1

Adaptive Millimeter Wave Beam-Alignment forDual-Polarized Broadcast MIMO Systems

Jiho Song, Junil Choi, Stephen G. Larew, David J. Love, Timothy A. Thomas, and Amitava Ghosh

Abstract—Fifth generation wireless systems are expected toemploy multiple antenna communication at millimeter wave(mmWave) frequencies for smaller cells within heterogeneouscellular networks (HCNs). The high path-loss of mmWave aswell as physical obstructions make it difficult to access mobileusers using mmWave. To compensate for the severe path loss,mmWave systems may employ a beam-alignment algorithm thatfacilitates highly directional transmission by aligning the beamdirection of multiple antenna arrays. This paper discusses ammWave system employing dual-polarized antennas. First, wepropose a practical soft-decision beam-alignment (soft-alignment)algorithm that exploits orthogonal polarizations. By sounding thedual-polarized channel over orthogonal polarizations in parallel,the equality criterion of the Welch bound for training sequencesis relaxed. Second, we propose a method to efficiently adaptthe number of channel sounding observations to the specificchannel environment based on an approximate probability ofbeam misalignment. Simulation results show the proposed soft-alignment algorithm with adaptive sounding time effectivelyscans the channel subspace of a mobile user by exploitingpolarization diversity.

Index Terms—Millimeter-wave wireless, Dual-polarized chan-nel, Beam-alignment algorithm.

I. INTRODUCTION

M ILLIMETER wave (mmWave) multiple-input multiple-output (MIMO) systems are a prime candidate to

allow future communication systems to provide the throughputenhancements needed to meet the expected demands for mobiledata [1]–[4]. Radio links operating over the wide bandwidthsavailable in the mmWave spectrum are a promising method ofproviding access for small cells within heterogeneous cellularnetworks (HCNs). When compared to lower frequencies,the higher expected path loss of mmWave requires greatersystem gains in the link budget [2], which may be attainedvia beamforming with multiple antenna systems. The smallwavelength of mmWaves allows for a dense packing of manyantennas in a small space. In effect, a steerable, phased-arrayarchitecture of many antennas can be controlled to form high-gain directional transmissions. However, mmWave systems mayhave only a small number of radio frequency (RF) chains due totheir high cost and power utilization. Therefore, beamformingin mmWave systems is performed mainly by using inexpensiveRF phase shifters in the analog domain [4].

In contrast to conventional cellular systems using one RFchain per antenna, analog beamformed or precoded mmWave

J. Song, J. Choi, S. G. Larew, and D. J. Love are with the School ofElectrical and Computer Engineering, Purdue University, West Lafayette, IN47907 (e-mail: {jihosong, choi215, sglarew, djlove}@purdue.edu).

T. A. Thomas and A. Ghosh are with Nokia Solutions and Networks, Arling-ton Heights, IL 60004 (e-mail: {timothy.thomas, amitava.ghosh}@nsn.com).

systems cannot observe the channel of each receive antennadirectly since the incident waves at each antenna are combinedin the analog domain. Moreover, the large number of antennasand the high path loss at mmWave frequencies make it difficultto get enough samples to compute meaningful channel estimatesfor each receive antenna. Instead, one approach is to soundthe channel subspace a finite number of times in an attempt toalign the beamformer to the channel subspace.

Several beam-alignment techniques for mmWave systemshave been developed based on hard-decision beam-alignment(hard-alignment) techniques [5]–[9]. In the hard-alignmentalgorithms, a set of candidate training vectors are used toscan the channel subspace. The chosen beamformer for datatransmission comes from the same set for scanning channelsubspace. Hard-alignment sampling algorithms taking a “ping-pong” beam search approach are studied for uniform lineararray (ULA) scenarios in [5], [6] and extended to dual-polarizedarrays in [7], [8], which show better performance than thestandardized technique in [9]. However, the hard-alignmenttechniques have fundamental limits on beam-alignment perfor-mance. In low signal-to-noise ratio (SNR) environments, thehard-alignment techniques exhibit a high probability of beammisalignment.

The dual-polarized channels and concepts discussed in [10]–[15] are expected to be incorporated with mmWave sys-tems. Measurements at mmWave frequencies show that theorthogonality of the polarizations remains high throughoutpropagation [16]. From a signal processing point-of-view, theindependent polarizations relax the Welch bound conditions thatconstrain the channel sounding time. Each desired polarizationcan be decoupled independently by using reference signal(RS) sequences which are designed to support high ranktransmission in systems such as LTE and LTE-Advanced [17]–[19]. Numerous advantages exist, including the ability to soundthe orthogonal polarizations in parallel for channel trainingas well as to multiplex information across the polarizationsor obtain a polarization diversity gain. Furthermore, a dual-polarized system may be more space efficient, allowing formore antennas in a small form factor. To the best of ourknowledge, outdoor mmWave systems over dual-polarizedchannels have not been reported yet except in our previouswork [7], [8].

For connecting small-cells and mobile users, we considera mmWave system employing dual-polarized antennas. Toalign the beam-direction of a large number of antennas, wedevelop a beam-alignment algorithm which operates well evenin the low SNR regime. First, with a simplified, dual-polarizedchannel model, an optimal set of combining and beamforming

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vectors at baseband and an analog beamforming vector underperfect channel state information (CSI) are discussed in orderto provide criteria for practical combining and beamformingsolutions. Then, we propose a soft-decision beam-alignment(soft-alignment) algorithm that exploits the dual-polarizedchannel. Based on maximum likelihood estimation (MLE),the beam-alignment algorithm estimates the channel subspacegiven our channel model assumptions, which consist of a sub-channel vector and a block matrix corresponding to propagationbetween two polarizations. One stage of sampling is performed,followed by a post-processing stage consisting of two roundsof beam-alignment. Based on the rough beam-direction of thesub-channel vector estimated in the first round, a more accuratebeam-direction is estimated in the second round.

Our second contribution is an adaptive algorithm whichselects the sounding time efficiently. We derive an approximateprobability of beam misalignment based on a beam patternanalysis of the codebook used for beam-alignment. With thisapproximate probability, the system is able to choose anefficient number of channel samples in order to ensure lowprobability of beam misalignment. Based on the formulation,the system changes the appropriate sounding time adaptivelyaccording to the channel environment to satisfy the predefinedcriterion. Despite the severe path loss at mmWave frequencies,the proposed algorithm effectively scans the channel subspacewith the minimum necessary number of sounding samples.

In Section II, we describe a mmWave system employing dual-polarized antennas. In Section III, a practical soft-alignmentalgorithm is proposed for dual-polarized mmWave systems. InSection IV, an adaptive sounding algorithm is developed for thesoft-alignment algorithm. In Section V, numerical results arepresented to verify the performance of the proposed algorithmsand Section VI details our conclusions.

Throughout this paper, C denotes the field of complexnumbers, CN denotes the complex normal distribution, Udenotes the uniform distribution, IN is the N × N identitymatrix,

[a, b]

is the closed interval between a and b, E[·] is theexpectation operator, (·)H is the conjugate transpose, 1[a,b) isthe indicator function, ‖ · ‖p is the p-norm, � is the Hadamardproduct, ⊗ is the Kronecker product, Aa,b, A(:, a), vec(A),∣∣A∣∣ denote (a, b)th entry, ath column vector, vectorization,and cardinality of the matrix A.

II. SYSTEM MODEL

A. System model

We consider a mmWave MIMO system over block fadingchannels between a base station and mobile users. The systememploys dual-polarized antennas where each array of antennasis divided evenly into two groups, one of vertically polarizedantennas and the other of horizontally polarized antennas. Thebase station has Mt transmit antennas and two RF chains,one for each polarization. Mobile users has Mr = 2 receiveantennas1 with an RF chain for each polarization. An overviewof the mmWave system is shown in Fig. 1.

1In this work, we consider a multiple-input single-output (MISO) channelsfor each polarization, e.g., single receive antenna for both vertical and horizontalreceive antenna groups.

Fig. 1. An overview of the mmWave system.

The input-output expression is represented by

y = zH(√ρHfs+ n

), (1)

where y is the received baseband signal, z ∈ CMr is theunit norm receive combining vector, ρ is the transmit SNR,H ∈ CMr×Mt is the block fading, dual-polarized channelmatrix, f ∈ CMt is the unit norm transmit beamforming vector,s ∈ C is the data symbol subject to the power constraintE[|s|2] ≤ 1 and n = [nv nh]T ∈ CMr ∼ CN (0, IMr

) is thenoise vector with independent and identically distributed (i.i.d.)entries for each polarization.

As will become clear when the dual-polarized channel isexamined, the transmit beamforming vector f is logically splitinto two unit-norm vectors corresponding to the analog beam-forming weights. Let fv ∈ C

Mt2 and fh ∈ C

Mt2 respectively be

the vertical and horizontal analog beamforming vectors. Thecomplete beamforming vector is then f = [vvf

Tv vhf

Th ]T , where

v = [vv vh]T ∈ C2 is a unit norm weight vector that performsbeamforming at baseband, such as maximum ratio transmission.The beam combining vector z = [zv zh]T ∈ C2 accounts forthe ability to perform arbitrary combining at baseband, suchas maximum ratio combining.

At the base station, the analog beamforming vectors areconstrained to a subset of C

Mt2 and the antennas themselves

are assumed to be arranged as a ULA. The set of verticalantennas forms a ULA with uniform element spacing and thehorizontal set is similarly arranged. The ability to control thegain and phase of each antenna at baseband is impractical dueto the high cost and power consumption of many individual RFchains [5]–[9]. Instead, analog beamforming is performed withRF phase shifters and no gain control [5]. The constrained setof analog beamforming vectors is

BN = {w ∈ CN : (wwH)`,` = 1/N, 1 ≤ ` ≤ N}. (2)

B. Dual-polarized channel model

The design goal of maximizing receive SNR is addressed byconsidering practical algorithms for choosing the combiningand beamforming vectors. The mmWave channel is differentthan conventional Rayleigh channel model which assumes arich scattering channel environment and hence an appropriatemodel is necessary [20], [21].

A general view of the dual-polarized channel follows frompartitioning the channel into a block matrix form as

H =

[hHvv hHvhhHhv hHhh

](3)

where hab is a sub-channel vector from polarization b to a.

3

(a) Dual-polarized MIMO scenario (b) Rotated-dual-polarized MIMO sce-nario

Fig. 2. Rotated dual-polarized MIMO channel.

A dual-polarized mmWave channel model is proposed in [7].However, this realistic channel model complicates analysis andhence a simpler channel model is needed. Due to the dominantsingle path observed in mmWave systems [16], we approximatethe mmWave channel with a line-of-sight (LOS) path whoseangle of departure (AoD) from the base station to the mobileuser is θ. Under the assumption that propagation between thedifferent polarization pairs only varies by a scalar, the blockmatrix representation in (3) may be written as

H ' A⊗ hH =

[αvv αvhαhv αhh

]⊗ hH , (4)

where h = aMt2

(θ) is the array response vector for the singlepath with AoD θ ∼ U

[θLB , θUB

]. The array vector set is

AN = {aN (θ) ∈ CN : θLB ≤ θ ≤ θUB} (5)

aN (θ) = [1 ej2πdλ sin θ · · · ej 2πd

λ (N−1) sin θ]T , (6)

where d is the antenna element spacing and λ is the wavelength.In (4), the polarization matrix A contains the gains and

phase shifts across polarizations as

A =

[ejϕvv cos ζ −ejϕvh sin ζejϕhv sin ζ ejϕhh cos ζ

], (7)

where ζ is the difference in orientation between the base stationand mobile users based on Malus’ law [12], [15] as shown inFig. 2 and ϕab is the phase of the ray from polarization b toa. We develop the sounding schemes based on the simplifiedchannel model in (4), while the realistic channel model in [7]is used for numerical simulations presented in Section V.

C. Channel sounding for beam-alignment

The system considered in this paper sounds the channel Ltimes with the training beamformers c[`] ∈ C

Mt2 , ‖c[`]‖22 = 1

2on each polarized antenna group. Denote the set of beamformersused for sounding as C =

{c[1], · · · , c[L]

}and let the matrix

of beamformers be C =[c[1] · · · c[L]

]∈ C

Mt2 ×L. The time

required to sound the channel should be much less than thecoherence time of the channel in order to maximize the usefultime for information transmission. In the `-th channel use,for the training signal on the vertical and horizontal antennagroups, the `-th beamforming vector c[`] is modulated byusing unit norm, precoded pilot sequences vv and vh. Underthe simplified channel in (4), the received training signals

on vertical and horizontal antenna groups before receive sidecombining are[

yHv [`]yHh [`]

]=√ρ(A⊗ hH

)([vHvvHh

]⊗ c[`]

)+

[nHv [`]nHh [`]

].

The precoded pilot sequences vv and vh can be constructedusing RS found in standards such as LTE and LTE-Advanced[17]–[19]. The receiver decouples the four pairs of polarizationsby using a different RS for each polarization. Demodulationis facilitated by the orthogonality of the RS. For example, thepopular Zadoff-Chu sequences could be used for the RSs [19].

The training sample from polarization b at the transmitter isrecovered by a filter matched to the RS sequence vector vb.The decoupled output for receive polarization a is

yab[`] = yHa [`]vb =√ραabh

Hc[`] + nab[`]. (8)

Let the observation vector for a polarization pair ab be

yab =[yab[1], · · · , yab[L]

]H=√ραHabC

Hh + nab ∈ CL, (9)

where the noise vector nab has i.i.d. CN (0, 1) entries.

III. BEAM-ALIGNMENT ALGORITHM

In order to make up for the deficit in the system linkbudget [5] due to various channel losses and low poweramplification, a mmWave system may employ beamformingand combining to increase the effective SNR. However, themethod of choosing a beamformer and combiner is non-trivialwhen the channel is not directly observable. Reliable operationof previously proposed beamforming algorithms [5]–[9] islimited to an unnecessarily high SNR range. Links that wouldotherwise be possible with good beamforming gain cannot beformed. The common factor that limits performance at lowSNR is the hard-decision being made in the algorithms.

It is natural to consider a soft-decision algorithm to improvebeamforming reliability and performance. Instead of choosingone of the beamformers from the set used for channel sounding(referred to as a hard-decision), a soft-decision algorithmmay choose any feasible beamformer. In the hard-decisionalgorithms, the probability of choosing the optimal beamformeris fundamentally limited by the SNR ρ. More observations leadto a higher probability of choosing a good beamformer, but theprobability of choosing the optimal beamformer is eventuallylimited by the SNR and not the number of observations. Onthe other hand, the performance of our soft-decision algorithmis not fundamentally limited by the SNR and hence can scalewith the number of channel sounding observations.

A. Full CSI beam-alignment

Before investigating practical mmWave beamforming so-lutions, we pause to consider optimal beamforming vectorsunder perfect CSI conditions for the simplified channel modelin (4). The full CSI beamforming solution gives an upperbound on performance for practical systems under the assumedsystem and channel model and it offers insight on the designof practical beamforming algorithms.

4

In order to maximize the achievable rate of the system,the magnitude of the effective channel gain should be maxi-mized [22] by choosing optimal beamforming and combiningvectors. The maximizers of the effective gain is

(z, f) = arg maxz∈C2,f∈CMt

|zHHf |2. (10)

Keep in mind that z and f are unit norm. The solution to (10)is well known to be the dominant left and right singular vectorsof H. Taking a closer look, the singular value decomposition(SVD) of H is

H = (UAΣAVHA)⊗ (UhΣhVH

h )

where UAΣAVHA is the SVD of A, and UhΣhVH

h is theSVD of hH . Considering the constraints on the basebandbeamforming vector fA, the analog beamforming vector fhand the channel model in (4), the constrained maximizers are(zA, fA, fh

)= arg max

(zA,fA,fh)∈C2×C2×BMt2

∣∣(zHAAfA)⊗ (hHfh)∣∣2 .

The optimal solutions for zA and fA are the dominant left andright singular vectors of A, respectively (i.e., zA = UA(:, 1),fA = VA(:, 1)), because the combining and beamformingvectors zA and fA at baseband have no equal gain constraint.On the other hand, fh is restricted by the equal gain constraint.An optimal solution is fh = h

‖h‖2 since h is assumed to be anarray manifold vector in (6), which has equal gain entries.

Optimal and feasible combining and beamsteering vectorsin (10) are given by

z = zA, f = fA ⊗ fh. (11)

The unit norm weight vectors z and v that perform combiningand beamforming at baseband and the analog beamformingvectors fv = fh at RF phase shifters on both verticaland horizontal antenna groups, defined in the input-outputexpression of (1), correspond to zA, fA, and fh in (11),respectively.

The beamformer for a sub-channel vector in the arraymanifold set is better described as a beamsteering vector.Furthermore, the algorithm of choosing a good beamsteeringvector is better described as a beam-alignment algorithm. Whenthe beamformer is chosen from the array manifold set, themainlobe of the gain pattern is being steered to maximize gainin a specific direction. In effect, narrow beams are created thatmust be aligned to maximize the effective channel gain.

B. Beam-alignment with noisy channel sounding

We now present a beam-alignment algorithm that seeks tomaximize the effective channel gain without direct knowledgeof the channel. The general idea is to observe the basebandoutput responses yab in (9) resulting from a set of trainingvectors C and then choose a beamformer in a finite set E giventhe observations.2

2The choice of beamformer in E for beam-alignment is based on themultilevel codebook in [5], [6].

1) Limits for hard-alignment: Previously proposed hard-alignment algorithms show good performance at high SNRbut are unreliable at low SNR [5]–[8]. In a hard-decisionalgorithm, the chosen beamformer is restricted to be one ofthe beamformers used for channel sounding (i.e., E = C). Theestimated beamforming vector f is chosen to be the trainingvector c[ˆ] ∈ E with test sample yab[`] in (8) as

ˆ= arg max`∈{1,...,L}

|√ραabhHc[`] + nab[`]|2 (12)

.= arg max`={1,...,L}

∣∣mhab[`] + nhab[`]

∣∣2,where mh

ab[`] denotes the correlation between c[`] and thesub-channel vector h to be estimated, and nhab[`] ∼ CN (0, 1)is additive random noise that hinders good beam-alignment.

The variance in the estimation of h in (12) is related to themean and variance of the test sample yab[`], i.e., E[mh

ab[`]]and E[|nhab[`]|2] = 1. To perform good beam-alignment, thevariance of noise nhab[`] should be small in comparison withthe power of mh

ab[`] which is given by

|mhab[`]|2 =

ρMt

2|αab|2|hHc[`]|2 ≤ ρMt

4|αab|2, (13)

where h = h‖h‖2

is the normalized sub-channel vector, ‖h‖22 =Mt

2 , and |hHc[`]|2 ≤ 12 . Notice that increasing the number of

samples L does not increase the upper bound. Thus, adaptivelychoosing L in a hard-alignment algorithm does not help at lowSNR, with fixed Mt.

2) Maximum likelihood soft-alignment: In the proposed soft-alignment algorithm, for beam-alignment, multiple soundingsamples yab in (9) are considered together, while each soundingsample yab[`] in (8) is considered separately in the hard-alignment algorithm. Specifically, the algorithm chooses thebeamformer in a predefined codebook E that maximizesthe likelihood function of the received samples. In the soft-alignment algorithm, training vectors and beamformers do notneed to be in the same set (i.e., E 6= C), in contrast to thehard-alignment case where E = C. The number Q of candidatecodewords in E may be much larger in size than the numberof sounding samples L, whereas Q ≤ L in hard-alignment.

In the proposed soft-alignment algorithm, the channelsounding and beam-alignment are conducted in separate stages.In the first stage, the algorithm performs subspace samplingusing the training vectors in the set C and collects the soundingsamples in the vectors yab for each polarization pair as in (9).In the second stage, the beamformer from the codebook E ischosen along with the baseband beamformer.

The training vectors in the set C are important but havenot yet been discussed. For our soft-alignment algorithm,we use omni-directional training vectors, in contrast to otherworks [5]–[9]. The set of training vectors C is designed tomeet the criterion of the Welch bound with equality trainingsequences for omni-directional sounding as in [23] (i.e.,CCH = L

MtIMt

2, L ≥ Mt

2 ). To satisfy the criterion, thetraining vectors are designed by using an array vector withuniform phase shift of 2π

L based on [24] and are given by

c[`] =1√Mt

[1 ej

2πL ` · · · ej 2π

L (Mt2 −1)`

]T

5

for ` = 1, . . . , L. Note that, in comparison with a single-polarized system, the criterion of the Welch bound with equalitytraining sequences is relaxed from L ≥Mt to L ≥ Mt

2 due tothe parallel operation of vertical and horizontal sounding.

In the second stage, the complex channel gains αab andthe sub-channel vector h in (4) are estimated by using theobservation vectors yab in (9). Typically, in the conventionalcellular systems, a minimum mean square error (MMSE)estimate of the channel is computed given the observationvectors [23]. However, using the MMSE estimate of the channelis limited due to little or more likely no control over the gainof each individual antenna in a mmWave system.

Our proposed soft-alignment algorithm maximizes the like-lihood function of the observation vector instead of estimatingthe channel with an MMSE estimator. The high Ricean K-factor exhibited in mmWave channels [20] limits the spaceof practical beamforming vectors that must be searched. Thisenables mmWave systems to effectively perform MLE.

Initially, we will derive the MLE for a single polarizationpair ab, which is given by

(αab, h) = arg max

αab∈C,h∈CMt2

f(yab|ρ, αab, h,C),

where f(yab|ρ, αab, h,C) is the probability distribution func-tion (pdf) for yab defined as

f(yab|ρ, αab, h,C) =1

πLexp (−‖yHab −

√ραabh

HC‖22).

The maximization of the pdf is equivalent to the minimization

(αab, h) = arg min

αab∈C,h∈CMt2

‖yHab −√ραabh

HC‖22 (14)

which needs to be minimized over αab and h.First, we consider the channel gain αab. To estimate αab,

we differentiate the objective function over αHab as

∂

∂αHab‖yHab −

√ραabh

HC‖22 = ραab‖CH h‖22 −√ρyHabC

H h.

Then, the channel gain MLE is derived as

αab =yHabC

H h√ρ‖CH h‖22

= αabhH h

‖h‖2+

√2

ρLnab. (15)

Note that the effective channel hH h influences the accuracyof the channel gain estimator. Moreover, the variance of noisecomponent decreases as L increases.

By plugging in the estimated channel gain αab, the MLEfor the sub-channel vector becomes

h = arg max

h∈CMt2

|yHabCH h|2

‖CH h‖22. (16)

The size of the feasible space for the MLE in (16) can besignificantly reduced from the full Mt

2 dimensional complexspace. Under the high Ricean K-factor and ray-like propagationassumptions, the sub-channel vector is simply an array responsevector (i.e., h ∈ AMt

2). This assumption simplifies the

maximization of the likelihood function by not requiring asearch over the large set of all feasible channel vectors. Instead,a practically sized set of equal-gain beamsteering vectors

E = {e1, . . . , eQ} ⊂ BMt2

may be used as the feasible setfor the maximization. With the feasible set E , the estimatedchannel vector in (16) is h = ‖h‖2eq where

q = arg maxq∈{1,...,Q}

|yHabCHeq|2

‖CHeq‖22.= arg maxq∈{1,··· ,Q}

|tab[q]|2. (17)

In (17), test sample tab[q] is divided into two components

tab[q] =

√ραabh

HCCHeq

‖CHeq‖2+

nHabCHeq

‖CHeq‖2

=

√ρL

2αabh

Heq +nHabC

Heq‖CHeq‖2

.= ms

ab[q] + nsab[q].

The mean and variance of tab[q] are E[msab[q]] and

E[|nsab[q]|2] = 1. The power of msab[q] is

|msab[q]|2 =

ρL

2|αab|2G[q] ≤ ρL

2|αab|2, (18)

where G[q].=∣∣hHeq

∣∣2 ≤ 1 is the beamforming gain betweenthe normalized sub-channel and the codeword vector.

At this point, it is interesting to compare the maximizationin (17) to the hard-decision maximization in (12). If L = Mt

2 ,the power of ms

ab[q] in (18) is the same as for the hard-decisionalgorithms in (13). However, L is a system parameter that maybe varied to control the upper bound in (18). A method ofefficiently selecting L will be developed in Section IV.

3) Optimal combining for MLE: Until now, the beam-alignment algorithm has only considered a single observationvector yab. The estimation of h may be improved by linearlycombining the observation vectors yab,

[yvv yvh yhv yhh]

= vec(A)H ⊗ (√ρCHh) + [nvv nvh nhv nhh] ∈ CL×4.

Let the linear combination of yab and some unit-norm vectorb give the combined observation vector y, explicitly written asy = [yvv yvh yhv yhh]b. The combined observation vectory is a Gaussian random vector. Then, the MLE for h is

h = arg max

h∈CMt2

|yHCH h|2

‖CH h‖22.

The question remains on how to choose the combiner b.Optimal combining is achieved when b = vec(A)

‖vec(A)‖2 . However,the optimal combiners rely upon some knowledge of A. If A isnot known or cannot be accurately estimated, the performanceof these combining schemes suffers especially at low SNR.

4) Joint MLE: Instead, we develop the joint MLE for Aand h. The likelihood function that will be maximized is

f(yvv,yvh,yhv,yhh|ρ,A,h,C) =∏ab

f(yab|ρ, αab,h,C),

which can be manipulated in a similar manner as in SectionIII-B2 to give the MLE for h = ‖h‖2eq , where q is

q = arg maxq∈{1,...,Q}

∑a,b∈{v,h}

|yHabCHeq|2

‖CHeq‖22(19)

= arg maxq∈{1,··· ,Q}

∑a,b∈{v,h}

|tab[q]|2.

6

A maximizer in (19) is found by searching through a codebookE as was done in (17). Finally, the joint MLE for the entriesof A is identical to (15) with h = h.

C. Extension to multi-round alignment

We fully elaborate our proposed soft-alignment algorithmbased on the previous results. In the beam-alignment stage,after receiving sounding samples yab in (9), the algorithmperforms multiple rounds of soft-alignment, all with the same Lsounding samples.3 Each successive round of beam-alignmentrefines the previous choice by using larger codebooks withnarrower beamformers.4 Note that multiple rounds of beam-alignment only increases the computational complexity andnot the overhead of channel sounding.

In the first round of beam-alignment, the system makes apreliminary decision for the beamforming vector correspondingto the sub-channel vector. The first level codebook E1 for thefirst round contains Q1 = Mt

2 orthogonal codewords whichare designed to satisfy eHc ed = 0, c 6= d. This assumptionguarantees the likelihood samples are uncorrelated sinceE[|tab[c]Htab[d]|2] = E[|eHc ed|2] = 0. Recall the assumptionthat the sub-channel vector h is accurately described by an arrayresponse vector in (6). Since the beam-width of the channelvector is quite narrow for large Mt, the channel vector might beorthogonal to all codewords except the properly estimated beam-alignment codeword and its neighbor codewords. We addressthe range that is covered by the optimal codeword eq and itsneighboring codewords eq±1 as a rough beam-direction. Notethat q = arg maxq∈{1,··· ,Q}

∣∣hHeq∣∣2 is the index for the code-

word which is optimally aligned under perfect CSI condition ofh. Beyond the rough beam-direction, we can approximate thepower of the other likelihood samples as |tab[q]|2 ' |nsab[q]|2since hHeq ' 0, q ∈ {1, · · · , Q1}/{q, q ± 1}.

In later rounds of beam-alignment, only beamformers whichcover the range of the rough beam-direction from the previousrounds are considered, allowing for more accurate beam-alignment over many iterations. For example, the secondlevel codebook E2 should be larger than the first, with eachbeamformer covering a smaller area. In this work, the number ofcodewords in E2 is set to Q2 = 2Mt. Only the set of codewordswhich cover the rough beam-direction from the previous roundare selected as the feasible set for beam-alignment. With theestimated sub-channel vector h and the block matrix A, mobileuser computes an optimal combining and beamforming vectoran identical way to the full CSI case in (11).

IV. ADAPTIVE CHANNEL SOUNDING FORBEAM-ALIGNMENT ALGORITHM

In the proposed soft-alignment algorithm, beam-alignmentperformance is proportional to the sounding time L. Toguarantee good beam-alignment performance, L should be

3In this work, the same L samples are used during all rounds of beam-alignment, while in [5], [6] a separate sampling is performed for each round.

4Having multiple rounds for beam-alignment may increase the beam-alignment performance. However, by considering practical issues, e.g., alimited length of the coherence time of the block fading channel and thecomputational complexity, we only consider two rounds of beam-alignment.

a large number. However, a large number of sounding samplesimposes a heavy burden on the overhead of the system,especially in the case of a fast fading channel. To handlethis trade-off problem, the base station needs to adaptivelyselect L based on the channel environment of the users. In thissection, an approximate probability of beam misalignment isderived to aid in choosing L. Based on the error probability,we propose an adaptive sounding algorithm which adjusts Laccording to the channel environment.

A. Probability of beam misalignment

In the proposed algorithm, a more accurate beam-directionis estimated in the second round based on the rough beam-direction estimated in the first round of beam-alignment, asdefined in Section III-C. When the estimated rough beam-direction does not contain the beam-width covered by theoptimal codeword, the system fails to align the beamformer tothe channel subspace. Thus, we define the event q 6∈ {q, q±1}as the beam misalignment event.

The probability of beam misalignment is defined as

Pmis.= Pr

({q 6∈ {q, q ± 1}}

)= Pr

(maxq,q±1

|t[q]|2 ≤ maxq∈{1,··· ,Q1}/{q,q±1}

|t[q]|2)

< Pr(|t[q]|2 ≤ max

q∈{1,··· ,Q1}/{q,q±1}|t[q]|2

).= Pup

mis,

where the upper bound Pupmis follows from maxq=q,q±1 |t[q]|2 ≥

|t[q]|2. Note that in this section, the polarization index ab andthe index s, which denotes the soft-alignment algorithm, aredropped for simplicity.

As discussed in Section III-C, we assume |t[q]|2 '|n[q]|2, q ∈ {1, · · · , Q1}/{q, q ± 1}. Under this assumption,Pup

mis is approximated as

Pupmis ' Pr(X− Z ≤ 0), (20)

X.= |m[q] + n[q]|2,

Z.= maxq∈{1,··· ,Q1}/{q,q±1}

|n[q]|2.

In (20), X is the power of the optimal likelihood sample andZ is the maximum noise power among Q1 − 3 noise samples.

First, X follows the noncentral chi-squared distribution withtwo degrees of freedom [25]. The cumulative distributionfunction (cdf) of X is

FX(x) =(

1−Q1

(√2µ2

x,√

2x))1[0,∞)(x), (21)

where µ2x = E[|m[q]|2] is the noncentrality parameter of X

and Q1(·, ·) is the Marcum Q-function.The power of each noise sample N

.=∣∣n[q]

∣∣2 follows thecentral chi-squared distribution with two degrees of freedom[25]. The cdf of N is defined as

FN(n) =(1− e−n

)1[0,∞)(n).

Then, the cdf of Z is derived with the binomial series expansion,

FZ(z) =(FN(z)

)(Q1−3)=

Q1−3∑q=0

(Q1 − 3

q

)(−1)qe−qz.

7

Based on the distribution of X and Z, an upper bound onthe probability of beam misalignment in (20) is derived as

Pupmis

.= 1− Pr(X− Z > 0)

= 1−∫ ∞

0

(∫ x

0

fZ(z)dz

)fX(x)dx

= 1−∫ ∞

0

(Q1−3∑q=0

(Q1 − 3

q

)(−1)qe−qx

)fX(x)dx

= 1−Q1−3∑q=0

(Q1 − 3

q

)(−1)q

∫ ∞0

e−qxfX(x)dx

=

Q1−3∑q=1

(Q1 − 3

q

)(−1)q+1MX(−q)

=

Q1−3∑q=1

(Q1 − 3

q

) (−1)q+1 exp(−µ2

xq1+q

)1 + q

, (22)

where MX(t) = E[etX]

=exp

µ2xt

1−t1−t for t < 1 is the moment-

generating function in [25].

B. Adaptive sounding algorithm for soft-alignment algorithm

In this work, the sounding time L for the proposed soft-alignment algorithm is adapted according to the probability ofbeam misalignment in (22). To develop an adaptive soundingalgorithm, the noncentrality parameter µ2

x in (21) for theprobability of beam misalignment should be derived first. Thenoncentrality parameter µ2

x is a function of the expectationof the channel gain entries in (7) (i.e., E[|α|2] = 1

2 ) and theexpectation of the beamforming gain defined as

G[q] = |hHeq|2. (23)

We derive an upper bound for the expected beamforming gainE[G[q]

]< ς2 in Lemma 2 of Appendix A. Then, a tighter

upper bound ν for the noncentrality parameter µ2x, which is

a function of ς2, is defined by substituting the upper boundς2 into E

[G[q]

]. In the proposed algorithm, we use this upper

bound ν ≥ µ2x as a noncentrality parameter instead of µ2

x.With a predefined target error probability ε, the sounding

time L is adaptively adjusted as a function of the SNR ρ.Specifically, the sounding time L should satisfy

L = arg maxL∈N

Pupmis(ρ, L) s.t. Pup

mis(ρ, L) < ε (24)

where ν is used as an approximation of µ2x in Pup

mis(ρ, L). Thefinal adaptive sounding time L must satisfy the Welch boundwith equality (L ≥ Mt

2 ) in [23] and is therefore chosen to be

L = max

(L,Mt

2

).

V. SIMULATION RESULTS

In this section, we present numerical performance resultsfor the proposed algorithm, which combines the soft-alignmentestimation algorithm with the adaptive channel sounding. Weconsider two performance metrics: 1) expected beamforming

(a) Mt = 32, Mr = 2 (b) Mt = 64, Mr = 2

Fig. 3. Data rate of soft and hard beam-alignment algorithm against soundingtime L (Mt = 32, 64, Mr = 2, χ = 0.2).

TABLE IBEAM-ALIGNMENT SCHEMES COMPARED IN FIG. 4.

ρ = [−10,−8,−6,−4,−2, 0, 2] dBProposed Alg. Mt = 32, L = [93 59 37 24 16 16 16]

w. adaptive sounding Mt = 64, L = [129 81 52 33 32 32 32]Hard-align. Mt = 32, L = [93 59 37 24 16 16 16]

w. adaptive sounding Mt = 64, L = [129 81 52 33 32 32 32]Hard-align w. multiple Fixed sounding time

round sampling [5] Mt = 32, 64, L = 60

gain GBF and 2) expected data rate R. The metrics are definedas

GBF = E[∣∣zHHf

∣∣2/∣∣zHHf∣∣2], (25)

R = E[

log2

(1 + ρ

∣∣zHHf∣∣2)], (26)

where z, f are the estimated combining and beamformingvectors and z, f are the optimal combining and beamformingvectors.

The numerical results were obtained from Monte Carlosimulations of over 10000 independent channel realizationswith the following parameters. For the simulations, we adoptthe realistic channel model in [7], [8]. We consider a streetgeometry [20] with a LOS path and three first order reflectedpaths from both the wall of buildings and the ground. Weassume the upper and lower bounds of the range of AoD areθUB = −θLB = π

3 [5] and ζ ∼ U[0, π2 ]. The Ricean K-factoris set to 13.5dB based on the channel measurements in [16].The reciprocal of the cross polar discrimination value, whichrepresents the ability to distinguish the polarization differenceof the antennas [26], is set to χ = 0.2. We assume 6 bit phasecontrol registers in the analog beamforming hardware.

In Figs. 3a and 3b, the data rate of the proposed soft-alignment algorithm is compared with that of the hard-alignment algorithm with multiple round ping-pong samplingin [5] against total sounding time L. In these simulations, theadaptive channel sounding, which adjusts the sounding time, isnot applied to the soft-alignment algorithm. In Figs. 3a and 3b,the simulation results are presented in different cases of SNR(ρ = −8,−10,−12 dB). It is clear that in the low SNR regime,the soft-alignment algorithm scans the channel subspace betterthan the hard-alignment algorithm.

In Figs. 4a and 4b, the beamforming gain of the beam-alignment algorithms is compared against SNR ρ. In theproposed algorithm, the sounding time L is computed basedon the inequalities of the adaptive sounding algorithm in (24)

8

(a) Mt = 32, Mr = 2 (b) Mt = 64, Mr = 2

Fig. 4. Beamforming gain of proposed and hard beam-alignment algorithmagainst SNR (Mt = 32, 64, Mr = 2, χ = 0.2).

(a) Mt = 32, Mr = 2 (b) Mt = 64, Mr = 2

Fig. 5. Beamforming gain of proposed algorithm against SNR with differenttarget error probabilities (Mt = 32, 64, Mr = 2, χ = 0.2).

taking ρ, Mt, and ε = 0.60 into account. In Table I, thecomputed sounding times based on the proposed algorithm andhard-alignment algorithm with adaptive sounding algorithmare summarized as a function of SNR, sorted in ascendingorder (from −10dB to 2dB). In addition, the sounding time ofthe hard-alignment algorithm with multiple round ping-pongsampling in [5] is set to L = 60 in our simulations. It isclear that the proposed algorithm shows better performancethan the hard-alignment algorithms, especially in low SNRenvironments.

In Figs. 5a and 5b, the beamforming gains of the proposedalgorithm with various target error probabilities are comparedagainst SNR. In the simulations, target error probabilities ε areset to 0.45, 0.50, 0.55, and 0.60. Note that sounding time Lis computed under different target error probabilities based onthe proposed algorithm. It is shown that the beamforming gainperformance is maintained according to the predefined targeterror probabilities. Note that sounding time is restricted bythe criterion of the Welch bound L ≥ Mt

2 for the SNR above−2dB in Fig. 5a and −4dB in Fig. 5b.

VI. CONCLUSION

In this paper, we propose practical beam-alignment algo-rithms for mmWave MIMO systems employing dual-polarizedantennas. We first propose a soft-alignment algorithm for adual-polarized wireless channel. The soft-alignment algorithmscans the channel subspace and relaxes the criterion of theWelch bound with equality training sequences by exploitingpolarization diversity. We also propose an adaptive soundingalgorithm that selects an efficient amount of sounding time

Fig. 6. Quantized sector for each codeword.

by considering the channel environment. It is shown that theproposed algorithm effectively samples the channel subspaceof mobile users even in low SNR environments.

APPENDIX AEXPECTATION OF BEAMFORMING GAIN

In this section, the expectation of beamforming gain E[G[q]

]is considered. As mentioned before, we design the codebookE1 with Q1 = Mt

2 codewords based on [5], [6]. In thecodebook, when Q1 = Mt

2 , each codeword is represented

by the normalized array response vector√

2Mt

aMt2

(θ) in(6). Therefore, the beamforming gain G[q] in (23) betweenthe normalized sub-channel vector h and the q-th codewordeq = 1√

NaN (θq) with N elements is given by [27]

G[q] =|aHN (θq + φq)aN (θq)|2

N2=

1

N2

sin2(πηN2 )

sin2(πη2 )

.= Γ(|η|)

(27)

where θq is the AoD beam-direction of the q-th codeword, φqis the beam-direction difference between the q-th codewordand the sub-channel vector and η .

= sin θq − sin(θq + φq) isthe beam-direction difference in the ψ-domain in Fig. 6.

In the codebook of [5], [6], the beam-width of each codewordis considered in the ψ-domain. To optimize the codebook, thebeam-widths of each codeword is equally divided in the ψ-domain as 2T

.= sin θUB−sin θLB

Q where θUB and θLB are theupper and lower bounds of the entire range, respectively. Then,the upper beam-width ∆u,q and the lower beam-width ∆l,q ofeach codeword is given by

sin(θq + ∆u,q)− sin θq = T, (28)sin θq − sin(θq + ∆l,q) = T. (29)

In the q-th quantized sector, ∆u,q and ∆l,q are dependent onthe beam-direction θq of the codeword eq . In order to computethe expected value of the beamforming gain, the beam-widthof each sector needs to be represented by the beam-directionθq. The beam-width of each codeword in the θ-domain isapproximated in the following lemma.

Lemma 1: The upper and lower beam-widths of each code-word may be approximated as

∆θq.= ∆u,q = ∆l,q '

T

cos θq,

9

when the beam-width is sufficiently small.Proof: First, we consider the upper beam-width of q-th

sector with eq . The upper beam-width ∆u,q is derived as

T = sin θq cos(∆u,q) + cos θq sin(∆u,q)− sin θq

' sin θq

(1−

∆2u,q

2

)+ cos θq∆u,q − sin θq

' cos θq∆u,q.

The first approximation is based on the small-angle approxima-tion of trigonometric functions (sin ∆ ' ∆, cos ∆ ' 1− ∆2

2 )since the half beam-width ∆u,q is designed to have a suffi-

ciently small value. Then, we drop the∆2u,q

2 term since ∆2u,q

approaches 0.The lower beam-width in (29) is approximated in the same

way and has an identical value. The upper and lower beam-width of the q-th sector in the θ-domain is approximated as

∆θq.= ∆u,q = ∆l,q '

T

cos θq.

The upper and lower beam-width are functions of the beam-direction θq and the half beam-width T in the ψ-domain.

With the approximate beam-width in Lemma 1, the expec-tation of beamforming gain E[Γ(|η|)] is upper bounded. Thebeamforming gain is defined with two random variables, θqand φq in (27). Because it is hard to derive E[Γ(|η|)] directly,we derive an upper bound of E[Γ(|η|)].

Lemma 2: An upper bound of E[Γ(|η|)] is

E[G[q]

]= E

[Γ(|η|)

]≤ Γ

(E[|η|]

)' Γ

(T

2

).= ς2.

Proof: In this proof, we drop the codeword index termq for simplicity. The upper bound of E[Γ(|η|)] is computedbased on Jensen’s inequality since Γ(|η|) is a concave functionwith regard to |η|. |η|, which represents the beam-directiondifference between two array manifold vectors, is written as

|η| ={

sin(θ + φ)− sin θ, 0 ≤ φ ≤ ∆θ

sin θ − sin(θ + φ), −∆θ ≤ φ < 0

where ∆θ ' Tcos θ is the approximate half beam-width which

is defined in Lemma 1. We assume φ ∼ U[−∆θ ∆θ].First, we compute the expected value of sin(θ + φ)− sin θ

when 0 ≤ φ ≤ ∆θ. The expected value is derived as follows

Eθ,φ[sin(θ + φ)− sin θ]

=

∫ θUB

−θUB

(∫ ∆θ

0

(sin(θ + φ)− sin θ

)f(φ)dφ

)f(θ)dθ

=

∫ θUB

−θUB

(2

∆θsin(θ +

∆θ

2) sin(

∆θ

2)− sin θ

)f(θ)dθ

'∫ θUB

−θUB

(sin(θ +

∆θ

2)− sin θ

)f(θ)dθ

=1

2θUB

∫ θUB

−θUB

(sin θ

(cos(

∆θ

2)− 1

)+ cos θ sin(

∆θ

2)

)dθ

' 1

2θUB

∫ θUB

−θUB

(sin θ

(∆θ

2 )2

2+ cos θ

∆θ

2

)dθ

' 1

2θUB

∫ θUB

−θUBcos θ

T

2 cos θdθ =

T

2.

(a) N = 16, θUB = −θLB = π3

(b) N = 32, θUB = −θLB = π3

(c) N = 16, Q = 32, θUB = −θLB (d) N = 16, Q = 32, θUB = −θLBFig. 7. Beamforming gain analysis of codebook [5], [6].

The first and second approximations are based on the smallangle approximation technique for trigonometric functions

(sin(∆θ

2 ) ' ∆θ

2 , cos(∆θ

2 ) ' 1− (∆θ2 )2

2 when ∆θ

2 is sufficiently

small). In the third approximation, the term (∆θ2 )2

2 is droppedsince (∆θ

2 )2 approaches 0. In the last approximation, ∆θ isreplaced by T

cos θ which is defined in Lemma 2.The approximation of E[sin θ − sin(θ − φ)] when −∆θ ≤

φ < 0 is derived in the same way and it is identical with thatof E[sin(θ + φ)− sin θ]. The expectation of |η| is then

E[|η|] ' T

2.

In Fig. 7, the approximate upper bound of E[Γ(|η|)] iscompared with the simulation results with the codebook. In Figs.7a and 7b, the beamforming gain of the codebook is comparedagainst the number of codewords under θUB = −θLB = π

3 .In Figs. 7c and 7d, the beamforming gain of the codebookis compared against the bounds of possible AoDs. Note thatthe simulation results for E[Γ(|η|)] are upper bounded by theformulation ς2 in Lemma 2.

REFERENCES

[1] A. Ghosh, T. A. Thomas, M. C. Cudak, R. Ratasuk, P. Moorut, F. W.Vook, T. S. Rappaport, G. R. MacCartney, Jr., S. Sun, S. Nie, “Millimeterwave enhanced local area systems: a high data rate approach for futurewireless networks,” to appear in IEEE Journal on Selected Areas inCommunications, 2014.

[2] Z. Pi, and F. Khan, “An introduction to millimeter-wave mobile broadbandsystems”, IEEE Commun. Mag., vol. 49, no. 6, pp.101 -107 June 2011.

[3] T. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G.Wong, J. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobilecommunications for 5G cellular: it will work!,” IEEE, Access, vol. 1,pp. 335-349, May. 2013.

[4] W. Roh, J. Seol, J. Park, B. Lee, J. Lee, Y. Kim, J. Cho, K. Cheun, and F.Aryanfar, “Millimeter-wave beamforming as an enabling technology for5G cellular communications: theoretical feasibility and prototype results,”IEEE Commun. Magazine, vol. 52, no. 2, pp. 106-113, Feb. 2014.

10

[5] S. Hur, T. Kim, D. J. Love, J. V. Krogmeier, T. A. Thomas, and A. Ghosh,“Millimeter wave beamforming for wireless backhaul and access in smallcell networks,” IEEE Trans. Commun., vol. 61, no. 10, pp. 4391-4403,Oct. 2013

[6] S. Hur, T. Kim, D. J. Love, J. V. Krogmeier, T. A. Thomas, and A. Ghosh,“Multilevel millimeter wave beamforming for wireless backhaul,” in Proc.IEEE Globecom Workshops, Dec. 2011.

[7] J. Song, S. G. Larew, D. J. Love, T. A. Thomas, and A. Ghosh, “Millimeterwave beam-alignment for dual-polarized outdoor MIMO systems,” in Proc.IEEE Globecom Workshops, Dec. 2013.

[8] J. Song, S. G. Larew, D. J. Love, T. A. Thomas, and A. Ghosh, “Millimeterwave beamforming for multiuser dual-polarized MIMO systems,” in Proc.IEEE Globalsip, Dec. 2013.

[9] IEEE, “IEEE802.11-10/0433r2, PHY/MAC complete proposal specifica-tion (TGad D0.1),” 2010.

[10] C. Oestges, V. Erceg, and A. J. Paulraj, “Propagation modeling of MIMOmultipolarized fixed wireless channels,” IEEE Trans. Veh. Technol., vol.53, pp. 644-654, May 2004.

[11] H. Asplund, J. Berg, F. Harrysson, J. Medho, and M. Riback, “Propagationcharacteristics of polarized radio waves in cellular communications,” inProc. IEEE Veh. Tech. Conf. Fall, Sep. 2007.

[12] L. Jiang, L. Thiele, and V. Jungnickel, “On the modelling of polarizedMIMO channel,” in Proc. Europ. Wireless Conf., Apr. 2007.

[13] M. Coldrey, “Modeling and capacity of polarized MIMO channels,” inProc IEEE Veh. Tech. Conf. Spring, May 2008.

[14] T. Kim, B. Clerckx. D. J. Love, and S. J. Kim, “Limited feedbackbeamforming systems in dual-polarized MIMO channel,” IEEE Trans.Commun., vol. 9, no. 11, pp. 3425-3439, Nov. 2010.

[15] J. Choi, B. Clerckx, and D. J. Love, “Differential codebook for generalrotated dual-polarized MISO channels,” in Proc. IEEE Globecom, Dec.2012.

[16] Z. Muhi-Eldeen, L. Ivrissimtzis, and M. Al-Nuaimi, “Modelling and mea-surements of millimeter wavelength propagation in urban environments,”IET Microwaves Antennas & Propagation, vol. 4, no. 9, pp. 1300-1309,2010.

[17] Y. -H. Nam, Y. Akimoto, Y. Kim, M. I. Lee, K. Bhattad, and A.Ekpenyong, “Evolution of reference signals for LTE-Advanced systems,”IEEE Commun. Mag., vol. 50, no. 2, pp. 132-138, Feb. 2012

[18] S. Sesia, I. Toufik, and M. Baker, LTE, the UMTS long term evolu-tion, 2nd ed. Wiley, 2011.

[19] 3GPP, TS 36.211 v10.1.0, Physical channels and modulation, Mar. 2011.[20] H. Zhang, S. Venkateswaran, and U. Madhow, “Channel modeling and

MIMO capacity for outdoor millimeter wave links,” in Proc IEEE WirelessCommun. and Networking Conf. (WCNC), pp. 1-6, Apr. 2010.

[21] T. A. Thomas, H. C. Nguyen, G. R. MacCartney, Jr., and T. S. Rappaport,“3D mmWave channel model proposal,” in Proc IEEE VTC-Fall, 2014.

[22] D. J. Love and R. W. Heath, Jr., “Equal gain transmission in multiple-input multiple-output wireless systems,” IEEE Trans. Commun., vol. 51,pp. 1102-1110, Jul. 2003

[23] W. Santipach and M. L. Honig, “Optimization of training and feedbackoverhead for beamforming over block fading channels,” IEEE Trans. Inf.Theory, vol. 56, no. 12, pp. 6103-6115, Dec. 2010.

[24] A. Sayeed and N. Behdad, “Continuous aperture phased MIMO: basictheory and applications,” in Proc. Allerton Conf., Sep. 2010.

[25] J. G. Proakis, Digital communications, 4th ed. McGraw-Hill, 2001.[26] G. Calcev, D. Chizhik, B. Goransson, S. Howard, H. Huang, A. Kogiantis,

A. F. Molish, A. L. Moustakas, and H. X. D. Reed, “A wideband spatialchannel model for system-wide simulations,” IEEE Trans. Veh. Technol.,vol. 56, no. 2, pp. 389-403, Mar. 2007.

[27] R. C. Hansen, Phased array antennas, Wiley-Interscience, 1998.