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Low-Complexity Robust Beamforming Based onConjugate Gradient Techniques

Luk Ldut+ Rodrgo de Lmret Le Wgt d r Hrd+

Communications Reseach Group, Department of Electronics, University of YorkYork YOO 5DD, United Kingdom

Emails:ltl502@york.ac.ukrcdI500@ohm.york.ac.uklw517@york.ac.uk+ Communications Research Laborato, lmenau University of Technology

P Box 100565, D-98684 lmenau, GermanyEmails:lukas.landau@tu-ilmenau.demartin.haardt@tu-ilmenau.de

bsrac propose low-complexity robust beamformingalgorithms based on the conjugate gradient method and theworst-case optimization based criterion. Unlike the existingrobust beamformers based on the worst-case optimization basedcriterion that use a second-order cone program, the proposed

algorithms employ a joint optimization strategy based on lowcomplexity conjugate gradient algorithms. The proposed algorithms are termed the Robust Constrained Minimum Variance Modied Conjugate Gradient (Robust-CMV-MCG) andthe Robust Constrained Constant Modulus Modied ConjugateGradient (Robust-CCM-MCG), which has an advantage for thespecial case of constant modulus signals. Simulations show thatthe performances of the proposed algorithms are equivalentor better than that of existing robust algorithms, whereas thecomplexity is more than an order of magnitude lower.

n robust adaptive beam forming, array steeringvector mismatch, low-complexity algorithms, conjugate gradientmethod,

I. NTRDUCTIN

Befoming has may applications in wieless communications ad sona medical imaging adio astonomy andothe aeas. In pactice cicumstances like local scatteingimpefectly calibated aays and impecisely known waveeldpopagation conditions can lead to a pefomace degadationof the conventional algoithms. the last decades someappoaches have been published to solve this poblem [][2] [3]. Some popula appoaches ae based on the wostcase (WC) citeion [4]. en these poblems end up in theso-called second ode cone pogam (SC) which c besolved with inteio point methods with ove cubic complexityas a nction of the numbe of senso elements. Due to its highcomputational eot thee is oom fo simplication and fothe development of low-complexity algoithms

The poposed obust adaptive beamfome taes advantageof a joint optimization of the obust constaint and thebefome design. This method uses a modied conjugategadient (MCG) algoithm pefoming only one iteation pesnapshot [5]. The novelty of ou appoach is that it benetsom pevious computations and employ a joint optimization stategy to compute the befome d the Lagangemultiplie that aises om the optimization poblem. Besides

764$ I

the minimum vaiace citeion which leads to the RobustCMV-MCG algoithm we combine the new method withthe constat modulus citeion [6] [7] which leads to theRobust-CCM-MCG algoithm poviding bette esults in caseof constat modulus signals. The complexity of ou algoithmsis quaatic with the numbe of senso elements while thepefoace is equivalent o bette tha that of the wok in[4].

The pape is ogaized as follows. The System model isintoduced in Section 2. Section 3 gives a bief eview ofaleady existing obust beafoming appoaches. In Section 4the poposed obust adaptive beamfoing appoach with itsjoint optimization method is shown. The adaptive ealization isdescibed in Section 5. Section 6 shows the simulation esults.A bief conclusion is given by Section 7.

II. YSTEM DEL

The output of a befome fo naowbad signals isdened as

(1)

whee x E CMX1 is the input vecto W E CMx1 isthe vecto of beamfoing weigths denotes the time indexad . H stands fo Heitian taspose. Fo simplicity aunifo linea senso aay with M elements is consideedeven though the poblem can be extended to abita sensoaays. The input vecto fo a snapshot with D impingingwaveonts ca be descibed as

x AOs + n (2)whee 0 [Bo, , BD-1 E IDx1 is the vecto with thediections of aival (DoA) AB [(B, , (BD)] ECMxD is the matix containing the ay steeing vectos(Bm) E CMX1 fo m , . . . D the following B1is the diection elated to the desied use which is oughlyknown by the system. The vecto s E CDx1 epesentsthe uncoelated souces. e vecto n E CMX1 is thesenso noise assumed as zeo-mea complex Gaussi. what follows the pesumed aay steeing vecto is denoted as (B e signal-to-intefeence plus noise atio(SINR)

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is dened as

(3)

wheeRs is the signal covaiace matix coesponding to thedesied use andRHn is the intefeence plus noise covaiancematix.

III. A EVIEW N BUST DAPTIVE EAMFRMINGThis section gives a bief eview of aleady existing obust

adaptive beamfoing appoaches. The most common wayto impove obustness of a adaptive befome is to adda diagonal loading tem to the signal covaice matix.Howeve it is not clea how to obtain the optimal value fothe diagonal loading facto. A pactical choice of its valueis 10 0; , whee 0; is the noise powe. Anothe populaappoach is the adaptive beafoe based on the wostcase pefomance optimization [4]. It is based on a constaintthat the ay esponse is always geate o equal to unityfo all vectos that belong to a pedened set of vectos

in the neighbouhood of the pesumed vecto. In [4] thepedened set is a sphee A = a + e, IIel2: E wheee epesents the aay steeing vecto mismatch and E is itsuppe bound. Finally the poblem ca be tasfomed into asecond ode cone poga. Similaly to that the appoachin [3] implies a pedened set of the aay steeing vectowhich c also be dened as a ellipsoid. Anothe notableobust appoach is given by the eigenspace-based beafome[8]. This technique implies the pojection of the pesumedaay steeing vecto onto the saple signal-plus-intefeesubspace. To benet om the eigenspace-based appoach theank of the signal-plus-intefee subspace needs to be low adknown. Anothe appoach which exploits this idea is given by

the educed-k beafoming method [9] which avoids aeigendecomposition of the signal subspace. Thee also existobust adaptive befoming appoaches taking adavatage ofstatistical infomation about the steeing vecto mismatch [10].Recently a obust adaptive befoming appoach has beendeveloped which estimates the mismatch using constainedenegy mimization [11]. hile the method equies no additional infoation about the mismatch its sequential quaaticpogamming equies a high computational eot.

IV. RPSED BUST ESIGN RITERIA AND JINTPTIMIZATIN APPRACH

A bus Cnsrnd nu Vrnc DsgnThe poposed beafoes e elated to the wost case ap

poach whose pefomance is aleady well established [2] [4].In case of the minimum vaiance design it ca be tansfomedin the following optimization poblem.

mwRxxw, . t {wa} -8 E IIwl2, (4)wheeRxx ={(i)(i)} is the covaiace matix of theinput signal. Accoding to [3] it is sucient to constain theeal pa in the constaint. It has been shown that this kind ofbefome belongs to the class of diagonal loading schemes

[4]. As aleady mentioned this poblem can be solved byinteio point methods while its complexity is moe than cubicwith the numbe of senso elements. Conta to that appoachthe poposed on-line algoithms te advantage of the peviouscalculations which leads to a complexity eduction of moetha an ode of magnitude. In ode to do so the constaintis slightly modied. Hee it is assumed that the use ofEIlwll;causes the se eect as the conventional constaint. Finallythe poposed design citeion fo the minimum vaiance caseis

mwRxxw, . t {wa} -8 EIlwll; (5)Using the method of Lagage mltiplies gives

LCMV(w,) =wHRxxw+[E wHw-e{wHa} +8] ,(6)

whee is the Lagange multiplie. Computing the gadientof(6) with espect to * ad equating it to zeo leads to

WCM =(Rxx + E) a/. (7)Because it is not clea how to obtain the Lagange multipliein a closed fom [4] we popose to adjust it in a paallelalgoithm. In this joint optimization the Lagage mltiplieis inteeted as a penalty facto. this case > 0 needsto hold all the time. The adjustment ises the penalty factowhen the constaint is not lllled deceases it otheise. case of a too small penalty the minimum vaiance design leadsto a non llled constaint. Fo that we devise the followingalgoithm

(i) =(i-1)+J Ellw(i)II;-{w(i)a} +8 ,(8)whee

J is the step size. In addition it is easonable to dene

boundaies fo the update te.

B. bus Cnsrnd Csn duus Dsgn

In case of constat modulus signals it has been shown thatthe constat modulus design pefoms bette than the minimumvaiace design [6][7]. Similaly the obust appoach can becombined with the constained constant modulus citeion. econstt modulus cost nction is dened by

(9)

Since it is a fouth ode nction with a moe complicatedstuctue a closed fom solution is not possible. Fo this eason

the poblem is solved iteatively which coesponds to thefollowing undelying objective nction

Jit =wRW -' dw , (10)wee R =IYI

2 ad d ={xy* ae computedusg the pevO snapshots. Finally the optimzation poblemis cast as

mwRW - '{wd} , (11). t {wa} -8 EIlwll; (12)

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Using the method of Lagange multiplies gives

.CCM (w, A) =wHRaW -2 1Re{wHd}

+ A wHw -Re {wH } + 5J (13)Computing the gadient of (13) with espect to w* adequating it to zeo leads to

WCCM = (R + A) d+ A/) (14)The adjustment of the Lagage multiplie A can be done inthe sae way as in the minimum vaice case.

DAPTIVE LGRITHMS

To take advantage of the joint optimization appoach an online modied conjugate gadient method with one iteation pesnapshot is used to solve the esulting poblem. Its deivationis based on [5] and it can be inteeted as an extension ofthe idea in [12].

A bus-CV-CG

In the algoithm an exponentially decayed data window isused to estimate Rxx

xx(i) = xx(i - + (i)(i)H ,whee is the fogetting facto. Accoding to [1]

Rxx (-) xx(i)

(15)

(16)

can be assume fo lage i Replacing Rxx in(6) intoducing'(i) = , leads towCMv(i) = [xx(i)+(i)](i)/Let us intoduce the CG weight vecto ( i) as followswCMv(i) = (i) 5i). e conjugate gadient algoithm solvesthe poblem by iteatively updating the CG weight vecto

(i) = (i + (i)p(i), (17)wheep(i) is the diection vecto d (i) is the adaptive stepsize. ne way [5] to ealize the conjugate gadient methodpefoming one iteation pe snapshot is the application of thedegeneated scheme. Unde this condition the adaptive stepsize(i) has to lll the convegence bound given by

:pH(i)g(i) : pH(i)g(i - (18)

To erange(18) the negative gadient vecto and its ecusiveexpession ae consideed as

g(i) = -[xx(i) + '(i)](i)

=( -) + g(i --[H + ('(i) -'(i -))](i --(i)[xx(i) + '(i)]p(i) (19)

Pemultiplying withpH(i) taking the expectation om bothsides and consideingp(i) uncoelated with (i) d(i- leads to

Hee it is assumed that the algoithm has aleady convegedwhich implies ( -) -[E {H} + '(i)](i - = whee equation(16) is taen into account ad (i) (i -Intoducing P = [xx(i) + A(i)]p(i) eaanging (20)ad plugging into (18) detemines the stepsize within itsboundaies as follows

(21)

whee : ' : The diection vecto is a linea combination om the pevious diection vecto d the negativegadient.

p(i + ) = p(i) + f(i)g(i), (22)

whee f( i) is computed fo avoiding the eset pocedue byemploying the Pola-Ribiee appoach [13].

f = [g(i -)Hg(i _)t[g(i) -g(i -)]Hg(i) (23)

The poposed algoithm which is teed Robust-CMV

MCG is descibed in Table I.

Table I: Proposed Robust-CMV-MCG algorithm

v(O)= ; p(l)= g(O)= (O)= 81; (0)= (1)= For each time instant i= 1,

xx(i)= xx(i -1) + (i)(i)HPR= [xx(i)+ '(i)E]p(i) v = ['(i) -(i -1)] Ea(i)= p(i)H PRr ( -" p(i)H g(i -1) (0: " : 0.5)v(i)= v(i -1) + a(i)p(i)g(i)= (1 -) + g(i -1) -a(i)pR

-(i)(i)H + v v(i -1)(i)

=g(i -1)H g(i -1)r [g(i) - g(i _1)]H g(i)

p(i+ 1)= g(i)+ (i)p(i)w(i)= (i)v(i)/

8> = >[E wV(i) e {WV(i)H } + 8]while 8> : -(i)r 8> 8>

8> = 8>/end

(i+ 1)= (i)+ 8>

Note that fo the paallel algoithm to adjust the Lagagemultiplie we divide the update-tem by 2 if the Lagagemultiplie is outside a pedened ange as it is descibed inTable I. The application of the poposed algoithm coesponds

to a computational eot which is quadatic with the numbeof senso elements M

B. bus-CC-CG

The adaptive algoithm in case of the constained constantmodulus citeion is developed analogously to the minimumvaiace case. The estimates ofRa ad d ae based on an

E {p(i)Hg( in E {pH(i)g(i -I)} exponentially decayed data window.

-E (i)E {pH(i)[xx(i) + (i)]p(i)} (i) P(i - + ly(i)12 (i)(i)H(20) d(i) d(i - + (i)y(i)*

(24)

(25)

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Following the steps while ting into account that

(-)(i)d (-)d(i)

(26)

(27)

leads to the adaptive algoithm. Note in contast to the CMVcase hee the befoming weight vecto is the sae asthe conjugate

_fadient weight vecto which means WCC

( + .) ("d+ ).a/21 e poposed algoithm whichis teed RObust-CCM-MCG , is descibed in Table II.

Table II: Proposed WC-CCM-MCG algorithm

p(l)= g(O)= a; (O)= 81; d(O)= (0)= (1)= ; WCCM = aiMFor each time instant i= 1,

(i)= (i -1) + y(i)2 m(i)m(i)HPR= [(i) + '(i)E]p(i) v = ['(i) -(i -1)] Ea(i)= p(i)H PRJ -1 ( -') p(i)H g(i -1); (0: ' : 0.5)wccM(i)= wCCM(i-1)+a(i)p(i)

g(i)= g(i -1) -a(i)pR -

y(i)2 m(i)m(i)H

wcc(i -1)

+m(i)y(i)* + v [al (E) - w(i -1)](i)= g(i -I)H g(i -I)J -1 [g(i) - g(i -1)]H g(i)p(i+ 1)= g(i) + (i)p(i)

8 = fE W(i) e {wc(i)Ha} + while 8 : -(i)or 8 28

8 = 8/end

(i+ 1)= (i)+ 8

V I. IMULATINS

Fo the simulations we use a unifo linea senso aaywith 10 senso elements. e fogetting facto 99 ischosen. e step sizes ae M CMV 8 and M CCM e update limitation is set to omx Fo theobust constaints holds. Accoding to thedieent constaint nctions the equatity is a special case foM Fo the loaded saple matix invesion beamfome

(loaded-SMI) the diagonal loading facto is chosen as a;.Besides the desied use (use 1) thee ae 4 intefeeswhose elative powes(P) with espect to the desied use addiections of aival(DoA) in degee ae detailed in Table III.At i the beam fomes ae cononted with a sceniochge.

The signal steeing vecto is coupted by local coheentscatteing whee

a + Lj

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25r-20

15

l 10:Z 5

o oadSopima SIN [2] p54

- - ropos RosCMVCG ropos RosCMMGWCCMV[4

-10L-15 - -5 oSNR (d8)

5

Fig. 2. SINR performance versus SNR,i = 1500

15

0.251====

ua

0

015

0.1

0.05

opos RobustCeeG opos RobsCMVCG

WCMV [4

oadSM

o 2 4 6SIN(d8)

Fig. 3. Probability density nction of the output SINR,i = 2000, SNR=OdB

comped to the wost-case optimzation based beafomewhich is solved with a second ode cone poga. epoposed Robust-CCM-MCG algoithm based on the ConstantModulus design citeion shows a bette pefomce whichtaes advatage of the knowledge of the signal amplitude ofthe desied use.

EFERENCES

[1] H. V Trees,Optimum Array Pcessing. John Wiley,2002.[2] J. Li and P. Stoica,Robust Adaptive Beam/orming. NJ: Wiley: Hoboken,

2006.[3] R. Lorenz and S. Boyd,"Robust minimum variance beamforming, IEEE

Trans. Signal Processing, vol. 53,pp. 1684 1696, May 2005.[4] S. A. Vorobyov, A. B. Gershman, and Z.-Q. Luo, "Robust adaptive

beamforming using worst-case performance optimization: A solution tothe signal mismatch problem, IEEE Tns. Signal Pcessing, vol. 51,pp. 3 13324, Feb. 2003.

[5] P. S. Chang and A. N. Willson, nalysis of conjugate gradient algorithms for adaptive ltering, IEEE Trans. Snal Processing, vol. 48,pp. 40941 8, Feb. 2000.

[6] R. C. de Lamare and R. Sampaio-Neto,"Blind adaptive code-constrainedconstant modulus algorithms for cdma interference suppression in multipath channels, IEEE Signal Pcessing Lett.,vol. 9,no. 4,pp. 334336,Apr 2005.

[7] R. C. de Lamare, M. Haardt, and R. Sampaio-Neto, "Blind adaptiveconstrained reduced-rank parameter estimation based on constant mod

ulus design for cdma interference suppression, IEEE Trans. SnalPcessing, vol. 56, pp. 24702482,Jun 2008.[8] D. D. Feldman and . J. Griths, "A projection approach to robust

adaptive beamforming, IEEE Trans. Signal Processing, vol. 42, pp.867876, Apr. 1994.

[9] R. C. de Lamare, . Wang, and R. Fa, "Adaptive reduced-rank lmvbeamforming algorithms based on joint iterative optimization of lters:

[10]

[ 1 1 ]

[12]

[13]

Design and analysis, Snal Processing, vol. 90, no. 2, pp. 640652,Feb 2010.S. A.Vorobyov, A. B. Gershman, and Y Rong, "n the relationshipbetween the worst-case optimization-based and probability-constrainedapproaches to robust adaptive beamforming, in Pc. IEEE ICASSP,vol. 2, Honolulu, HI, Apr. 2007,pp. 977980.A. Hassanien and S. A.Vorobyov,"Robust adaptive beamforming usingsequential quadratic programming: iterative solution to the mismatchproblem, IEEE Signal Processing Lett., vol. 15, pp. 733736,2008..Wang and R. C. de Lamare,"Constrained adaptive ltering algorithms

based on conjugate gradient techniques for beamforming, ET SignalProcessing, vol. 4, pp. 686697, Dec 201 0.D. G. Luenberger,Linear and Nonlinear Pgramming 2nd e Reading,: Addison-Wesley, 1984 .