array beamforming synthesis

3878 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 9, SEPTEMBER 2015

Array Beamforming Synthesis for Point-to-PointMIMO Communication

Farnaz Karimdady Sharifabad, Student Member, IEEE, Michael A. Jensen, Fellow, IEEE,and Adam L. Anderson, Senior Member, IEEE

AbstractWhile transmit and receive beamforming for themultiple-input multiple-output (MIMO) point-to-point channel isa well-studied topic, existing algorithms cannot specify effectivebeamformers for certain situations. To address this deficiency,we present an iterative algorithm that can establish beamformersbased on different channel information available to the transmitterand receiver, different receiver architectures, and different con-straints on the power radiated by the transmit array. Simulationresults based on modeled and measured channels demonstrate thatthe algorithm effectively accommodates a wide range of conditionsand that the performance achieved with the approach matchesthat obtained using existing algorithms that are applicable onlyto specific conditions.

Index TermsAntenna arrays, antenna radiation patternsynthesis, multiple-input multiple-output (MIMO) systems.

I. INTRODUCTION

P RIOR work has established that antenna array beam-forming in multipath propagation environments enablesmultiple spatial degrees of freedom [1], [2] that can be exploitedby multiple-input multiple-output (MIMO) signaling in whichmultiple data streams, each weighted by a unique beamformer,are simultaneously communicated [3], [4]. However, this simul-taneous data transmission leads to sophisticated considerationsregarding the radio system that impact the array synthesisproblem. For example, while linear beamforming maximizesthroughput in a point-to-point MIMO system if both transmitterand receiver have accurate channel state information (CSI) [3],nonlinear successive decoding must be added to the receiverif the transmitter CSI is inaccurate or outdated or the trans-mitter only has channel distribution information (CDI) in theform of a spatial covariance matrix [5], [6]. Furthermore, whilemost algorithms assume a sum power constraint (SPC) thatlimits the total transmitted power, physical power amplifiers(PA) favor a per-antenna power constraint (PAPC) that lim-its the average power transmitted from each antenna [7]. The

Manuscript received March 07, 2015; revised May 11, 2015; accepted June11, 2015. Date of publication June 18, 2015; date of current version September01, 2015. This work was supported by the U.S. Army Research Office underGrant W911NF-12-1-0469.

F. K. Sharifabad was with the Electrical and Computer EngineeringDepartment, Brigham Young University, Provo, UT 84602 USA. She isnow with Qualcomm Technologies, San Diego, CA 92121 USA (e-mail:[email protected]).

M. A. Jensen is with the Electrical and Computer Engineering Department,Brigham Young University, Provo, UT 84602 USA (e-mail: [email protected]).

A. L. Anderson is with the Tennessee Technological University, Cookeville,TN 38505 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAP.2015.2447035

difficulty with these practical considerations is that existingalgorithms for transmit beamformer synthesis focus on maxi-mization of the mutual information and, therefore, inherentlyassume specific radio capabilities. Since the optimal trans-mit beamforming weights depend on the receiver capabilitiesand the power constraint, existing algorithms cannot constructoptimal beamformers for many practical radio architecturesthat differ from the assumptions inherent in this mutual infor-mation maximization. Table I summarizes the availability ofbeamforming synthesis algorithms for different communicationscenarios.

These observations suggest that in developing beamformingstrategies for the point-to-point MIMO link, the optimiza-tion cost function should incorporate practical system designdecisions. Recent work on multiuser MIMO communicationdemonstrates how to write the achievable rate to incorporatelinear transmit processing [10] and/or linear receive process-ing [11]. More recent work provides an iterative solution todetermine the optimal beamformers for cooperative multiuserMIMO links based on linear or nonlinear transmit and receiveprocessing [12]. While these prior developments teach somefoundational principles in MIMO beamforming, they are posedfor multiuser MIMO channels and do not accommodate all ofthe practical radio topologies listed in Table I.

This work develops a general framework for transmit andreceive beamformer synthesis by adapting the iterative algo-rithm detailed in [12] to maximize the achievable commu-nication rate for strictly point-to-point MIMO channels withdifferent types of channel information (CSI or CDI) available atthe transmitter, linear or nonlinear receiver capabilities, and dif-ferent power constraints. Simulations based on a simple channelmodel and on experimental MIMO channel data reveal thatthe approach is highly effective and is able to generate trans-mit beamformers whose performance with reduced-complexityradios and practical power constraints rivals that achieved withmore sophisticated radio capabilities.

II. POINT-TO-POINT MIMO SIGNALINGA. System Model

Consider a MIMO communication link based on Nt trans-mit antennas and Nr receive antennas. The Nr 1 receivedsignal vector is given by y = Hx+ , where H is an Nr Nt channel matrix and is an Nr 1 noise vector whoseentries are i.i.d. zero-mean complex Gaussian random variables

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SHARIFABAD et al.: ARRAY BEAMFORMING SYNTHESIS 3879

TABLE IEXISTING MIMO TRANSMIT BEAMFORMING ALGORITHMS

with variance 2 . We assume that the system communicatesK min(Nt, Nr) data streams represented by the K 1 datavector x0 with covariance E

{x0x

0

}= I, where E {} is the

expectation, {} is the conjugate transpose, and I is the iden-tity matrix. The Nt 1 transmit vector x is then formed fromx = Bx0, where B is the Nt K matrix whose columns arethe transmit beamforming weights for each stream.

The receiver applies an Nr 1 unit-length beamformingvector wk when detecting the kth element of x0, or x0k =wkyk where yk represents a possibly modified version of thereceived vector y. For example, if the receiver uses only linearbeamforming, then yk = y. However, if the receiver also usesnonlinear successive interference cancellation (SIC), then ykrepresents y modified such that the contributions of the previ-ously detected symbols are removed. For notational simplicity,we let wk represent the kth column of the Nr K matrix Wand bk represent the kth column of B.

Under these assumptions and with B specified, for an opti-mal receiver the achievable system rate is [13]

Copt = log

I+ 12HBBH (1)

where | | is the matrix determinant. If the transmitter has noknowledge of the channel information, then for a total transmitpower P we have K = Nt and BB = (P/Nt)I, so that (1)becomes the capacity for an uninformed transmitter.

B. Power ConstraintsConstructing the transmit beamforming matrix B that max-

imizes the communication rate requires that we constrainthe transmit power. Most work assumes a SPC that limitsthe total power transmitted from all antennas, or E

{xx

}=

E{tr(xx

)}= tr

(BB

)= P , where tr() is the trace.

However, in most radios, the power transmitted from eachantenna is limited by the PA. If the SPC is used, each PAmust be able to transmit the total power P despite the factthat, on average, it must only accommodate its proportionalshare (P/Nt). Therefore, we also consider a PAPC wherethe average power radiated from the ith antenna is specifiedas Pi. The constraint becomes E

{xix

i

}= E

{bix0x

0b

i

}=

bibi = Pi, where xi and bi represent the ith element of x and

ith row of B, respectively.

C. Existing Beamforming MethodsOur objective is to develop a framework for generating trans-

mit beamforming vectors under the assumptions in Table Iwhere currently no algorithms exist. In preparation for thisdevelopment, it is instructive to first review existing algorithmsfor two of the scenarios listed.

1) CDI-Based Beamforming With SPC: If the transmitterpossesses CDI in the form of the transmit spatial covarianceRt = E

{HH

}/Nr, the beamformer should be constructed

as B = Ut [5], [14] where is a real, nonnegative, diago-nal power allocation matrix constrained such that tr

(

)=

P (SPC) and Ut is the unitary matrix of eigenvectors of Rt.Recent work has proposed an iterative solution for the powerallocation assuming that the full channel spatial covariance isseparable into a Kronecker product [6], a technique we call CDIKronecker. Alternatively, if we assume that the channel matrixentries are wide-sense stationary random variables and take theexpectation of (1) [11], [15], by Jensens inequality this averagerate becomes

Copt = E {Copt} logI+ Nr2 BRtB

. (2)The accuracy of using Jensens inequality to achieve this upperbound has been studied previously [16][18], with the findingthat the accuracy depends on the level of element correlation.However, the tightness of (2) as a bound is not as importantas whether or not the value of B that maximizes the bound alsoapproximately maximizes the expectation of (1). Using a water-filling approach to maximize (2) yields a second approach forspecifying [19], a technique we call CDI waterfilling. Ournumerical results that follow help to establish the validity ofthis approach.

2) CSI-Based Beamforming With PAPC: The Drop-Rankalgorithm is an iterative solution for finding the transmit beam-formers that maximize the rate, which inherently assumes theavailability of nonlinear detection at the receiver [7]. Themethod assumes that the transmitter possesses CSI, and theredoes not appear to be a straightforward way to extend theconcept to linear receivers or to CDI-based beamforming.

III. ITERATIVE BEAMFORMING

When choosing transmit beamformers to maximize theachievable rate in (1) or its upper bound in (2), in most cases,the receiver must combine linear minimum mean squared error(MMSE) beamforming with nonlinear SIC to realize the rate[13], [20], [21]. Because the prior work for CDI-based beam-forming under an SPC or CSI-based beamforming under aPAPC focuses on capacity maximization, the transmit beam-formers are derived under the assumption of MMSE-SIC atthe receiver and therefore are inappropriate for use when thereceiver uses only linear beamforming.

This observation motivates development of an alternateframework for constructing beamforming matrices based onthe available channel information, receiver capabilities, andpower constraint. To develop our iterative beamforming (IBF)


approach, we incorporate the receive beamforming into theexpression for the achievable rate to obtain [10], [12]

C =K

k=1

log (1 + SINRk) (3)

SINRk =|wkHbk|2

2 +

iLk |wkHbi|2

=nkdk

(4)

where (4) represents the signal-to-interference-plus-noise ratio(SINR) for the kth data stream.

Because the second term in the denominator of (4) repre-sents the interference to one data stream caused by the otherdata streams, the set Lk of integers used in the sum dependson the receiver architecture. Specifically, for linear receivebeamforming, the kth stream experiences interference from allother streams, or Lk = {1 i K, i = k}. In contrast, if thereceiver uses SIC, then we assume that the kth stream expe-riences interference only from streams that have not yet beendetected. For simplicity in notation, we assume that the receiverdetects streams in numerical order of their indices so that Lk ={k < i K}. Section III-D indicates how this choice of order-ing impacts initialization of the optimization. Throughout thedevelopment, we use the notation CSIT and CSIR (CDIT andCDIR) to indicate that CSI (CDI) is available at the transmitterand receiver, respectively.

A. CSIT and CSIRTo incorporate design decisions into the rate in (3), we can

write the SINR given in (4) usingnk = b

kAkbk (5)

dk = 2 +

iLk

biAkbi (6)

Ak = Hwkw

kH. (7)

To optimize the rate, we form the cost function = C f ,where f generically represents a Lagrange multiplier term thatwill, in the following, be developed for both the SPC and thePAPC. We then take the gradient j = jC jf withrespect to bj and set the result equal to zero, where {}represents a conjugate. The gradient of the rate C is

jC = Cbj

=K

k=1

nkdk

nk [nk + d

k]

dk [nk + dk](8)

where

nk = jnk ={2Akbk, j = k0, j = k (9)

dk = jdk ={2Akbj , j Lk0, j Lk (10)

and we have used that wk is unit length (wkwk = 1).Substitution of (9) and (10) into (8) yields

jC = 2dj

Ajbj 2j Lk

nkdk(nk + dk)

Akbj (11)

where the summation is over all values of k such that the fixedindex j is within the set Lk = Lk k.

We will see that for the power constraints considered, we canexpress the gradient of the Lagrange multiplier term as

jf = 2fbj (12)where f is a diagonal matrix. Use of (11) and (12) leads to

j = 2dj

Ajbj 2f +

j Lk

nkdk(nk + dk)

Ak

bj .(13)

We now construct the matrix f for the SPC and PAPC.1) SPC: For IBF with the SPC, we have

f =

(K

k=1

bkbk P)

(14)

where is a Lagrange variable, leading to jf = 2bj orf = I. (15)

The Lagrange variable can be determined by recognizingthat since we will set j = 0, we have bjj = 0 and

j bjj = 0. Since P =

j b

jbj , use of (13) leads to

=1

P

Kj=1

bjFjbj (16)

Fj =1

djAj

j Lk

nkdk(nk + dk)

Ak (17)

which is equivalent to

=Kj=1

njdj(nj + dj)

2P

. (18)

2) PAPC: For IBF with the PAPC, we have

f =

Nti=1

i

(bib

i Pi

)(19)

where we now have Nt different Lagrange variables i. Thisleads to jf = 2diag()bj , where = [1, 2, . . . , Nt ]Tand diag() produce a diagonal matrix from the vector argu-ment. Therefore, we have

f = diag(). (20)The Lagrange variables are constructed as discussed above forthe SPC after first prescaling with indicator matrices Ji thatare defined as all-zero matrices with a 1 on the ith diagonalelement, leading to

i =1

Pi

Kj=1

bjJiFjbj . (21)

Now that we have expressions for f for both power con-straints, we can set (13) to zero with the goal of solving


for the beamforming vector bj . However, we recognize thatbecause the beamformers appear explicitly within nk and dk,directly solving for bj is difficult. Instead, we can rearrange theresulting equation into the form

bj =1

djG1j Ajbj (22)

Gj = f +

j Lk

nkdk(nk + dk)

Ak. (23)

Before discussing the solution of this equation, we note thatthe above procedure can be applied to determine the receivebeamformers W. Specifically, incorporating the Lagrange mul-tiplier f =

(wjwj 1

), taking the gradient of the cost

function with respect to wj , and setting the result equal to zeroleads to

wj = G1j Ajwj (24)

Gj =nj

nj + dj

2I+ kLj

Ak

(25)Ak = Hbkb

kH

. (26)

However, we recognize that expansion of Aj on the right-hand side of (24) leaves a right-most product of bjHwj thatis simply a constant, and since wj should have unit length,this constant is removed through normalization. Therefore, forCSIR, (24) reduces to the MMSE beamformer (with or withoutSIC) [11], [13], [20], [22]

wj =

2I+ kLj

HbkbkH

1Hbj . (27)Examination of these equations reveals that the following:1) bj appears explicitly on both sides of (22), as well as

implicitly on the right of (22) through Gj , and2) (22) depends on the vectors wj and (27) depends on the

vectors bj .These observations suggest that we first initialize B and theniteratively compute W using (27) and B using (22) untilboth matrices have converged [11]. Since the Lagrange multi-plier formulation does not guarantee enforcement of the powerconstraint, in each iteration, the transmit beamformers are nor-malized to explicitly enforce the constraint. If we use thesuperscript {}(n) to indicate a quantity computed at the nthiteration, then Fig. 1 shows a simple flow diagram for perform-ing the iteration, where the symbol TC indicates the thresholdfor terminating the iteration that is chosen as TC = 104 for allsimulations in this paper.

B. CDIT and CSIRThis beamformer synthesis technique can be extended to the

case where the transmitter has CDI. We take the expectation of

Fig. 1. Flow diagram showing implementation of IBF for CSIT and CSIR.

the rate in (3) over the randomly varying channel H and applyJensens inequality to obtain [15]

C = E {C} K

k=1

log (1 + E {SINRk}) . (28)

As discussed in Appendix A, we further make the approxi-mation E {SINRk} E {nk}/E {dk} = nk/dk. We can nowperform the procedure of Section III-A using nk and dk inplace of nk and dk, respectively. This leads to the same iterativeequation (22) but with the replacement

Aj E{Hwjw

jH

}. (29)

When implementing this algorithm, however, we must care-fully consider approximation of the expectation in (29). If weassume a single value of W is valid over the time windowduring which the transmit beamformer B based on CDI isassumed valid, Aj can be computed based on knowledge ofthe beamforming matrix W and the full covariance matrixR = E

{hh

}where h represents H stacked columnwise, as

illustrated in Appendix B. In practice, the receiver computes anew value of W each time it estimates a new channel matrix H,and, therefore, it is arguably more accurate to estimateAk usinga sample mean with the true values of H and W over the timewindow. Naturally, this latter approach is more computationallycostly, since we must compute this sample mean at each step inthe iterative computation. Our analysis with the experimentaldata discussed in Section IV shows that the accuracy improve-ment associated with this more costly approach provides verylittle performance improvement.

C. CDIT and CDIRThe IBF framework can also specify receive beamform-

ers based on CDI using the iterative computation for wjrepresented in (24)(26) and making the substitution Aj E{Hbjb

jH

}

. This is a problem that has been consideredfrom an information theoretic standpoint, but since practicaltechniques for coherent symbol demodulation without CSI at


the receiver remain an unsolved problem, we simply point outthe development for completeness and ignore this particularscenario in the remainder of the paper.

D. Beamformer Initialization and SIC OrderingAs indicated in Section III-A, the iteration begins with initial-

ization of B. Since transmitting multiple streams is generallybest accomplished using orthogonal beamformers, one optionis to form a matrix with independent, zero-mean, unit-variancecomplex Gaussian random variables and use its unitary singularvectors (SVs) as the initial value of B. Since, our goal is simplyto obtain a variety of different unitary matrices, other randommatrix realizations are likely effective. Regardless, since thecost function, in general, is not convex and convergence to thebeamformers that achieve the global optimum rate is sensitiveto the choice of initialization, we run the iteration with a numberof random initializations and choose the result that achieves thehighest rate. Alternatively, if using IBF with CSIT or CDIT, wecan initialize B as the right SVs of the channel matrix H or theeigenvectors of the transmit covariance matrix Rt, respectively.

This initialization discussion also relates to the specificationof the order in which symbols are detected when implement-ing SIC at the receiver. Specifically, because we have assumeddetection in the order of the stream index, achieving the optimalrate means that the iterative algorithm orders the beamformingvectors in the optimal way. The algorithms ability to achievethis ordering depends on the initialization. Since, optimal order-ing typically means that the stream with the highest SINRshould be detected first, we have found that the proper orderingof the initializing unitary matrix corresponds to the case wherethe singular values (or eigenvalues) are placed in decreasingorder.

IV. RESULTS

A. Simulation ApproachThe performance of the IBF algorithm is demonstrated here

through simulations. In all cases, the MIMO channel matrix His normalized so that

H2F = NrNt (30)where F represents the Frobenius norm. With this normal-ization, the ratio P/2 represents the single-input single-outputsignal-to-noise ratio (SNR) [23].

Most of the analysis is performed with MIMO channels mea-sured at a carrier frequency of 2.45 GHz for linear transmitand receive arrays of monopoles with half-wavelength elementspacing. The transmitter is held stationary while the receivermoves at a constant velocity of approximately 30 cm/s, andchannels are sampled at an interval of 2.5 ms correspondingto a distance moved by the receiver of 0.0062, where isthe wavelength at the carrier frequency. All measurements aretaken within an open area between several buildings on theBrigham Young University campus with the transmitter andreceiver positioned on either side of a dense stand of trees.Details concerning the measurement system used to collect thechannel matrices can be found in [24] and [25]. The simulations

use an adjacent subset of antennas in the transmit and receivearray to generate the Nr Nt channel matrix.

Use of this experimental data does not facilitate systematicevaluation of the effect of the channel directivity, or equiv-alently multipath richness, on algorithm performance. We,therefore, also use a two-ring channel model to analyze theimpact of multipath richness on the performance of differenttransmit beamforming techniques. In this model, the transmit-ter and receiver each lie within distinct circles of radius 40.A fixed number of scatterers lie on each circle, with the angleof each scatterer location being specified as a uniform ran-dom variable on [0, 2). Scatterers on the transmit and receivecircles are paired so that each propagation path has a singledeparture and arrival angle as well as a complex gain specifiedas a zero-mean, unit-variance circularly symmetric complexGaussian random variable. The linear transmit array remainsstationary at the center of its circle while the linear receivearray moves along a straight line within its circle. Normalizedchannel matrices are easily computed in this two-dimensionalpropagation model using established techniques [23].

Prior work has demonstrated that optimally allocating powerto different beamforming vectors offers little advantage overequally allocating power across the beamformers as the SNRincreases [26]. Therefore, unless otherwise specified the sim-ulations use a transmit power of P = 1 and noise variance2 = 1 for an SNR of 0 dB, with other default parametersbeing Nt = Nr = K = 4. When the PAPC is applied, we usePi = P/Nt, 1 i Nt. When average results are provided,for each value of the swept parameter (number of antennas,number of paths, etc.), we compute the rate for each chan-nel sample and then compute the average of these rates. Thechannel samples obtained from the experiments and model arenaturally correlated, with the objective that the results showwhat would happen as a radio moves through the environ-ment. When using CDIT, the required covariance is estimatedusing a sample mean of these correlated channel observationsto approximate R over a specified window size (in terms ofreceiver motion) of 1, and this covariance is used to con-struct the transmit beamforming matrix B that is fed back tothe transmitter at a regular feedback interval of 2.5.

B. CSIT and CSIRTo begin, for each of the experimentally obtained channel

matrices, we compute the capacity using the waterfilling solu-tion [3] and the rate using the beamformers obtained with theIBF algorithm with the SPC (based on an MMSE receiver)assuming CSIT and CSIR. While the results are not shown forthe sake of brevity, the rate obtained using IBF is indistinguish-able from the waterfilling capacity over a variety of systemparameters. This demonstrates that for this simple case, the IBFalgorithm provides the expected result.

While this simple demonstration is encouraging, it is moreinteresting to explore IBF performance for the PAPC. Fig. 2shows the rate averaged over the experimental channels as afunction of the number of antenna elements in the transmit andreceive arrays for the IBF algorithm with a PAPC and the previ-ously published Drop-Rank algorithm [7] with both MMSE and


Fig. 2. Rate averaged over all experimentally obtained channel matrices forCSIT and CSIR with the PAPC as a function of the number of antennas in thetransmit and receive arrays (Nr = Nt).

MMSE-SIC receivers. For the MMSE-SIC receiver, the per-formance of the two algorithms is almost identical. However,when an MMSE receiver is used, the rate achieved with theDrop-Rank beamformer that is designed for a nonlinear receiverdegrades significantly, while that obtained using the IBF beam-former remains almost unchanged. The key point is that the IBFcan accommodate reduced receiver capabilities and, at least forthis case, do so with little performance degradation.

Because IBF requires an iterative computation, it is impor-tant to consider the convergence rate of the iteration as well aswhat is required to achieve the global optimum solution to thenonconvex optimization. To explore the convergence rate, foreach iteration, we compute the fractional difference

(n) =

Cfinal C(n)Cfinal

(31)

where C(n) and Cfinal denote the rate at the nth iteration and atfinal convergence, respectively. Fig. 3 plots this convergencemetric as a function of the iteration index n for both powerconstraints and for two different array sizes. The algorithm isinitialized with the channel right SVs and 20 randomly gener-ated vectors in the top and bottom plots, respectively. As canbe seen, initialization with the channel SVs is highly effective,even with the PAPC. Furthermore, regardless of the initializa-tion, on average the iteration converges within 20 iterations.Fig. 4 plots the cumulative distribution function (CDF) of thenumber of iterations required for convergence for the same sce-narios considered in Fig. 3, where convergence is defined asthe value of n for which (n) 104. These results confirmthat when initializing with the channel SVs, fewer than 20 iter-ations are needed to essentially guarantee convergence, whilesignificantly more iterations are required when the computa-tion is randomly initialized. Fig. 4 also shows the result for theDrop-Rank algorithm for Nr = Nt = 8 (similar results occurfor Nr = Nt = 4), revealing that for most cases, IBF convergesmore rapidly than Drop-Rank.

To explore achievement of the global optimum solution, foreach experimental channel observation, we seed the iterationwith NI random matrices and choose the outcome of the iter-ation that achieves the highest rate. Fig. 5 plots the achieved

Fig. 3. Fractional difference (n) between the converged rate and the rate ateach iteration index n in the IBF algorithm for CSIT and CSIR with an MMSE-SIC receiver averaged over all experimentally obtained channel matrices fordifferent algorithm initializations.

Fig. 4. CDF of the number of iterations required for the IBF algorithm to con-verge for CSIT and CSIR with an MMSE-SIC receiver using all experimentallyobtained channel matrices for different algorithm initializations, array sizes,and power constraints.

rate averaged over all experimental channel observations asa function of NI. The triangular symbols show the averagerate achieved when the channel SVs are used for the initial-ization. As can be seen, when using the SPC or the PAPCwith an MMSE-SIC receiver, we achieve the maximum in justa few random initializations, and more importantly the chan-nel SV initialization achieves the same result without requir-ing multiple iterative optimizations. However, when using thePAPC with an MMSE receiver, more random realizations arerequired. Furthermore, while the channel SV initialization withan MMSE receiver is not as effective for the large array, the ini-tialization still achieves a result that is within 1.5% of the rateachieved with the more costly random initializations.

C. CDIT and CSIRWe next compare the performance achieved with IBF to that

obtained using the two previously reported methods for beam-forming based on CDIT [6], [19]. Fig. 6 plots the rate averagedover the ring model realizations as a function of the number


Fig. 5. Rate averaged over all experimentally obtained channel matrices forCSIT and CSIR with the different power constraints and array sizes as afunction of the number of random initializations and, therefore, iterative opti-mizations. The average rate obtained when initializing with the channel rightSVs is also shown.

Fig. 6. Average rate achieved using different transmit beamformers for CDITand CSIR (MMSE-SIC) as a function of the number of paths in the two-ringchannel model.

of multipaths for the three transmit beamforming algorithmsassuming MMSE-SIC at the receiver. As can be seen, the per-formance of IBF with the SPC matches that achieved withthe previously developed CDI-based beamforming algorithms(that also use the SPC). Since both CDI waterfilling and IBFmaximize the upper bound on the rate while CDI Kroneckermaximizes the rate, their nearly identical performance suggeststhat maximizing the upper bound yields acceptable results.Furthermore, the considerable difference between the rate foran uninformed transmitter and that for CDI-based beamformingillustrates the improvement offered by transmit beamform-ing, although the benefit is reduced as the channel directivitydecreases (more paths).

When the channel is highly directive (few paths), very spe-cific transmit beamforming is required to fully exploit thechannel. Since the PAPC limits the range of beamformers thatcan be achieved, IBF with the PAPC performs about the same asor slightly better than the uninformed transmitter. As the num-ber of paths increases, however, the results show that it becomes

Fig. 7. Average rate achieved using different transmit beamformers for CDITand CSIR (MMSE-SIC and MMSE) with the SPC as a function of SNR usingexperimentally obtained channel matrices.

Fig. 8. Average rate achieved using different transmit beamformers for CDITand CSIR with the PAPC as a function of SNR using experimentally obtainedchannel matrices.

easier to form beams that can take advantage of the propaga-tion environment while still satisfying the PAPC, although theseresults still fall short of those achieved with the more flexibleSPC.

Fig. 7 shows the rate achieved using different techniqueswith the SPC as a function of SNR over the experimentallyobtained channels using MMSE-SIC and MMSE receivers. Thesimulation demonstrates that the algorithm works effectively athigh SNR. Furthermore, all results show that the effectivenessof transmit beamforming relative to uninformed transmissiondecreases with increasing SNR, as suggested in Section IV-A.Once again, all three algorithms achieve nearly identical per-formance for an MMSE-SIC receiver. However, when using theMMSE receiver, the performance of CDI waterfilling and CDIKronecker degrades significantly, while IBF is able to main-tain competitive performance. Fig. 8 plots similar results for theIBF algorithm with the PAPC using MMSE-SIC and MMSEreceivers. As there are no existing algorithms for this case, onlythe uninformed transmitter performance is shown for compari-son purposes. The results are quite similar to those in Fig. 7 for


the SPC, both in trend and in achieved performance. If we com-pare the results in Figs. 7 and 8 at 0 dB SNR to those in Fig. 6for a large number of paths, we see that the values are very sim-ilar. This comparison suggests that the experimental channelsgenerally have a large number of multipath components.

V. CONCLUSIONThis paper proposes an iterative algorithm that specifies

transmit and receive beamformers based on different typesof channel information, different capabilities of the receiver,and different power constraints, thereby offering beamformingsolutions for specific situations where solutions have not yetbeen available. Simulations in measured and modeled MIMOchannels show that the performance achieved with the tech-nique matches that obtained with existing algorithms whenused in conjunction with optimal receiver architectures butthat it further allows construction of near-optimal transmitbeamforming for simple MMSE receivers. The simulations fur-ther demonstrate the impact of different system parameters oncommunication performance.

APPENDIX AAPPROXIMATING THE EXPECTED SINR

The transmit beamformer for CDIT relies on the approxima-tion E {SINRk} E {nk}/E {dk}. While rigorously justify-ing this assumption is difficult, it can be shown that

E

{nkdk

} E {nk}

E {dk} cov(nk, dk)

E {dk}2+

var(dk)E {nk}E {dk}3

(32)

where cov(x, y) is the covariance of x and y and var(x) isthe variance of x. Examining (4), if the transmit beamformersare nearly orthogonal, then it is reasonable that cov(nk, dk) issmall. Furthermore, over a block that might be used for covari-ance estimation, it is likely that the variance of dk is also small.These observations support the approximation. Furthermore,even if the approximation is poor, as long as finding the beam-formers that maximize the rate under this approximation alsomaximize the actual expected rate, then use of the approx-imation is justified. Since, our results demonstrate that theperformance of the algorithm with this approximation matchesthat of other algorithms (for an MMSE-SIC receiver), use ofthis approximation has merit.

APPENDIX BEXPRESSING A AND A IN TERMS OF R

It is straightforward to express A and A in terms of elementsof the full covariance matrix R. Specifically,

Ak = reshape(tvec(wkw

k), Nt, Nt

)(33)

Ak = reshape(rvec(bkb

k), Nr, Nr

)(34)

where vec() stacks the matrix argument columnwise into a vec-tor, reshape(,m, n) reshapes the vector into a m n matrix,and

t = E{HT H} (35)

r = E {H H} (36)

where {}T and represent a transpose and a Kronecker prod-uct, respectively. The elements of t and r correspond toelements of the full covariance matrix R.

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Farnaz Karimdady Sharifabad (S12) received theB.S. degree in electrical engineering from AmirkabirUniversity of Technology (Tehran Polytechnic),Tehran, Iran, in 2007, the M.S. degree in wire-less communications from Lund University, Lund,Sweden, in 2010, and the Ph.D. degree in electri-cal and computer engineering from Brigham YoungUniversity, Provo, UT, USA, in 2013.

She is currently a Systems Engineer withQualcomm Technologies Incorporated, San Diego,CA, USA. Her research interests include covari-

ance modeling for multipath propagation channels and multiantenna signalprocessing.

Michael A. Jensen (S93M95SM01F08)received the B.S. and M.S. degrees from BrighamYoung University (BYU), Provo, UT, USA, in 1990and 1991, respectively, and the Ph.D. degree fromthe University of California, Los Angeles, CA, USA,in 1994, all in electrical engineering.

Since 1994, he has been with the Electrical andComputer Engineering Department, BYU, where heis currently a University Professor. His researchinterests include antennas and propagation for com-munications, microwave circuit design, multiantenna

signal processing, and physical layer security.Dr. Jensen is currently a President-Elect of the IEEE Antennas and

Propagation Society. He was previously the Editor-in-Chief of the IEEETRANSACTIONS ON ANTENNAS AND PROPAGATION as well as an AssociateEditor for the same journal and for the IEEE Antennas and WirelessPropagation Letters. He has been a Member and Chair of the Joint MeetingsCommittee for the IEEE Antennas and Propagation Society, a member of thesociety AdCom, a member of the society Publications Committee, and Cochairor Technical Program Chair for six society-sponsored symposia. In 2002, hewas the recipient of Harold A. Wheeler Applications Prize Paper Award inthe IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He was ele-vated to the grade of the IEEE Fellow in 2008 in recognition of his research onmultiantenna communication.

Adam L. Anderson (S00M10SM15) receivedthe B.S. and M.S. degrees from Brigham YoungUniversity, Provo, UT, USA, in 2002 and 2004,respectively, and the Ph.D. degree from the Universityof California at San Diego, La Jolla, CA, USA, in2008, all in electrical engineering.

He was a Research Assistant Professor with theUniversity of South Florida, Tampa, FL, USA, andis currently an Assistant Professor with TennesseeTechnological University, Cookeville, TN, USA.

Dr. Anderson was the winner of the 2014 DARPASpectrum Challenge, recipient of the 2014 Leighton E. Sissom Award forCreativity and Innovation, and a 2015 ORAU HERE Faculty Fellow at OakRidge National Laboratory.

array beamforming synthesis

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