Iterative Joint Beamforming Training with Constant-Amplitude Phased Arrays in Millimeter-Wave Communication
aa r X i v : . [ c s . I T ] N ov Iterative Joint Beamforming Training withConstant-Amplitude Phased Arrays inMillimeter-Wave Communications
Zhenyu Xiao,
Member, IEEE , Lin Bai,
Member, IEEE , and Jinho Choi,
Senior Member, IEEE
Abstract —In millimeter-wave communications (MMWC), inorder to compensate for high propagation attenuation, phasedarrays are favored to achieve array gain by beamforming, wheretransmitting and receiving antenna arrays need to be jointlytrained to obtain appropriate antenna weight vectors (AWVs).Since the amplitude of each element of the AWV is usuallyconstraint constant to simplify the design of phased arrays inMMWC, the existing singular vector based beamforming trainingscheme cannot be used for such devices. Thus, in this letter,a steering vector based iterative beamforming training scheme,which exploits the directional feature of MMWC channels, isproposed for devices with constant-amplitude phased arrays.Performance evaluations show that the proposed scheme achievesa fast convergence rate as well as a near optimal array gain.
Index Terms —Millimeter wave, beamforming, phased array,beamforming training, 60 GHz.
I. I
NTRODUCTION W HILE AROUSING increasing attentions in bothacademia and industry due to abundant frequencyspectrum [1], [2], millimeter-wave communications (MMWC)face the challenge of high propagation attenuation caused bythe high frequency. To remedy this, phased arrays can beadopted at both the source and destination devices to exploitarray gains and compensate for the propagation loss [1], [2].To achieve a sufficiently high array gain, the antenna weightvectors (AWVs) at both ends need to be appropriately set priorto signal transmissions, which is the joint
Tx/Rx beamforming.If the channel state information (CSI) is available at bothends, the optimum AWVs can be directly found under well-known performance criteria, e.g., maximizing receiving signal-to-noise ratio (SNR) [1], [2]. Unfortunately, since the numberof antennas is generally large, the channel estimation becomestime-consuming in MMWC. In addition, the computationalcomplexity is high, because the matrix decomposition, e.g., thesingular vector decomposition (SVD), is generally required.Owing to this, the beamforming training approach, whichhas a lower complexity, becomes more attractive to find theAWVs [3]–[5]. Generally, there are two types of joint Tx/Rxbeamforming training schemes. One is switched beamformingtraining based on a fixed codebook [3]. The codebook contains
This work was partially supported by the National Natural Science Foun-dation of China under Grant Nos. 61201189, 91338106, and 61231011, aswell as GIST Colleges 2013 GUP Research Fund.Z. Xiao and L. Bai are with the School of Electronic and InformationEngineering, Beihang University, Beijing 100191, P.R. China.J. Choi is with the School of Information and Communications, GwangjuInstitute of Science and Technology (GIST), Korea.Corresponding author is Dr. Z. Xiao (Email: [email protected]). a number of predefined AWVs. During beamforming training,the AWVs at both ends are examined according to a certainorder, and the pair that achieves the largest SNR is selected.The other one is adaptive beamforming training [4], [5], whichdoes not need a fixed codebook. The desired AWVs at bothends are found via real-time joint iterative training. It is clearthat the switched beamforming training is simpler, while theadaptive one is more flexible.Most adaptive beamforming training schemes adopt thesame state-of-the-art approach, i.e., finding the best singularvector via iterative training without a priori
CSI at both ends[4], [5]. This singular vector based training scheme (SGV)requires that both the amplitudes and phases of the AWVare adjustable. On the other hand, in MMWC, phased arraysare usually implemented with the approach that only phasesof the antenna branches are adjustable; the amplitudes areset constant to simplify the design and reduce the powerconsumption of phased arrays [3], [6], [7]. In fact, even ingeneral multiple-input multiple-output (MIMO) systems [8],antenna branches with constant amplitude (CA) are also anoptimization objective to reduce implementation complexity[9], [10]. In such a case, SGV becomes infeasible due to theCA phased array. The schemes proposed in [9], [10] cannotbe used here either, because these schemes are designed fortransmitting beamforming with full or quantized a priori
CSIat only the source device, but for MMWC with CA phasedarrays, joint beamforming is required without a priori
CSI atboth the source and destination devices.In this letter, a steering vector based joint beamformingtraining scheme (STV), which exploits the directional featureof MMWC channels, is proposed. Performance comparisonsshow that for line-of-sight (LOS) channels, both STV andSGV have fast convergence rates and achieve the optimal arraygain. On the other hand, for non-LOS (NLOS) channels, STVachieves a faster convergence rate at the cost of a slightly lowerarray gain than SGV, which can also achieve the optimal arraygain. In conclusion, STV achieves a fast convergence rate anda near optimal array gain under both LOS and NLOS channels,which highlights its applicability in practice.II. S
YSTEM AND C HANNEL M ODELS
Without loss of generality, we consider a MMWC systemwith half-wave spaced uniform linear arrays (ULAs) of M and N elements at the source and destination devices, respectively.The ULAs are phased arrays where only the phase can becontrolled. A single RF chain is tied to the ULA at each of the Algorithm 1
The SGV Scheme
1) Initialize:
Pick an initial transmitting AWV t at the sourcedevice. This AWV may be chosen either randomlyor with the approach specified in Section V.
2) Iteration:
Iterate the following process ε times, and thenstop.Keep sending with the same transmitting AWV t at the source device over N slots. Meanwhile, useidentity matrix I N as the receiving AWVs at thedestination device, i.e., the n th column of I N asthe receiving AWV at the n th slot. Consequently,we receiving a vector r = I H N Ht + I H N n r = Ht + n r , where n r is the noise vector. Normalize r .Keep sending with the same transmitting AWV r at the destination device over M slots. Meanwhile,use identity matrix I M as the receiving AWVs atthe source device. Consequently, we receiving anew vector t = I H M H H r + I H M n t = H H r + n t ,where n t is the noise vector. Normalize t .
3) Result: t is the AWV at the source device, and r is theAWV at the destination device.source and destination devices. According to the measurementresults of channels for MMWC in [1], [11], only reflectioncontributes to generating multipath components (MPCs), whilescattering and diffraction effects are negligible due to theextremely small wave length of MMWC. Thus, the MPCsin MMWC have a directional feature, i.e., different MPCshave different physical transmitting steering angles φ t ℓ andreceiving steering angles φ r ℓ . Consequently, the channel modelis expressed as [12], [13] H = √ N M X L − ℓ =0 λ ℓ g ℓ h H ℓ , (1)where L is the number of multipath components, ( · ) H is the conjugate transpose operation, λ ℓ are the chan-nel coefficients, and g ℓ and h ℓ are the receiving andtransmitting steering vectors [12], [13] that are givenby g ℓ = { e jπ ( n − r ℓ / √ N } n =1 , ,...,N and h ℓ = { e jπ ( m − t ℓ / √ M } m =1 , ,...,M , respectively. Note that Ω t ℓ and Ω r ℓ represent the transmitting and receiving cosine anglesof the ℓ th MPC, respectively [13], i.e., Ω t ℓ = cos( φ t ℓ ) and Ω r ℓ = cos( φ r ℓ ) .Given the transmitting AWV t and the receiving AWV r ,where k t k = k r k = 1 , the received signal y is given by y = r H Ht s + r H n , where s is the transmitted symbol, n isthe noise vector. The target of beamforming training is to findappropriate transmitting and receiving AWVs to obtain a highreceiving SNR, which is given by γ = | r H Ht | /σ , where σ is the noise power.III. S INGULAR -V ECTOR B ASED S CHEME
Let us introduce the SGV scheme first. It is known that theoptimal AWVs to maximize γ is the principal singular vectors of the channel matrix H [4], [5] . Denote the SVD of H as H = U Σ V H = X Kk =1 ρ k u k v H k , (2)where U and V are unitary matrices with column vectors(singular vectors) u k and v k , respectively, Σ is an N × M rectangular diagonal matrix with nonnegative real values ρ k on the diagonal, i.e., ρ ≥ ρ ≥ ... ≥ ρ K ≥ and K =min( { M, N } ) . The optimal AWVs are t = v and r = u .In the common case that H is unavailable, iterative beam-forming training can be adopted to find the optimal AWVs.According to [4], [5], H m ∆ = ( H H H ) m = P Kk =1 ρ mk v k v H k ,which can be obtained by m iterative trainings utilizing thereciprocal feature of the channel. When m is large, H m ≈ ρ m v v H1 . Thus, the optimal transmitting and receiving AWVscan be obtained by normalizing H m t and H × H m t ,respectively.The SGV scheme is described in Algorithm 1. The iterationnumber ε depends on practical channel response, which will beshown in Section V. It is clear that although the SGV schemeis effective, it is required that both the amplitudes and phasesof AWV are adjustable, which cannot be satisfied when CAphased arrays are used, where only phases are adjustable.IV. S TEERING -V ECTOR B ASED S CHEME
In fact, the SGV scheme is a general one suitable for anarbitrary channel H . It does not use the specific feature ofMMWC channels. In MMWC, the channel has a directionalfeature, i.e., H can be naturally expressed as in (1), whichis similar to the expression in (2). The difference is thatin (1), the vectors { g ℓ } and { h ℓ } are CA steering vectors,not orthogonal bases, but in (2), { u k } and { v k } are strictnon-CA orthogonal bases. Nevertheless, according to [13], | g H m g n | and | h H m h n | are approximately equal to zero given that | Ω r m − Ω r n | ≥ /N and | Ω t m − Ω t n | ≥ /M , respectively, i.e.,the receiving and transmitting angles can be resolved by thearrays, which is the common case in MMWC. Consequently,as a suboptimal approach, the steering vectors of the strongest MPC can be adopted as the transmitting and receiving AWVsat the source and destination devices, which leads to theproposed STV scheme. The advantage of STV is that theelements of the steering vector have a constant envelope,which is suitable for the devices with CA phased arrays.Moreover, although the transmitting and receiving angles arerequired to be resolved by the arrays in the following analysis,the STV scheme can work even when there exists angles thatcannot be resolved, because two or more MPCs associatedwith sufficiently close angles that cannot be resolved actuallybuild a single equivalent MPC.Assuming that H is available in advance, the background ofSTV is presented as follows. Using the directional feature ofMMWC channels, we have H m ≈ P Lℓ =1 |√ M Nλ ℓ | m h ℓ h H ℓ ,for a positive integer m . Suppose the k th MPC is the strongestone. For ℓ = k , | λ ℓ | m / | λ k | m exponentially decreases. Thismeans that the contribution to the matrix product H m fromthe the other L − MPCs exponentially decreases, compared The SVD on H gives a set of orthogonal transmitting and receiving AWVpairs, as well as the energies projected to these AWV pairs. Send with Receive with :Send withSignature estimation:Receive with :Source device Destination deviceSignature estimation: H H H r
N N
H H r
N N r F Ht F n exp(j ( )) N N g ut e es exp(jexp(j e F r at o : ( )) N N (( rr r e H H H t
M M
H H H t
M M t F H r F n exp(j ( )) M M exp(j g (exp(j e F t ( )) M M (( tt t e t H r H H H H
N N N H H r
N N r I Ht I n
H H H
M M M H H H t
M M t I H r I n N F M F r t Fig. 1. Process of the proposed STV scheme. with the strongest one. Therefore, we have lim m →∞ H m = |√ M N λ k | m h k h H k . Thus, for given a sufficiently large m andan arbitrary initial transmitting AWV t , we have H m t = |√ M Nλ k | m h k h H k t = (cid:16) |√ M Nλ k | m h H k t (cid:17) h k , which is h k multiplied by a complex coefficient. It is notedthat h k is a constant-envelope steering vector. Hence, thedesired transmitting AWV can be obtained by the signatureestimation where e t = exp(j ∡ ( H m t )) / √ M is to be esti-mated. Here, ∡ ( x ) represents the angle vector of x in radian.In fact, the signature estimation can be carried out by theentry-wise normalization on H m t .In addition, we have H × H m t = (cid:16) λ k √ M N | λ k √ M N | m h H k t (cid:17) g k . (3)Thus, the desired receiving AWV can be obtained by thesignature estimation of e r = exp(j ∡ ( H × H m t )) / √ N .It is clear that given full CSI, the AWVs steering alongthe strongest MPC in both ends can be obtained. In practicalMMWC, however, H is basically unavailable in both ends;thus we propose the joint iterative beamforming training pro-cess of STV, which is shown in Fig. 1, and the correspondingalgorithm is described in Algorithm 2. The iteration number ε depends on practical channel response. According to thesimulation results in Section V, ε = 2 or 3 can basicallyguarantee convergence.It is noted that STV is tailored for MMWC devices withCA phased arrays based on SGV. Thus, STV and SGV havecommon features, e.g., both schemes need iteration. However,their mathematical fundamentals are different. SGV is to findthe principal singular vectors of the channel matrix H , whichis optimal and applicable for arbitrary channels, while STVis to find the CA steering vectors of the strongest MPC byexploiting the directional feature, which is sub-optimal andonly feasible under MMWC channels. Thus, in each iteration,STV requires the signature estimation, which is to estimatethe CA steering vector of the strongest
MPC. Meanwhile, inorder to make STV feasible for CA phased arrays, it adopts theDFT matrices in transmitting and receiving training sequences,because the entries of them have a constant envelope.V. P
ERFORMANCE E VALUATION
In this section we evaluate the performances of STV, in-cluding array gain and convergence rate, and compare them There are other approaches to obtain the desired AWV here. The presentedone is a simple one in implementation.
Algorithm 2
The STV Scheme
1) Initialize:
Pick an initial transmitting AWV t at the sourcedevice. This AWV may be chosen either randomlyor with the approach specified in Section V.
2) Iteration:
Iterate the following process ε times, and thenstop.Keep sending with the same transmitting AWV t at the source device over N slots. Meanwhile,use discrete Fourier Transform (DFT) matrix F N as the receiving AWVs at the destination device,i.e., the n th column of F N as the receiving AWVat the n th slot. Note that I N cannot be usedfor the receiving AWVs here, due to its non-constant-envelope entries, but other unitary ma-trices with constant-envelope entries are feasible.Consequently, we receiving a vector r = F H N Ht + F H N n r , where n r is the noise vector. Estimate thesignature e r as e r = exp(j ∡ ( F N r )) / √ N andassign e r to r .Keep sending with the same transmitting AWV r at the destination device over M slots. Meanwhile,use DFT matrix F M as the receiving AWVs at thesource device. Consequently, we receiving a newvector t = F H M H H r + F H M n t , where n t is thenoise vector. Estimate the signature e t as e t =exp(j ∡ ( F M t )) / √ M and assign e t to t .
3) Result: t is the AWV at the source device, and r is theAWV at the destination device.with those of SGV via simulations. In all the simulations,the channel is normalized as E ( P Lℓ =1 | λ ℓ | ) = 1 . Thetransmitting SNR is thus γ t = 1 /σ , and the array gainbecomes the ratio of receiving SNR to transmitting SNR,i.e., η = γ/γ t = | r H Ht | . The initial transmitting AWVsin the two schemes are selected under the principle that itspower evenly projects on the M basis vectors of the receivingmatrices at the source, i.e., I M and F M , respectively. Thus, theinitial transmitting AWV for SGV is M / √ M , while that forSTV is a normalized constant-amplitude-zero-autocorrelation(CAZAC) sequence with length M .The array gain is empirically found using the ratio ofthe average receiving SNR to the average transmitting SNRover realizations of channels. Furthermore, the SVDupper bound is obtained by averaging the squares of theprincipal singular values of these channel realizations. Channelrealizations are generated under the Rician and Rayleighfading models for the LOS and NLOS channels, respec-tively. For the LOS channel, the power of the LOS MPC is | λ | = 0 . , and the average powers of the NLOS MPCsare E ( {| λ ℓ | } ℓ =2 , , ) = [0 . . . . For theNLOS channel, E ( {| λ ℓ | } ℓ =1 , , , ) = [0 .
25 0 .
25 0 .
25 0 . .The transmitting and receiving steering angles are randomlygenerated within [0 2 π ) in each realization.The left and middle figures in Fig. 2 show the achieved A rr a y G a i n ( d B ) STV, ε =1STV, ε =2STV, ε =3SGV, ε =1SGV, ε =2SGV, ε =3SVD Bound 0 5 10 15 20 25 30121416182022 SNR (dB) A rr a y G a i n ( d B ) STV, ε =1STV, ε =2STV, ε =3SGV, ε =1SGV, ε =2SGV, ε =3SVD Bound 0 2 4 6 820222426283032 Iteration Number ε A rr a y G a i n ( d B ) STVSGVSVDBLOS, M=N=32NLOS, M=N=32LOS, M=N=16NLOS, M=N=16
Fig. 2. Comparison of array gain between SGV and STV under the LOS (left) and NLOS (middle) channels with different numbers of iterations, where M = N = 16 , and that of convergence rate (right), where SVDB represents SVD bound. array gains of SGV and STV under the LOS and NLOSchannels, respectively, with different numbers of iterations,where M = N = 16 . The right figure in Fig. 2 shows thecomparison of convergence rate between SGV and STV underthe LOS and NLOS channels with a high transmitting SNR,i.e., 25 dB, in the cases of M = N = 16 and M = N = 32 ,respectively. From the left and right figures, it is found thatunder the LOS channel, both schemes achieve fast convergencerates and approach the optimal array gain, i.e., the SVD upperbound. From the middle and right figures, it is observed thatunder the NLOS channel, both the two schemes have slowerconvergence rates, and STV achieves a faster convergence rateat the cost of a slightly lower array gain than SGV that alsoapproaches the SVD upper bound. It is noted that, althoughnot shown in these figures, similar results are observed with asmaller or larger number of antennas.Explanations for these observations are as follows. Underthe LOS channel, there is one and only one strong MPC, andthe steering vectors of this MPC are almost the optimal AWVs.Thus, STV can achieve the optimal array gain. But under theNLOS channel, there are several MPCs with different steeringangles (or steering vectors) and the STV scheme obtains oneof them as an AWV, which is not optimal. Hence, STV cannotachieve the optimal array gain in such a case. On the otherhand, since the SGV is based on the principal singular vector,it can surely achieve the SVD upper bound once convergencehas been achieved. Besides, the fact that STV achieves a fasterconvergence rate in NLOS channel indicates that the signatureestimation in each iteration of STV is more robust againstnoise, while the AWV estimation of SGV is more sensitive tonoise.In brief, although STV is tailored for MMWC devices withCA phased arrays, where SGV is infeasible, it has comparableperformances to SGV in terms of the convergence rate andarray gain, under both the LOS and NLOS channels. Onthe other hand, it is noted that a single iteration consumes M + N training slots, which may significantly degrade thesystem efficiency, especially when the number of antennasis large. Hence, even if there is no CA constraint, i.e., bothphase and amplitude are adjustable and thus SGV is feasible,STV may still be favored in the case that the iteration numberis constrained to be 1 or 2 to save training time, because itachieves a higher array gain according to the right figure of Fig. 2. VI. C ONCLUSIONS
Since the existing SGV scheme cannot be used in MMWCwith CA phased arrays, the STV scheme has been proposedin this study, which effectively exploits the directional featureof MMWC channels. Performance comparisons showed thatunder LOS channel, both the schemes achieve fast convergencerates and achieve the optimal array gain; under NLOS channel,STV achieves a faster convergence rate at the cost of aslightly lower array gain than SGV that can still approachthe optimal array gain. In summary, while the proposed STVscheme is well-suited to MMWC with CA phased arrays, it hascomparable performances to SGV in terms of the convergencerate and array gain under both the LOS and NLOS channels.R
EFERENCES[1] S. K. Yong, P. Xia, and A. Valdes-Garcia, . Wiley, 2011.[2] Z. Xiao, “Suboptimal spatial diversity scheme for 60 GHz millimeter-wave WLAN,”
IEEE Communications Letters , vol. 17, no. 9, pp. 1790–1793, 2013.[3] J. Wang, Z. Lan, C. Pyo, T. Baykas, C. Sum, M. Rahman, J. Gao, R. Fu-nada, F. Kojima, and H. Harada, “Beam codebook based beamformingprotocol for multi-gbps millimeter-wave WPAN systems,”
IEEE Journalon Selected Areas in Communications , vol. 27, no. 8, pp. 1390–1399,2009.[4] P. Xia, S. Yong, J. Oh, and C. Ngo, “A practical SDMA protocol for60 GHz millimeter wave communications,” in
Asilomar Conference onSignals, Systems and Computers . IEEE, 2008, pp. 2019–2023.[5] Y. Tang, B. Vucetic, and Y. Li, “An iterative singular vectors estimationscheme for beamforming transmission and detection in MIMO systems,”
IEEE Communications Letters , vol. 9, no. 6, pp. 505–507, 2005.[6] A. Valdes-Garcia, S. T. Nicolson, J.-W. Lai, A. Natarajan, P.-Y. Chen,S. K. Reynolds, J.-H. C. Zhan, D. G. Kam, D. Liu, and B. Floyd, “Afully integrated 16-element phased-array transmitter in SiGe BiCMOS for60-GHz communications,”
IEEE Journal of Solid-State Circuits , vol. 45,no. 12, pp. 2757–2773, 2010.[7] E. Cohen, C. Jakobson, S. Ravid, and D. Ritter, “A thirty two elementphased-array transceiver at 60GHz with RF-IF conversion block in 90nmflip chip CMOS process,” in
IEEE Radio Frequency Integrated CircuitsSymposium (RFIC) . IEEE, 2010, pp. 457–460.[8] L. Bai and J. Choi, “Lattice reduction-based MIMO iterative receiverusing randomized sampling,”
IEEE Transactions on Wireless Communi-cations , vol. 12, no. 5, pp. 2160–2170, 2013.[9] X. Zheng, Y. Xie, J. Li, and P. Stoica, “MIMO transmit beamformingunder uniform elemental power constraint,”
IEEE Transactions on SignalProcessing , vol. 55, no. 11, pp. 5395–5406, 2007.[10] J. Lee, R. U. Nabar, J. P. Choi, and H.-L. Lou, “Generalized co-phasing for multiple transmit and receive antennas,”
IEEE Transactionson Wireless Communications , vol. 8, no. 4, pp. 1649–1654, 2009. [11] A. Maltsev, R. Maslennikov, A. Sevastyanov, A. Lomayev, A. Khoryaev,A. Davydov, and V. Ssorin, “Characteristics of indoor millimeter-wavechannel at 60 GHz in application to perspective WLAN system,” in
European Conference on Antennas and Propagation (EuCAP) . IEEE,2010, pp. 1–5.[12] M. Park and H. Pan, “A spatial diversity technique for IEEE 802.11 adWLAN in 60 GHz band,”
IEEE Communications Letters , vol. 16, no. 8,pp. 1260–1262, 2012.[13] D. Tse and P. Viswanath,