Some Aspects on Hybrid Wideband Transceiver Design for mmwave Communication Systems

Transcription

1 Some spects on Hybrid Wideband Transceiver Design for mmwave Communication Systems Marcin Iwanow 12, Nikoa Vučić 1, Mario H. Castañeda 1, Jian Luo 1, Wen Xu, 1 and Wofgang Utschick 2 1 Huawei Technoogies Duessedorf GmbH, Munich, Germany Emai: {iwanow.marcin, nikoa.vucic, mario.castaneda, jianuo, wen.dr.xu}@huawei.com 2 ssociate Institute for Signa Processing, Technische Universität München, Germany Emai: utschick@tum.de bstract In this paper, the probem of designing a muticarrier transceiver for singe-ink mmwave transmission is considered. ssuming hybrid beamforming with mutipe data streams, it is shown how the common wideband anaog beamformer can be optimized together with digita mutipe-input, mutipe-output precoders and receive equaizers which operate on per-subcarrier basis. For this purpose, two approaches originay constructed for narrowband systems are extended to wideband, and severa simpifications for soving the optimization probems are made. Their evauation is performed on a proposed statistica mmwave channe mode and by utiizing measured channes. I. INTRODUCTION In recent years, the utiization of new frequency bands in the mmwave region for the 5th generation 5G) of ceuar networks has drawn significant attention in the industry and academia [1], [2]. Expoitation of arge channe bandwidths at high frequencies is seen as one of the key enabers in addressing the extremey chaenging 5G requirements, particuary in terms of anticipated data rates [3]. However, moving to higher portions of the radio spectrum requires considerabe changes in both ower and higher ayers of the protoco stack. Currenty, the first commercia products in the icense-free V-band appear in the market foowing the IEEE 82.11ad standard [4]. The new IEEE82.11ay standard is expected to provide significant improvements in achievabe data rates and supported use cases. In parae, severa research initiatives have started with the goa of expoiting other mmwave frequency bands K a band, E-band, etc.) and designing an integrated 5G ceuar wireess system c.f. [5]). In this work, we anayze the probem of jointy designing the transmitter and the receiver in a muti-stream, muticarrier mmwave system. The underying architecture foows the hybrid beamforming principe, where anaog processing beamforming) is performed in order to reduce the number of required anaog-to-digita and digita-to-anaog chains DCs/DCs) [6]. The considered system mode is recognized in the community as one of the candidate soutions for 5G mmwave radio access networks RN) and backhau.. State of the rt on mmwave Precoding/Equaization s a consequence of the high frequencies, arge bandwidths and the high number of antennas at both the transmitter and receiver expected in mmwave systems, performing fuy digita beamforming might not be practica. though digita beamforming offers the maximum degree of fexibiity for the transceiver design, it resuts in high compexity, cost and energy consumption when operating at mmwave frequencies due to the need of a arge amount of radio frequency ) chains incuding high resoution converters DCs/DCs) with a high samping rate. To address this aspect, one can consider decreasing the resoution of the converters, for instance of the DCs at the receiver side [7]. nother possibiity consists of reducing the number of chains eading to ess converters) by empoying hybrid beamforming [8] [11]. By performing part of the beamforming operations in the anaog domain, hybrid beamforming is abe to provide a trade-off between performance and compexity/energy consumption. The hybrid beamforming architecture has aso been considered for massive MIMO systems [12] [14] due to the use of a massive amount of antennas at the base station. the works assume, however, a narrowband mode, i.e., a frequency fat channe. s pointed out in [15], the transceiver design for broadband systems is sti an open research probem. The design of hybrid beamforming for muticarrier systems has not been studied extensivey yet it has been ony considered in a few works [16] [19]. simpe extension of the hybrid beamforming agorithms for the narrowband system is not possibe for a practica reason the anaog beamformer cannot vary among the subcarriers, i.e., it is fixed for a the entire bandwidth [17] [19]. In the foowing we briefy review the prior work for broadband systems. In [16], the anaog beamforming at the transmitter and at the receiver are chosen with the aim of maximizing the average over a subcarriers) of the argest singuar vaue of the effective channe matrix on each subcarrier. The effective channe matrix on each subcarrier consists of the product of the anaog receive beamforming matrix, the mutipe-input mutipe-output MIMO) channe on each subcarrier and the anaog transmit beamforming matrix. For the proposed singe stream transmission, the digita beamforming for each subcarrier at the transmitter and receiver are chosen such that they correspond to the right and eft singuar vectors of the effective channe matrix on each subcarrier, respectivey. The design of hybrid beamforming is studied for a MIMO- OFDM system for singe stream transmission in [17], where the anaog and digita beamformers are chosen from quan-

2 tized codebooks. The optimum choice for the transmit anaog and digita beamformers and the receive anaog beamformers invoves an exhaustive joint search over a possibe combinations. To simpify the exhaustive search, a sequentia searching agorithm is aso proposed in [17], where the best transmit/receive anaog beam pairs are first chosen without considering the transmit digita beamforming, which is ater seected based on the chosen anaog beamformers. In [18], a mutiuser muticarrier system is considered. The authors aim at the hybrid beamforming at the transmitter such that it perfecty matches the performance of a system with fuy digita frontend. The authors propose QR-based factorization of the coection of the digita beamforming matrices over a subcarriers and show that a hybrid beamforming architecture with r t chains and 2r t N tx r t + 1) infinite resoution phase shifters can match the performance of an system with fuy digita frontend. r t and N tx are the rank of the digita beamforming matrices and the number of transmitter antennas, respectivey. Note that for a standard hybrid structure ony one phase shifter per antenna is assumed for each chain. The design of the hybrid beamforming with mutipe streams for wideband mmwave systems has been studied aso in [19] under the assumption of imited feedback. The anaog beamformers are taken from a codebook. The case when the digita beamformers are aso considered to be taken from a codebook is aso investigated. The work in [19] provides insight into the design of the codebooks. The concept of Joint Spatia Division and Mutipexing JSDM) which consists of two-ayer or two-stage precoding has been recenty proposed assuming singe-antenna users [2], [21], where a pre-beamformer is designed based on the channe covariance matrices of the users. By impementing the pre-beamforming in the anaog domain, JSDM can be performed with a hybrid architecture at the transmitter. s pointed out in [2], the channe covariance matrix can be considered in genera to be frequency fat, such that JSDM coud be extended to the muticarrier case where the anaog pre-beamformer is constant across a subcarriers. B. Contributions Differenty from the previous works which consider singe stream transmission or assume codebook based approaches, we aim at optima inear muticarrier MIMO transceiver design based on hybrid beamforming. The probem of interest is invoved due to specific constraints on the anaog fiters, which are composed of phase shifters and shoud be fixed for the entire bandwidth. The setting we study is in compiance with the often assumed time-division-dupex TDD) assumption for a sma mmwave ce. We show how two agorithms originay constructed for the narrowband case, namey the compressed sensing based CS) and the bock coordinate descent BCD) strategies [22], [23], respectivey, can be extended to support the muti - carrier case. Having in mind specific mmwave sparse channe characteristics, we anayse severa ow-compexity aternatives to the sub)-optimized designs, and examine the gap with respect to the fuy digita transceiver design. statistica spatio-tempora channe mode is used in our anaysis to evauate the proposed soutions. Finay, the performance of the agorithms is aso tested with measured channes. C. Outine of the Paper The rest of the paper is organized as foows. Section II gives a detaied description of the system mode. The assumed mmwave channe mode is eaborated in Section III. In Section IV, the mathematica descriptions of the considered probems are presented, foowed by the proposed soutions. Finay, Section V shows simuation resuts for the assumed statistica channe mode, as we as for a set of channe measurements. D. Notation We use ower case bodface characters to denote coumn) vectors and capita bodface characters to denote matrices. We denote N N identity matrix with I N. ) T, ) H, and ) denote transpose, conjugate transpose, and Moore-Penrose pseudoinverse, respectivey. We use tr), F, 2 for the trace, the Frobenius norm, and the 2-norm of the matrix, respectivey. The m, n)-th entry of the matrix is denoted with [] m,n). The expected vaue of a random variabe is denoted with E[ ], b N C a, ) means that the random variabe b foows a circuar compex Gaussian distribution with mean a and covariance matrix. b U[a 1, a 2 ] means, that random variabe b foows the uniform distribution within [a 1, a 2 ]. Ω is the cardinaity of Ω; otherwise, it stands for the absoute vaue. II. SYSTEM MODEL We consider a point-to-point OFDM setup consisting of one transmitter Tx) with N tx antennas and one receiver Rx) with N rx antennas. The frontend of both Tx and Rx has a hybrid structure as shown in Fig. 1. The number of chains in the transmitter and in the receiver is Ntx and Ntx N tx, Nrx N rx. and Nrx, respectivey The system is operating within a bandwidth substantiay exceeding the coherence bandwidth of the channe, i.e., the channe is wideband frequency seective). In the time domain, this means that the channe impuse response CIR) consists of mutipe taps [24, Chapter 2]. If the channe is samped with a samping rate f s = 1 T s, it can be seen as an FIR fiter. If the order of the fiter determined by the maximum deay spread) is D and the channe is assumed to be inear time-invariant LTI), the symbo received at the time instance nt s reads as y[n] = D H[d]x[n d] + η[n], 1) d= where y[n] C Nrx is the vector received at the time instance nt s, x[n d] C Ntx the vector transmitted at the time instance n d)t s, H[d] C Ntx Ntx the CIR samped at dt s, and η[n] C Nrx is the white noise with η[n] N C, σηi 2 Nrx ).

3 N tx Tx antennas N rx Rx antennas s OFDM Processing: chain 1 IFFT,CP insertion # 1 naog Digita. Tx.. Tx Precoder Precoding. s OFDM Processing: chain Nsubc IFFT,CP insertion # N tx Channe H[d]. naog Rx Equaizer chain # 1. chain # N rx OFDM Processing: CP deetion,fft. OFDM Processing: CP deetion,fft Digita Rx Equaizer. s 1 s Nsubc Fig. 1. System mode Using mutipe subcarriers for transmission aows to paraeize the frequency-seective channe into a set of frequencyfat channes [25]. Then, for each subcarrier k we have y k = H k x k + η k, k {1,..., }, 2) where y k C Nrx is the vector received on the k-th subcarrier, x k C Ntx the vector transmitted on the k-th subcarrier, H k C Nrx Ntx the channe on the k-th subcarrier, and η k C Nrx is the white noise with η k N C, σηi 2 Nrx ). In the paper, we consider inear precoding at the transmitter and inear combining at the receiver. If the transmission carries N s streams, the estimated symbo vector can be written as s k = G H k H k P k s k + G H k η k, k {1,..., }, 3) where x k in 2) has been repaced by the symbo vector s k C Ns precoded with P k C Ntx Ns. The matrix G k C Nrx Ns is used for ineary combining the received signas. The per-subcarrier power constraint is ensured by the condition P k 2 F Ptx, where P tx is the tota transmit power. The hybrid structure imposes additiona constraints for the precoding and combining matrices. The digita precoding at the transmitter unit is foowed by anaog precoding and anaog combining is foowed by digita combining at the receiver cf. Fig. 1). The anaog precoding and combining is usuay reaized in hardware by means of interconnecting a matrix of Ntx N tx and Nrx N rx phase shifters, respectivey. Equivaenty, the entries of the anaog precoding and combining matrices have equa magnitude. Without oss of generaity, we set it here to 1. Then, the precoder and combiner for the k-th subcarrier can be written as P k = P k, P k,d, P k, P, 4) G k = G k, G k,d, G k, G, 5) where P k, Ntx N C tx, Gk, Nrx N C rx are the anaog precoding and combining matrices, P k,d C N tx Ns and G k,d C N rx Ns are the digita precoding and combining matrices for the k-th subcarrier. The sets P and G are defined as P = G = } Ntx N {Ξ C tx : [Ξ] i,j) = 1 i, j) 6) } Nrx N {Ξ C rx : [Ξ] i,j) = 1 i, j). 7) t this point, we assume different anaog precoding and combining matrices for each subcarrier in order to derive a baseine soution. We consider the constraint of the anaog precoding and combining matrices constant for a subcarriers in Section IV-B. Specificay for the hybrid transceiver structure, 3) can be rewritten as s k = G H k,dg H k,h k P k, P k,d s k + G H k,dg H k,η k, 8) k {1,..., }, with P k, P k,d 2 F Ptx. We assume avaiabiity of idea channe state information CSI) both at the receiver and transmitter some comments on channe estimation methods appicabe for mmwave systems are given in the foowing section). The objective of the precoding and combining design is the achievabe rate of the ink with Gaussian signaing cf. [26]) R P k,, P k,d, G k,, G k,d ) = 1 N subc 1 og N 2 det I Ns + R 1 k,η H k,effh H ) k,eff subc k= where the effective channe for k-th subcarrier reads 9) H k,eff = G H k,dg H k,h k P k, P k,d, 1) the noise covariance matrix R η can be expressed as R k,η = σ 2 ηg H k,dg H k,g k, G k,d, 11) and the normaization by is due to the ength of the OFDM symbo. III. CHNNEL MODEL Currenty, there is no common agreement about the parameters for mmwave outdoor channe. The foowing assumpions are frequenty used though e.g., in [22], [27] [31]): The number of paths is significanty ower than for the sub-6ghz frequency band. The paths propagate in space and time custers. There seem to be different views whether propagation in the time custer impies propagation in the space custer. The works [22], [29], [3] take this assumption whie in [27] it is argued that this is not necessariy the case. Because of the high number of channe parameters and no common agreement on their modeing, we choose to define a simpe mode which is compatibe with the above two features. The channe parameters are based on [22] and [27] and are

4 described in detais in Section V. We use the extended Saeh- Vaenzuea mode [19], [32], i.e., the MIMO channe samped at time instance dt s is written as a superposition of individua, custered paths as H[d] = N Nrx N c N tx path β α r, pdt s τ τ r )a rx θ r, )a H L txφ r, ), =1 r=1 12) where N c is the number of time custers, Npath is the number of paths in the -th custer, α r, is the path gain for the r-th path in the -th custer incuding the antenna gain), τ is the deay of the -th custer, τ r is the reative deay w.r.t. the custer deay of the r-th path within the custer. θ r, = [κ θ r,, ζθ r, ] is the direction of arriva Do) vector of the r-th path within the - th custer, composed from eevation ange κ θ r, and the azimuth ange ζr, θ. φ r, = [κ φ r,, ζφ r, ] is the direction of departure DoD) vector of the r-th path within the -th custer, composed from the eevation ange κ φ r, and the azimuth ange ζφ r,. L expresses the path oss between the transmitter and [ the receiver, and D ] β is a normaization factor such that E d= H[d] 2 F = N rx N tx. The contribution of the r-th path in the -th custer for the channe at the time instance dt s is evauated by samping the transfer function p of the puse-shaping fiter at dt s τ τ r. The vectors a rx and a tx are the antenna array response vectors for the receiver and the transmitter, respectivey. For the rest of our paper, we assume uniform inear arrays UL) at both the transmitting and receiving sides. The array response vector for UL reads as a UL [κ, ζ]) = 1 [1, e j 2π λ d sinζ),..., e jm 1) 2π λ d sinζ)] M 13) where M denotes the number of antennas, λ the waveength of the transmitted/received wave, and d is the spacing of the antenna eements. In our work, we assume haf-waveength antenna eements spacing d = λ 2. The response is independent of the eevation ange κ and depends ony on the azimuth ange ζ. For the muticarrier setup, the channe matrix H k at the k-th subcarrier is expressed as D 1 1 ) j2πk H k = H[d] exp d, 14) Nsubc d= where we assume that the number of subcarriers is arger than the number of taps in the CIR, i.e., > D. s the focus of this paper is on transceiver design, it wi be assumed in the seque that the mmwave channe 12) is known to the transmitter and the receiver of the considered system. We remark, however, that the probem of channe estimation itsef is a non-trivia one in mmwave systems, due to the arge anntenna arrays which are utiized and the hardware constraints. In [9], [23], some soutions expoiting specific mmwave channe characteristics sparsity, ow rank) are given for narrowband channe estimation. Whie these approaches can be appied with some modifications in the wideband systems considered here, an optimized wideband channe estimation is eft as an interesting open topic for the future work. IV. WIDEBND TRNSCEIVER DESIGN In this section, we discuss the design of the hybrid wideband precoding and combining. First we present in Section IV- a baseine soution which is a straightforward extension of agorithms for the narrowband channe. though it provides good resuts, it is not reaizabe in practice. To this end, in Section IV-B we propose an agorithm which aims on finding a practica hybrid wideband beamforming strategy with affordabe compexity.. Baseine Soution s a consequence of setting an individua power constraint for every subcarrier, each k-th component of the sum in the rate expression 9) is a function of an individua set of optimization variabes. Consequenty, the precoders and combiners are obtained by maximizing the sum rate 9) from the foowing optimization soution where R k,max = R max = 1 N subc 1 R k,max 15) k= max og 2 det INs + R 1 P k,,p k,d, k,η H k,effh H ) k,eff G k,,g k,d s.t. P k, P k,d 2 F P tx, P k, P, G k, G. 16) We can thus restrict further to soving the rate maximization for the k-th subcarrier individuay, namey maximizing R k,max. The probem woud have an optima soution if the precoding and combining were not decomposed into the anaog and digita parts. Namey, the precoding and combining shoud be based on the singuar vaue decomposition SVD) of the channe matrix H k, foowed by waterfiing at the transmitter [33]. In the foowing we assume uniform power aocation throughout N S streams. To sove 16), we first reax it by discarding the constraints P k, P and G k, G. In this way, we can obtain the suboptima 1 precoding and combining matrices as P k = V k [ INs Ns N tx N s)] T, G k = U k [ INs Ns N rx N s)] T, 17) where U k C Nrx Nrx and V k C Ntx Ntx are unitary matrices containing the eft and right singuar vectors of H k, respectivey, i.e., H k = U k Σ k Vk H. The singuar vaues in Σ k are given in descending order. 1 The soution is not optima due to the constraint on the number of streams and since no waterfiing is considered.

5 It is usuay not possibe to decompose Pk and G k such that Pk = P k, P k,d and G k = G k, G k,d with the constraints P k, P and G k, G. We use the same arguments as in [22] to justify that for the mmwave channe we can decompose the precoders and combiners by soving the foowing optimization probems: P k,, Pk,D ) = arg min P k,,p k,d and P k P k, P k,d F s.t. P k, P k,d 2 F P tx, G k,, G ) k,d = arg min G k,,g k,d G k G k, G k,d F P k, P 18) s.t. G k, G. 19) For soving these optimization probems, we use two agorithms. First one is the orthogona matching pursuit OMP) agorithm described in [22, gorithm 1], whose assumptions match we the spatia, sparse structure of the channe from 12). The second one is the modified bock coordinate descent BCD-SD) agorithm [23, Section IV], which is a we known approach to this cass of probems. We give a detaied description of both agorithms in ppendix. Note, that after the competion of the BCD-SD agorithm we perform the normaization of the digita precoding matrix P k,d = B. Practica soutions Ptx P k,d Pk, P k,d. 2) F In the previous subsection, we outined the baseine strategy for designing the hybrid precoding and combining. We observe that soving the optimization probem 16) for each subcarrier k resuts in different anaog precoding and combining matrices for each subcarrier. For the hardware impementation this means a separate set of phase shifters for each frequency bin, which is not practicabe in rea technica systems. Here we propose a suboptima soution by adding an additiona constraint to 18) and 19), i.e., Pk, and G k, remain constant for each subcarrier k. In order to sove this probem, we propose to jointy design an anaog precoder matrix Pk, and an anaog combiner matrix G k, for a certain set of subcarriers Ω {1,..., }. We consider a reduced set of subcarriers in order to reduce the computationa compexity. For this sake, we modify 18) and 19) as foows P Ω,, PΩ,D ) = arg min P Ω,,P Ω,D s.t. P Ω P Ω, P Ω,D F P tx P Ω, P Ω,D 2 F Ω P Ω, P, 21) and G Ω,, G ) Ω,D = arg min G Ω,,G Ω,D where ] PΩ = [P ω1,..., P ω Ω ] G Ω = [G ω1,..., G ω Ω P Ω,D = [ ] P ω1,d,..., P ω Ω,D G Ω,D = [ ] G ω1,d,..., G ω Ω,D Ntx N P Ω, C tx G Ω G Ω, G Ω,D F s.t. G Ω, G, 22) C Nrx Ω Ns), C Ntx Ω Ns), C N tx Ω Ns), C N rx Ω Ns) Nrx N G Ω, C rx 23) with ω Ω, {1,..., Ω }. Note that the per-subcarrier power constraint in 18) has been repaced by a sum power constraint in 21) in order to faciitate the cacuation of the anaog part of the transmitter. The optimization probems in 21) and 22) have the structure from equations 18) and 19) and can thus be soved with the OMP and BCD-SD agorithms described in Section IV-. Soving 21) and 22) provides aso with the corresponding digita precoding and combining matrices for the subcarriers from the set Ω. We scae them by using 2). For the subcarriers not in Ω, we propose to obtain them by the best approximation orthogona projection) of the optima precoding/combining soution by means of the anaog precoding/combining matrix found for the set Ω P,D = Ptx P Ω, P, Ω, P k P, Ω, P k ) F G,D = G, Ω, G k, {1,..., } \ Ω. 24) Both PΩ, and G Ω, are ta matrices, as N rx Nrx N tx Ntx. V. SIMULTION RESULTS and In our simuations, we use the system mode from Section II and the channe mode defined in 12), with parameters in Tabe I. We aso present resuts from the channes measured in the measurement campaign at the Imenau University of Technoogy [31]. We define the sectorized antenna pattern for the transmitter s antenna eements as in [22, Eq. 5)]. The azimuth width of the sector is 12 and the eevation width of the sector is 6. The receiver s antenna eements are omnidirectiona. We notice that because of the sectorized antenna pattern much ess than Nc =1 N path paths contribute to the channe. This is consistent with the principe of ow-rank mmwave channes. For sake of our simuations, we define the receive signato-noise ratio SNR) as SNR = P tx Lση 2. 25)

6 TBLE I TBLE OF SIMULTION PRMETERS TBLE II TBLE OF SIMULTED TRNSMISSION ND RECEPTION STRTEGIES Parameter Vaue Comments N rx 16 N tx 64 Nrx 4 Ntx N s 3 N c 8 Number of custers N path 1 [1, N c ] α r, N C, 1) pt) sinπt)/πt) sinc puse τ, τ r s in [27] θ r, Do) Eevation: κ θ r,, azimuth: ζr, θ φ r, DoD) Eevation: κ φ r,, κ θ r, ζ θ r, κ φ r, ζ φ r, κ mean,θ, κ mean,φ ζ mean,θ κ mean,θ ζ mean,θ κ mean,φ ζ mean,φ azimuth: ζ φ r, + κ rem,θ r, + ζ rem,θ r, + κ rem,φ r, + ζ rem,φ r, U[ π/2, π/2], ζ mean,φ U[, 2π] κ rem,θ r,, ζ rem,θ r,, κ rem,φ r,, ζ rem,φ r, L, 7.5 ) Lapace distribution with mean and standard deviation σ rem = 7.5 Rate [bit/c.u.] # ST Description Fu digita setup. Linear precoding with P k and inear combining with G k for each subcarrier k Hybrid transceiver structure. Linear precoding with Pk, P k,d and inear H combining with G k, k,d) G for each subcarrier. Not practica - not possibe with an anaog beamformer common for a subcarriers. ST3 Hybrid transceiver structure. Linear precoding with PΩ, P k,d and inear H combining with G Ω, k,d) G for ST4 each subcarrier. Practica - both the n transmitter and receiver requires ony one set of phase shifters in order to reaize both PΩ, and G Ω, which ST5 are constant across a the subcarriers) ST3 ST4 3 ST5 Comments Upper bound Ω = 2 - set containing ony the centra subcarrier Baseine not practica) Ω- set containing n subcarriers, symmetricay chosen w.r.t. the centra subcarrier Ω = {1,..., } considering a subcarriers) The figure of merit is the achievabe rate as defined in 9). We perform our simuations with 4 different transmission and reception strategies ST), as summarized in Tabe II. The resuts of numerica simuations are presented in Figures 2 with BCD-SD agorithm) and 3 with OMP agorithm), where the rates achieved with hybrid beamforming and combining -ST5) and fu digita precoding and combining ) are compared. In both Fig. 2 and Fig. 3 we observe that increasing the cardinaity of the set Ω eads to resuts coser to the baseine soution which requires an individua set of phase shifters for each subcarrier. Moreover, aready with a sma set of subcarriers Ω = 1 for the OMP decomposition and Ω = 3 for the BCD-SD decomposition), the considered agorithm resuts in good system performance. The gap to the baseine rate curve is negigibe. We expain the noticeabe gap between the baseine and Ω = 1 rate curves for BCD-SD decomposition in Fig. 2 with as foows. The OMP decomposition is taking the structure of the channe into account and is therefore more ikey to find the cose to optima soution based on the channe at ony one subcarrier. On the other hand, the BCD-SD agorithm is independent on the channe mode. The performance of the baseine soution with OMP is SNR [db] Fig. 2. Numerica evauation of the ink rates for different transmit and receive strategies defined in Tabe II. BCD-SD agorithm is used for decomposition of the suboptima precoder and combiner matrices. sighty worse than with the BCD-SD, since the set of avaiabe anaog precoders is more restricted with the OMP agorithm compared to the BCD-SD. Finay, Figs. 4 and 5 show resuts of the same simuations performed for a set of channes from the measurement campaign [31]. though ony a imited set of measurements were avaiabe in [31], we can observe the same trend as in Fig. 3 and 2. For the practica hybrid beamforming schemes, we observe that the gap to the upper bound and the baseine case is arger compared to empoying the channe mode 12). VI. CONCLUSION We showed how common wideband anaog transmit and receive beamformers can be jointy designed with per-subcarrier

7 Rate [bit/c.u.] ST3 ST4 3 ST5 Rate [bit/c.u.] ST3 ST4 3 ST SNR [db] Fig. 3. Numerica evauation of the ink rates for different transmit and receive strategies defined in Tabe II. OMP agorithm is used for decomposition of the suboptima precoder and combiner matrices SNR [db] Fig. 5. Numerica evauation of the ink rates for the existing set of channe measurements for different transmit and receive strategies defined in Tabe II. OMP agorithm is used for decomposition of the suboptima precoder and combiner matrices. Rate [bit/c.u.] ST3 ST4 3 ST4 1 ST SNR [db] Fig. 4. Numerica evauation of the ink rates for the existing set of channe measurements for different transmit and receive strategies defined in Tabe II. BCD-SD agorithm is used for decomposition of the suboptima precoder and combiner matrices. transmit and receive digita fiters in a muti-carrier, mutistream mmwave system. The performance of the investigated agorithms is seen to exhibit a rather sma oss compared to the soutions based on a fuy digita architecture, with the evauations being performed using a statistica mode as we as a set of measurement data. The oss was sighty arger with evauation being done on measurement data. By considering the fact that the beamformers are infuenced mainy by the setup geometry, the sub)-optimized soutions can be further simpified, e.g., by considering just a subset of subcarriers for the design of the anaog precoding/combining matrices. This wi resut in ower computationa compexity of the agorithms. Future work wi incude the investigation of channe estimation and muti-user aspects for the assumed hybrid architecture. PPENDIX. Detaied Description of the OMP and BCD-SD gorithms In gorithm 1, C Nant Ncod is the codebook matrix with, e.g., array steering vectors corresponding to equay spaced anges in the coumns and e i,l = [ 1 i 1) 1 1 L i) ] T. N ant is equa to N tx or N rx if the input to the agorithm is Pk or G k, respectivey. In gorithm 2, i stop = i + 1 after the ast iteration of the for oop. gorithm 1 The OMP agorithm Require: Pk or G k ) 1: F res = F = Pk or G k ) 2: for i Ntx do 3: Ψ = H F res [ 4: k = arg max ] j={1,...,ncod} ΨΨ H j,j) 5: F = [F e k,ncod )] 6: F D = F F 7: F res = F F F D F F F D F 8: end for 9: if the input is Pk then 1: Pk,D = P tx 11: P k, = F 12: ese 13: G k,d = F D F D F F D F 14: G k, = F 15: end if 16: return P k,d and P k, or G k,d and G k, ) CKNOWLEDGEMENT The research eading to these resuts received funding from the European Commission H22 programme under grant agreement n mmmgic project). The authors woud aso ike to thank Michae Joham Technische Universität München) for fruitfu discussions.

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