptive operations in theprevious step, we can assume that the interfering signalshave been suppressed. Thus, we can rewrite (34) as

r(t) ≃ s(t)Bv us( )+ Bn(t) (35)


Owing to the constrain in (31), we can rewrite (35) as

r(t) = s(t)vz us( )+ Bn(t) (36)

Next, we define a weight vector wHz for the optimum

estimation of the DS in the form of wHz r(t). The definition is

as the following problem

minwz

wHz E Bn(t)nH(t)BH{ }

wz

subject to wHz vz us

( ) = 1(37)

Problem (37) can be rewritten as below

minwz

wHz BB

Hwz subject to wHz vz us

( ) = 1 (38)

Therefore the optimum vector is given by

wz =BBH( )−1

vz us( )

vz us( )H

BBH( )−1

vz us( ) (39)

After the sub-array beamforming, wz performs the secondstage of beamforming. However, similar to the SMIbeamformer, we should replace the real DOA of the DSwith the presumed one. Thus, we have

wz =BBH( )−1

vz up

( )vz up

( )HBBH( )−1

vz up

( ) (40)

3.2 Close-form formula of the proposed algorithm

As explained in the previous part, we primarily performbeamforming in the sub-array level; afterwards, thesub-arrays signals are combined with the weight vector wz.Based on the procedure explained in steps 3 and 4, themain weighting vector (beamforming vector of the mainarray) can be written as

w = BH BBH( )−1vz up

( )(41)

whose normalisation constant has been omitted.Fig. 4 shows the block diagram of our algorithm. A short

description of the proposed algorithm is as below:

† Applying the blocking matrix G to the input data in orderto make data free from the DS† Calculating the sub-array beamforming vector and buildingthe sub-array beamforming matrix B† Beamforming vector calculation through (41)

In summary, in the proposed method we generate theDS-free data for the sub-array beamformer. The sub-arraybeamforming vector is estimated using the DS-free data.Consequently, the sub-array beamformer has a highconvergence rate. The beamforming vector of the proposedmethod (41) is exclusively calculated using the sub-arraybeamforming vector because the matrix B is constructed by


Fig. 4 Block diagram of the proposed algorithm

Fig. 5 Output SINR against training sample size M (SNR = 20dB); first example

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the sub-array beamforming vector. Accordingly, the highconvergence rate of the sub-array beamformer directly leadsto the high convergence rate of the proposed algorithm.The proposed method has a close-form formula, which is a

highly desirable feature for the practical implementation. Themain parts of the calculation of the beamforming vector (41)are the calculation of the sub-array beamforming vector (32)and the calculation of (BBH)− 1. Our algorithm is suitable forthe array which has its number of elements sufficiently largerthan the desired degree of freedom. Hence, the overallcomplexity of our beamformer is O((N− L + 1)3). Thecomplexity of the SMI algorithm is O(N3). Although ouralgorithm is less complex, but the SMI algorithm has acomputational advantage in the online mode, where theRLS algorithm can be employed to update the SMIbeamformer weights with the computational complexityO(N2) per updating step.

4 Simulations

In our simulations, a uniform linear array with N = 30omnidirectional sensors spaced half a wavelength apart isassumed. The additive noise is modelled as a complexGaussian zero-mean spatially and temporally white processwhich has identical variances in each array sensor. Weassume two interference sources with DOA 80° and 130°,respectively and their interference-to-noise ratio is equal to30 dB. The DS is always present in the sample data and thepresumed DOA of the DS is equal to 90°. In the proposedalgorithm, the interval [88°, 92°] is considered as theuncertainty region and the desired degree of freedom isequal to 10.The most steering mismatch occurs at the boundary of the

uncertainty region; thus the ε parameter in theworst-case-based beamformer can be defined as

1 = max v 90◦( )− v 92◦

( )∥∥ ∥∥, v 90◦( )− v 88◦

( )∥∥ ∥∥( ) = 8.04

In [12], the non-convex problem of the worst-case-basedbeamformer has been reformulated in the following convexform

minw

wH R_

w subject to wHa ≥ 1 w‖ ‖ + 1

Im wHa{ } = 0


where Im{wHa} is the imaginary part of wHa. Therefore, forthe feasibility of the first constrain, ε should satisfy thefollowing equation

1 w‖ ‖ + 1 ≤ max wHa( )

In our simulations, the norm of vector a is equal to��30

√;

therefore the above equation can be rewritten as

��30

√− 1

( )w‖ ‖ ≥ 1

Hence, the maximum feasible value of ε is equal to��30

√. In

our simulations, we define ε to be equal to 5.2, a little lowerthan the maximum feasible value. In the eigenspace-basedbeamformer, the signal-plus-interference space is alwaysbuilt using the three first dominant eigenvectors. Thediagonal loading factor λ = 10σ2 is taken in the LSMIbeamformer and in our method we take β = 4σ2. The CVXMATLAB toolbox has been employed to solve theoptimisation problem in the worst-case-based algorithm andthe beamformer of [14].

Example 1: Exactly known signal steering vector. In thisexample, we assume that the exact DOA of the DS isknown. Therefore, the real and presumed DOA of the DSis equal to 90°. The main purpose of this simulation is acomparison of the convergence rate of the proposedalgorithm with the other beamformer algorithms. Fig. 5displays the mean output SINR against the number ofsnapshots for the fixed single-sensor SNR = 20 dB. It isobserved that the proposed algorithm has the highestconvergence rate. This excellent performance is the resultof the DS elimination from training snapshots andperforming the adaptive operations in the sub-array level.In the proposed method, we generate the DS-free data forthe sub-array beamformer. The sub-array beamformingvector is estimated using the DS-free data. Consequently,the sub-array beamformer has a high convergence rate. Thebeamforming vector of the proposed method (41) isexclusively calculated using the sub-array beamformingvector because the matrix B is constructed by the sub-arraybeamforming vector. Accordingly, the high convergence


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rate of the sub-array beamformer directly leads to the highconvergence rate of the proposed algorithm. Fig. 6 displaysthe mean output SINR against SNR for the fixed trainingsize M = 100. One can observe that unlike otheralgorithms, the performance of the proposed algorithm isnot sensitive to the increase in the SNR since in theproposed algorithm before the estimation of thebeamforming vector, the DS is omitted from the trainingsnapshots. Hence, the presence of the DS in the trainingdata cannot affect its performance.
Fig. 7 Output SINR against training sample size M (SNR = 30dB); second example

Example 2: Signal look direction mismatch. In this example,the robustness of the proposed algorithm against DOAmismatch is studied. We assume that both the presumedand actual signal spatial signatures are plane wavesimpinging from the DOAs 90° and 91°, respectively. Fig. 7demonstrates the performance of the methods tested againstthe number of training snapshots for the fixed SNR = 30dB. One can observe that the proposed algorithm has thehighest convergence rate. However, in this case theproposed algorithm does not reach the optimal SINR. Thisloss is due to the DOA mismatch, because the peak of thebeam-pattern is not in the direction of the DS. In thissimulation, the amount of the diagonal loading is notsufficient to prevent the signal cancellation phenomenon,thus the SINR of the LSMI beamformer is seriouslydegraded. The DS is always present in the training data;hence, the subspace of the three first dominant eigenvectorscontains the real steering vector of the DS. Consequently, inthe eigenspace-based algorithm, the steering vectormismatch is removed by the projection of the presumedsteering vector to this subspace. But in this simulation,the strong presence of the DS seriously degrades theperformance of the eigenspace-based and theworst-case-based beamformers.

The performance of these algorithms against the SNR forthe fixed training data size M = 100 is illustrated in Fig. 8.Because of the elimination of the DS from trainingsnapshots, the performance of the proposed algorithm is notsensitive to the presence of the DS. One can observe thatthe performance of the LSMI beamformer is degraded forthe high SNR cases because its value of the diagonalloading factor is not sufficient for the high SNR cases. In

Fig. 6 Output SINR against SNR (M = 100); first example


fact, the worst-case-based beamformer shows the benefit ofchoosing the optimum value of the diagonal loading factor[12].

Example 3: SINR against mismatch angle. In this example,the presumed DS arrival angle θp is equal to 90° and theactual arrival angle ranges from θs = 88° to θs = 92°. Fig. 9shows the performance of the methods tested againstdifferent mismatch angles for the fixed SNR = 20 dB and M= 100. It can be observed that the proposed algorithm hasexcellent performance; however, the only associatedshortcoming is the loss of DOA mismatch which occurssince the peak of the beam-pattern is not in the direction ofthe DS.

The beamformer of [14] exactly estimates the steeringvector of the DS but similar to the SMI beamformer, thepresence of the DS strongly reduces its convergence rate. Inthis simulation we have a relatively high SNR, so thepresence of the DS leads to a significant degradation ofperformance of the beamformer of [14].

Example 4: Quality of the blocking matrix. In this example,the effect of the length of the sub-array (the desired degree

Fig. 8 Output SINR against SNR (M = 100); second example


Fig. 9 Output SINR against mismatch angle (SNR = 20 dB, M =100); third example

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of freedom) is studied. To accomplish this goal, theperformance of the proposed algorithm is compared forthree different sub-array lengths. We assume that the exactDOA of the DS is known. Thus, the real and presumedDOA of the DS is equal to 90°. In this example, we addanother interference. The DOA and interference-to-noiseratio of the third interference are equal to 85° and 30 dB,respectively. Fig. 10 displays the mean output SINR againstthe number of snapshots for the fixed single-sensor SNR =10 dB. One can observe that the beamformers with L = 10and L = 16 reach the optimal SINR. One can notice that theproposed beamformer with L = 28 has a comparatively poorperformance. This poor performance is not due to the signalcancellation phenomenon or the low convergence rateproblem. In fact, this weak performance is due to the poorquality of its blocking matrix because in the proposedalgorithm when L = 28, then the length of the synthesiserarray is equal to 3. Consequently, vector wo does not haveadequate degree of freedom to attain the properties in (20).In this circumstance, the generated blocking matrix has aslow transition from the rejection region to the pass region.As a result, besides the DS, the interference sources may be

Fig. 10 Output SINR against training sample size M (SNR = 10dB); fourth example


attenuated by the blocking matrix. If the blocking matrixeliminates an interference signal, the beamformer will notsense it and consequently will not try to suppress it. In thisexample, the blocking matrix of the last beamformer(beamformer with L = 28) strongly attenuates the interferingsignal with DOA 85°. Accordingly, the beamformer cannotplace an adaptive null in its direction.

5 Conclusion

In this study, a new beamformer, which is robust againstDOA mismatch, has been introduced. This method is basedon the DS elimination from the training snapshots and thesub-array beamforming technique. In this algorithm, ablocking matrix is synthesised which converts input data tothe DS-free data. Using the DS-free data, the sub-arraybeamforming vector is calculated and based on it, the mainbeamforming vector is calculated. As a result of theDS-elimination and performing adaptive operations inthe sub-array level, the proposed algorithm is robust againstthe DOA mismatch and yields an excellent convergence rate.

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