Full duplex (FD), which allows bidirectional transmission over the same resources, has the potential to reduce latency and double spectral efﬁciency. Recently, FD has been studied in 5G LTE millimeter wave cellular communications for New Radio in 3GPP releases 15–17. The primary drawback of FD is self-interference (SI). SI arises in the receiver for system 1 because it receives transmissions from itself and system 2. Because system 1 is much closer, SI can be 100-1000x the received power from system 2, thereby severely degrading communication. In this paper, we investigate a FD relay extending mmWave coverage to a single user. We propose to use alternating projections in designing the precoder and combiner to maximize the sum of the uplink and downlink spectral efﬁciency while bringing the SI below the noise ﬂoor. Our contributions include (1) all-digital and hybrid beamforming design algorithms for SI cancellation; and (2) communication performance analysis in terms of spectral efﬁciency, energy efﬁciency and outage probability. We compare the proposed algorithms against beam steering, singular value decomposition, and angle search techniques. Index Terms Full-Duplex, Self-Interference, Single-User MIMO, MmWave Cellular, Hybrid Beamforming I. I NTRODUCTION For a couple of decades, wireless networks have been evolving to meet the exponential growth in demand for mobile data. In cellular networks, long-term evolution (LTE) standards have been increasing bandwidths, spectrum reuse, antennas, and modulation size accordingly since their launch in 2008. Fourth-generation (4G) LTE standards (2011-2018) use frequency bands below 6 GHz and can provide up to 100 MHz in effective bandwidth depending on the Elyes Balti and Brian L. Evans are with the Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712 USA (e-mails: ebalti@utexas.edu, be- vans@ece.utexas.edu). 2 Data Center: Microcell coverage area : Macrocell coverage area /axisshort/axisshort/arrowaxisright : MmWave signaling /axisshort/axisshort/arrowaxisright : Sub-6 GHz signaling /axisshort/axisshort/arrowaxisright : FSO laser signaling Fig. 1: Outdoor heterogeneous mmWave cellular relaying network with free space optics (FSO) backhauling wherein the base stations are playing the key role of relays to improve coverage. Sub-6 GHz communications take place within the macrocells (large area) where reliable links require high power to maintain coverage. In microcells, where cell area is small, mmWave signaling can reliably support high data rates to the users [3]. deployment. 5G LTE standards (2018-) not only continue using the 4G sub-6 GHz bands but also introduce millimeter wave (mmWave) frequency bands in the 10 to 300 GHz range1. Designing beamforming (spatial ﬁltering) algorithms for massive antenna arrays can overcome the high propagation losses in mmWave bands. 5G New Radio (NR) can achieve 10x increase in peak and average bit rates over 4G due to the large mmWave bandwidths, e.g. 700 MHz in the 24 GHz band and 850 MHz in the 28 GHz band. In 2012, prior to the 5G LTE standards, the Wi-Fi IEEE 802.11ad standard adopted usage of the unlicensed 57-64 GHz mmWave band to achieve high bit rates [1], [2]. Full duplex (FD) could potentially reduce latency and double spectral efﬁciency in mmWave communication systems. These improvements make FD mmWave transceivers a potential candi- date for applications such as platooning, advanced driving assistance system, autonomous driving, and vehicular clouds, which require huge bandwidth, high data rate and low latency. Full duplex has been under study for New Radio by 3GPP Releases 15–17 for LTE standardization [4]. Fig. 1 illustrates an example FD cellular system. Because FD systems transmit and receive using the same resource blocks, FD transceivers are subject to the near-far problem. The near- far problem can be illustrated using point-to-point communications. In system 1, the receiver receives a signal from its transmitter that could be 100-1000x stronger than the transmitter in system 2 due to propagation losses over the longer distance. The near-far problem can result 1Although a rigorous deﬁnition of mmWave frequencies would place them between 30 and 300 GHz, industry has loosely deﬁned them to include the spectrum from 10 to 300 GHz. 3 in severe communication degradation. In order to make FD systems practical, designing robust beamforming and interference cancellation techniques are critical. A. Self-Interference Cancellation Techniques 1) Antennas Array Architectures: Interference cancellation can be realized by using antenna separation, isolation, polarization [5]–[8], directional antennas [9]–[11] or antennas placement to create null space at the receive arrays [12], [13]. The application of each technique depends solely on the hardware constraints and the application itself. For example, the application of interference passive cancellation using isolation and separation is limited for mobile devices due to their small dimensions. Fortunately, interference suppression can be achieved in relaying systems because the transmit and receive arrays are not necessarily collocated. In addition, [7] showed that directional arrays with a range of 4–6m of antenna isolation can achieve a hefty amount of SI reduction as high as 80 dB. This extent of isolation can be applicable in relaying systems; however, mobile devices cannot support such isolation due to their relatively small dimensions. 2) Analog Circuitry: This approach aims to suppress the SI before the low noise ampliﬁer (LNA) and analog-to-digital converter (ADC) by subtracting an estimate of the received SI from the received signal. The radio-frequency (RF) transmitted signal can be extracted at the transmit power ampliﬁer (PA) output, processed in the RF SI cancellation stage and subtracted from the received signal. Analog RF SI-cancellation can be ﬁrst applied [10], [11], [14] to suppress only the internal coupling and reﬂections modeled by a programmable analog tapped-delay line (TDL) transversal ﬁlter. Further adaptive digital RF cancellation can also be applied to suppress the SI components coming from the random external reﬂections by using a digital symbol-synchronous ﬁnite impulse-response (FIR) ﬁlter [9], [15], [16]. Traditionally, the analog RF cancellation uses the knowledge of the transmitted SI to cancel it before the receive LNA. A copy of the transmitted signal is obtained from the PA output and passed through a canceling circuit to reconstruct a copy of the received SI. The signal at the PA output includes the distortions of the transmitter (TX), which are reduced by the analog RF cancellation stage. The design of the canceling circuit is highly related to the nature of the SI channel. The SI channel can be divided into internal reﬂections with a smaller number of paths, shorter delays and stronger amplitudes compared to the external (far-ﬁeld) reﬂections. The internal reﬂections 4 are static as they depend on the internal components and the structure of the transceiver, while the external reﬂections vary according to the surrounding environment. Since it is difﬁcult to adapt the analog circuits with the variations of the external reﬂections, the analog RF cancellation stage reduces only the static internal reﬂections. A canceling circuit based on a balanced transformer, such as a QHx220 chip, is used in [9], [16]. The chip takes the transmitted SI as input, changes its amplitude and phase to match the received SI, and subtracts the resulting signal from the received signal. This method achieves about 20 dB reduction in received SI [16]. 3) Digital Circuitry: Processing the SI in the digital domain facilitates the use of adaptive ﬁltering for a large number of reﬂected paths due to the external environment. The digital SI cancellation is based on the general transversal symbol-synchronous FIR structure where the constant tap-delay is equal to the signal sampling period and implemented as a D-ﬂip ﬂop clocked by the sampling clock. Here, only the tap-coefﬁcients need to be speciﬁed from an estimate of the SI channel and thus we avoid the interaction between the delays and the attenuations as the case for the analog TDL. As a result, the digital processing can deal with a larger number of taps than the analog TDL to adapt to the varying external environment. The resulting canceling signal can be subtracted from the received signal at the RF input of the LNA/ADC to further reduce the SI resulting from the external reﬂections and to keep the LNA/ADC from being overloaded. This operation requires an additional digital-to-analog (DAC) converter and an upconverting radio chain to generate the RF signal. The additional components will slightly change the generated SI leading to residual SI. This RF cancellation stage can provide 30 dB of SI cancellation [14], [15], which, on top of the previously obtained 45 dB, still leaves a large amount of SI [17]–[24]. The baseband cancellation stage represents the last line of defense against the SI by reducing it after the ADC. For this reason, we should estimate the TX nonlinearities and residual SI channel resulting from the difference between the actual SI channel and the equivalent channel generated by the previous cancellation stages. In addition, related work has proposed circuits for joint analog and digital SI cancellation. Digital SI cancellation is particularly suitable for MIMO systems as the cross-interference between antennas increases the number of taps needed to reduce the SI considerably. In the same context, an all-digital SI cancellation based on a new FD transceiver structure signiﬁcantly reduces transceiver impairments [25]. This technique consists of an intermediary receiver (RX) chain to obtain a digital replica of the transmitted SI signal that will be used to cancel the SI signal and TX imperfections. The results show the combination of this digital technique and the passive RF cancellation signiﬁcantly reduces the 5 SI to be 3 dB higher than the noise ﬂoor, resulting in 67-76 %rate enhancement compared to the conventional half duplex (HD) systems operating at 20dBm of transmit power [26]–[38] 4) Spatial Beamforming: Another technique, spatial suppression, has been effective in miti- gating SI for FD MIMO system [39], [40]. This technique leverages available degrees of freedom (DoF) of the multiple antennas to cancel SI while maintaining an acceptable multiplexing gain. Analog beamforming architectures are cost-efﬁcient because they only require phase shifters. However, such architecture is primarily constrained by the low quality and limited DoF of the phase shifters where the beam pattern is managed only through the phase shifters while keeping constant amplitudes (CA). The CA constraint introduces additional performance loss. Similar work [41] proposed a design to ﬁnd the optimal analog beamformers; however, signiﬁcant performance losses are introduced when projecting the solution onto the subspace of the CA constraint. This result was explained by the fact that the projection onto the CA subspace violates the interference cancellation constraint. In addition, related work proposed various hybrid beamforming designs for FD systems. For example, the work in [42] proposed an iterative optimization algorithm to minimize the SI power and improve the spectral efﬁciency while maintaining reasonable number of iterations for the convergence. Moreover, [43] proposed beamforming design algorithm to maximize the spectral efﬁciency for a dual-hop FD relaying system wherein the performances are compared to HD and upper bound as benchmarking. B. Contributions We consider a wideband mmWave cellular system in a single-user (SU) scenario where an FD base station (BS) independently communicates with uplink and downlink user equipments (UEs). The goal is to design robust hybrid beamformers to cancel the loopback SI, avoid ADC saturation and provide the uplink UE with acceptable spectral efﬁciency relative to the downlink UE which is interference-free. The contributions follow: We aim for minimizing the SI power by jointly designing the full-digital beamformers for the base station and the UEs by using the alternating projection based on the idealized simplifying assumption of having perfect channel state information (CSI). We design in the second stage the analog beamformers by using alternating projection and then deriving closed-form solutions of the digital beamformers. With sufﬁcient number of iterations, the hybrid beamforming algorithm mitigates the SI power and nearly approaches the interference-free downlink UE performance. 6 We present quantitative comparisons of our proposed algorithms through simulation, wherein our proposed design achieves better performance gains than beam steering, Singular Value Decomposition (SVD) and angle search techniques. We analyze communication performance in terms of spectral efﬁciency, energy efﬁciency and outage probability. For spectral efﬁciency, we provide the full-digital beamformer and an upper bound to serve as benchmarking tools for the proposed hybrid beamformer. C. Organization The rest of the paper is organized as follows: Section 2 discusses the system and channel models, while the beamforming design for full-digital and hybrid architectures are detailed in Section 3. Energy efﬁciency and outage probability are analyzed in Section 4, whereas numerical results and their discussions appear in Section 5. Section 6 concludes the paper. D. Notation We use the following notation throughout the paper. We denote ()T,(),()yandP[]as the Transpose, Hermitian, Pseudo-Inverse and the probability measure operators, respectively. Moreover,kkFanddet()are the Frobenius norm and determinant, respectively. HandHare the discrete-time channel and its Fast Fourier Transform (FFT), respectively. In addition, INis the identity matrix with size NN. II. S YSTEM MODEL Fig. 2 describes the wideband mmWave full-duplex system with hybrid analog/digital beam- forming. Wideband transmission uses OFDM signaling with Ksubcarriers. At the k-th subcarrier, the symbolss[k]are transformed to the time domain using the K-point IDFT. The cyclic preﬁx (CP) of length ( Lc) is then appended to the time domain samples before applying the precoder. The OFDM block is formed by the CP followed by the Ktime domain samples and the data symbols follows E[s[k]s[k]] = KNsI, whereis the total average transmit power for the data i.e., without considering the CP, per OFDM symbol. We assume the maximum delay spread of the channel is within the CP duration. This description applies equally for uplink and downlink transmissions. 7 FBBDAC DACRF Chain RF ChainFRFWRF UplinkRF Chain RF ChainADC ADCWBB FBBDAC DACRF Chain RF ChainFRF WRFDownlink RF Chain RF ChainADC ADCWBB Ns NsLoopbackSelf-Interference Full-Duplex Base Station Uplink User Equipment Downlink User Equipment Fig. 2: Hybrid architecture of dual-hop FD relaying system. The uplink user equipment (UE) sends the data to the base station (BS) independently from the data intended to the downlink UE sent from the BS. Since the BS transmits and receives simultaneously at the same resource blocks, we model the SI leakage by the wideband channel matrix with the `-th tap Hs[`]. For uplink, the received signal at the BS in the k-th subcarrier is given by yuplink[k] =puW BS[k]Hu[k]FUE[k]su[k]+ psW BS[k]Hs[k]FBS[k]sd[k]+ W BS[k]nBS[k](1) where WBS[k];FBS[k];FUE[k]being thek-th full-digital combiner, precoder at BS and k-th full-digital precoder at the uplink UE, respectively. Hu[k],Hs[k]being thek-th uplink and SI subcarriers, respectively, while su[k],sd[k]andnBS[k]being the UE data sent to BS, BS data sent to downlink UE and the additive white Gaussian noise (AWGN) at BS with zero-mean and variance 2 u, respectively. Note that uandsare the average transmit power at BS and SI power, respectively. For downlink scenario, the received signal at UE and at the k-th subcarrier is expressed by ydownlink [k] =pdW UE[k]Hd[k]FBS[k]sd[k] +W UE[k]nUE[k] (2) where WUE[k]is thek-th fully-digital combiner at the downlink UE, dis the UE received power, nUE[k]is the AWGN at the UE with zero-mean and variance 2 dandHd[k]is thek-th 8 downlink subcarrier. Unlike the downlink scenario, the uplink received signal is corrupted by the loopback SI at the FD BS. A. Channel Model In this work, we assume the uplink and downlink MIMO channels are wideband, with a delay tap lengthLin the time domain. The `-th delay tap of the channel is represented by a NRXNTX matrix,`= 0;:::;L1, which, assuming a geometric cluster and ray based channel model given by [44] H[`] =s NRXNTX C1X c=0Rc1X rc=0 rcp(`Ts c rc) aRX(c+#rc)a TX( c+'rc)(3) whereTsis the signaling interval, cis the mean time delay of the cluster, and cand care the angles of arrival (AoA) and departure (AoD), respectively. Each ray has a relative time delay rc, relative AoA ( #c) and AoD ( 'c) shifts, and a complex gain rc. Here, is the pathloss andp( )is the raised cosine pulse shape evaluated at . In addition, aRX()andaTX( )are the antenna array response vectors of the RX and TX, respectively. The RX array response vector is aRX() =1pNRXh 1;ej2d sin();:::;ej2d (NRX1) sin()iT (4) The channel at the k-th subcarrier is given by H[k] =L1X `=0H[`]ej2k K`(5) wherek= 0;1;:::K1. B. Self-Interference Channel Model As illustrated in Fig. 3, the SI leakage at the BS is modeled by the channel matrix Hs. The SI channel is decomposed into line-of-sight (LOS) component modeled by Hlosand a non-line- of-sight (NLOS) leakage described by Hnloswhich follows the channel model (3). The LOS SI leakage matrix [Hlos]pq=1 dpqej2dpq  (6) 9 wheredpqis the distance between the p-th antenna in the TX array and q-th antenna in the RX array at BS. The aggregate `-th tap SI channel Hs[`]can be expressed as [45, Eq. (2)] Hs[`] =r + 1Hlos |{z} Near-Field+r 1 + 1Hnlos[`] |{z} Far-Field(7) whereis the Rician factor. C. Transceiver Impairments 5G mmWave cellular systems operate in 24, 28, 37, 39 and 47 GHz bands. In US FCC auctions, the corresponding transmission bandwidths are 0.7, 0.85, 1, 1, and 1.4 GHz broken into 100 MHz subbands. MmWave systems need very large antenna arrays to overcome high propagation losses. The massive jump in the number of antenna elements, transmission bandwidth, and data converter conversion rates for 5G mmWave systems has lead to new basestation architectures for energy efﬁciency, such as hybrid analog/digital beamforming. Introducing full-duplex transceivers also has several design challenges to achieve doubling spectral efﬁciency. High power ampliﬁer (HPA) nonlinearity can severely degrade the system performance such as the creation of irreducible outage/error ﬂoor and/or spectral efﬁciency saturation. In addition, the ampliﬁer produces intermodulation products that translate into spectral regrowth (or spectral shoulder) resulting in interference in adjacent subcarriers and loss of information. Conventional techniques to compensate nonlinear effects include those based on Bussgang Linearization. Moreover, heuristic HPA nonlinearity models have been proposed such as soft ω RX Array12 qpTX Array dpqLOS NLOS d Fig. 3: Relative position of TX and RX arrays at BS. Given that the TX and RX arrays are collocated, the far-ﬁeld assumption that the signal impinges on the antenna array as a planar wave does not hold. Instead, for FD transceivers, it is more suitable to assume that the signal impinges on the array as a spherical wave for the near-ﬁeld LOS channel. 10 envelope limiter, traveling wave tube ampliﬁer, and solid state power ampliﬁer [3], [46], [47]. Furthermore, mmWave transmitter impairments may affect SI cancellation for full-duplex systems. In order to subtract the received SI signal, any modiﬁcations that occur on the transmitter side have to be captured. This includes channel features and analog components such as the power ampliﬁer and IQ mixer. The transmitted SI is slightly modiﬁed as it moves through the transmit chain and such modiﬁcations are relatively negligible compared to the desired signal; however, they are of signiﬁcant magnitude compared to the intended signal and will limit the performance of full-duplex systems. In practical implementations, the inband image resulting from the transmit IQ mixer is about 30 dB lower than the direct signal. In the presence of strong SI of about 50 dB higher than the transmit signal, this IQ image represents additional interference for the intended signal and also has to be reduced. Previous work selects transceiver component impairments for full-duplex transceivers [48]–[50]. Alternate high-speed DACs provide a direct conversion of the baseband signal to RF, also known as a direct RF transmitter. This technique can avoid many nonlinear distortions related to up-conversion. The work in [51] consists of two full-duplex machine-to-machine nodes communicating over a bidirectional link. The direct and self-interference channels are narrowband. They propose full-digital and hybrid beamformers, and analyze spectral efﬁciency. In the next section, we modify the algorithms to support a full-duplex basestation communicating simultaneously with half-duplex uplink and downlink UEs; extend the framework to wideband uplink, downlink and self-interference MIMO channels; and investigate outage probability and energy efﬁciency in addition to spectral efﬁciency. Remark. Nonlinearities and other imperfections in mmWave analog/RF hardware have sig- niﬁcant impact on full-duplex transceiver communication performance. Modeling transceiver hardware impairments is out of the scope of this work; however, we treat these impairments as additional sources of SI. For example, the aggregate SI power used in this work is around 120 dB. The near-far problem incurs an SI of about 20–60 dB (depending on whether the UE is near the BS, at mid-range or at the cell edge) and the remaining SI comes from transceiver impairments. 11 III. B EAMFORMING DESIGN A. Full-Digital Beamforming In the full-digital domain, the uplink spectral efﬁciency is I(u) =1 KK1X k=0log det ( INs+uW BS[k]Hu[k]FUE[k] Qu[k]1F UE[k]H u[k]WBS[k](8) where Qu[k]is the SI plus noise covariance matrix at the k-th subcarrier given by Qu[k] =sW BS[k]Hs[k]FUE[k]F UE[k]H s[k]WBS[k] +2 uW BS[k]WBS[k]:(9) For downlink scenario, the spectral efﬁciency is given by I(d) =1 KK1X k=0log det ( INs+dW UE[k]Hd[k]FBS[k] Qd[k]1F BS[k]H d[k]WUE[k](10) where Qd[k]is the noise covariance matrix for the k-th subcarrier. The objective of this design is to construct robust beamformers to provide the uplink UE with acceptable spectral efﬁciency compared to the downlink UE, given the former is affected by SI. We decompose the beamforming design into two phases. The ﬁrst stage maximizes the rate for the uplink user, and the downlink user beamforming is designed in the second stage. Starting with the uplink scenario, we adopt a sub-optimal approach in which the zero-forcing constraint is imposed, thereby leading to the following optimization problem: max WBS[k];FBS[k];FUE[k];WUE[k] k=0;:::;K1I(u) +I(d) (11) 12 Subject to W BS[k]WBS[k] =INs F BS[k]FBS[k] =INs F UE[k]FUE[k] =INs W UE[k]WUE[k] =INs W BS[k]Hs[k]FBS[k] =0(Zero-Forcing Constraint ) k= 0;:::;K1 We observe that the BS beamformers have to be jointly designed to maximize the beamformed received power and minimize the SI power simultaneously. In other terms, the BS beamformers are coupled in the uplink and downlink optimization. To cope with this problem, we ﬁx the precoder and solve for the combiner and then ﬁx the combiner and solve for the precoder, and this process will be repeated iteratively until the cost function is maximized. This process is also known as Zero-Forcing cyclic maximization and a convergence criterion should be triggered to stop the iterative process. The convergence is guaranteed since the cost function is bounded as the beamformers are normalized and the transmit power is constrained. Accounting for the Zero-Forcing constraint, the optimization problem has the following generic form max Xlog det ( IN+XAAX) (12) Subject to XX=IN XC=0 where X2CMNis the beamformer variable, A2CMNis a given beamformed subcarrier and C2CMPis a given beamformed SI channel that spans the SI subspace. Note that this generic form is to design the beamformer for a single subcarrier. Thereby, this routine will be repeated for all the system subcarriers. The solution to this problem is summarized by Algorithm 1. Note that the proof to the solution is given by [51, Appendix A]. Remark. Given the solution matrix X2CMN, which is the digital beamformer solution to the problem,MandNare the numbers of antennas and spatial streams, respectively. Mshould be large enough to sustain Nspatial streams and the remaining P=MNdegrees of freedom should be used to cancel the SI. 13 Algorithm 1 Full-Digital Beamforming 1:function BASEBAND (A;C;N) 2: P? ICCy 3: X NDominant left singular vectors of P?A 4: return X 5:end function 6:Input H s[k];Hu[k];Hd[k]; k= 0;:::;K1 7:Initialize F BS[k];FUE[k]; k= 0;:::;K1 8:fork 0 :K1do 9: fort 1 :Nouterdo 10: WBS[k] BASEBAND (Hu[k]FUE[k];Hs[k]FBS[k];Ns) 11: FUE[k] NsDominant left singular vectors of H u[k]WBS[k] 12: WUE[k] NsDominant left singular vectors of Hd[k]FBS[k] 13: FBS[k] BASEBAND (H d[k]WUE[k];H s[k]WBS[k];Ns) 14: end for 15:end for 16:return F BS[k];WBS[k];WUE[k];FUE[k]; k= 0;:::;K1 B. Hybrid Beamforming In this part, we decompose the full-digital beamformers into their equivalent analog and digital factors. The analog beamformers are restricted by the constant amplitude (CA) hardware constraint. This constraint restricts the control of the array beam by tuning the phases shifters while keeping the amplitudes constant. The CA constraint deﬁnes the set of feasible solutions or equivalently the CA subspace. Like the full-digital beamforming design, we adopt the same Zero-Forcing cyclic maximization process. Starting with the formulation of the optimization problem as follows: max WBS,RF;FBS,RF;FUE,RF;WUE,RF WBS,BB[k];FBS,BB[k];FUE,BB [k];WUE,BB [k] k=0;:::;K1I(u) +I(d) (13) 14 Subject to WBS[k] =WBS,RFWBS,BB[k] W BS[k]WBS[k] =INs FBS[k] =FBS,RFFBS,BB[k] F BS[k]FBS[k] =INs FUE[k] =FUE,RFFBS,BB[k] F UE[k]FUE[k] =INs WUE[k] =WUE,RFWUE,BB[k] W UE[k]WUE[k] =INs W BS,RFGsFBS,RF =0 WBS,RF;FBS,RF2VNBSNRF WUE,RF;FUE,RF2VNUENRF k= 0;:::;K1 where Gsis the SI subcarrier with the lowest energy, VMLis the set of feasible analog solutions or the CA subspace. Note that the Zero-Forcing constraint is deﬁned in the analog and not in the digital domain in order to cancel the SI before the downconversion, sampling and quantization and hence avoiding the ADC saturation by the high SI levels. Like the full-digital case, the optimization problem for hybrid beamforming has the following generic form max XBB;XRFlog det ( IN+X BBX RFAAXRFXBB) (14) Subject to X BBX RFXRFXBB=IN X RFC=0 XRF2VML Note that XRF2CMLandXBB2CLNare the optimization variables, A2CMN,C2 CMPare given matrices. The constraints show that the analog beamformer should satisfy both the Zero-Forcing and CA constraints. In fact, imperfect SI cancellation may introduce severe degradation into the spectral efﬁciency and this might occur because the analog beamformers need not only to be projected onto the SI null subspace but also projected onto the CA subspace. 15 The projection onto the latter subspace may violate the Zero-Forcing constraint and hence the SI may not be perfectly eliminated. For example, the authors in [52] discussed how the CA constraint ( XRF2VML) violated the Zero-Forcing condition ( X RFC=0) and they quantiﬁed the losses in terms of spectral efﬁciency. To circumvent this limitation, a common approach to solve such problem is to use the Alternating Projection method [53] which has been shown to yield better results for analog-only architecture [54]. In this work, we alternately project the analog beamformers between CA and Zero-Forcing subspaces to ﬁnd a common subspace that optimally satisﬁes both conditions. The convergence after successive iterations of alternating projection provides the optimal subspace that eliminates SI and maximizes the sum spectral efﬁciency. To satisfy both conditions, we propose a two nested iterative loops wherein the outer loop handles the Zero-Forcing cyclic maximization (like full-digital case) while the inner loop deals with the alternating projection. Because closed-form analog solutions do not exist for (14), the optimal digital solution X BBcan be expressed in terms of the analog beamformers. The proof of the optimal digital solution is provided in [51, Appendix B]. More detailed analysis of the hybrid beamforming design is reported in Algorithm 2. IV. P ERFORMANCE ANALYSIS A. Energy Efﬁciency The energy efﬁciency, expressed in bits/s/Hz/Watt or bits/J/Hz, is deﬁned as the ratio between the spectral efﬁciency and the total power consumption. It is expressed as J() =I() Total(15) whereTotalis the total power consumption. We adopt full-digital and hybrid combiners (DC and HC) where the total power consumption model for each architecture is deﬁned by DC Total=NRX(LNA+RF+ 2ADC) (16) HC Total=NRX(LNA+SP+NRFPS) +NRF(RF+C+ 2ADC) (17) whereRFis the power consumption per RF chain which is deﬁned by RF=M+LO+LPF+BBamp (18) 16 Algorithm 2 Hybrid Beamforming 1:function ANALOG (A;C;L) 2: XRF LDominant left singular vectors of A 3: P? ICCy 4: fork 1 :Ninnerdo 5: Y P?XRF 6: fori 1 :Mandj 1 :Ldo 7: (XRF)ij (Y)ij j(Y)ijj 8: end for 9: end for 10: return X RF 11:end function 12:function DIGITAL (XRF;A;N) 13: Compute SVD XRF=URFSRFV RF 14: Q NDominant left singular vectors of U RFA 15: XBB VRFS1 RFQ 16: return X BB 17:end function 18:Input H s[k];Hu[k];Hd[k]; k= 0;:::;K1 19:Select uplink/downlink subcarrier of highest energy 20:k?= max k=0;:::;K1kH[k]k2 F 21:Select SI subcarrier of lowest energy 22:k?= min k=0;:::;K1kHs[k]k2 F 23:SetGu Hu[k?];Gd Hd[k?]andGs Hs[k?] 24:Initialize F RF,BS;FRF,UE;FBB,BS[k];FBB,UE[k]; k= 0;:::;K1 25:SetFUE[k] FRF,UEFBB,UE[k]; k= 0;:::;K1 26:SetFBS[k] FRF,BSFBB,BS[k]; k= 0;:::;K1 27:fork 0 :K1do 28: fort 1 :Nouterdo 29: WRF,BS ANALOG (GuFUE[k];GsFRF,BS;NRF) 30: WBB,BS[k] DIGITAL (WRF,BS;Hu[k]FUE[k];Ns) 31: SetWBS[k] WRF,BSWBB,BS[k] 32: WRF,UE NRFDominant left singular vectors of GdFBS[k] 33: WBB,UE[k] DIGITAL (WRF,UE;Hd[k]FBS[k];Ns) 34: SetWUE[k] WRF,UEWBB,UE[k] 35: FRF,UE NRFDominant left singular vectors of G uWBS[k] 36: FBB,UE[k] DIGITAL (FRF,UE;H u[k]WBS[k];Ns) 37: SetFUE[k] FRF,UEFBB,UE[k] 38: FRF,BS ANALOG (G dWUE[k];G sWRF,BS;NRF) 39: FBB,BS[k] DIGITAL (FRF,BS;H d[k]WUE[k];Ns) 40: SetFBS[k] FRF,BSFBB,BS[k] 41: end for 42:end for 43:return F BS[k];WBS[k];WUE[k];FUE[k]; k= 0;:::;K1 17 TABLE I POWER CONSUMPTION OF EACH DEVICE [55]. Device Notation Value Low Noise Ampliﬁer (LNA) [56] LNA 39 mW SplitterSP 19.5 mW Combiner [56] C 19.5 mW Phase Shifter [57], [58] PS 2 mW Mixer [59] M 16.8 mW Local Oscillator [60] LO 5 mW Low Pass Filter [60] LPF 14 mW Baseband Ampliﬁer [60] BBamp 5 mW ADCADC Table III B. Outage Probability Once a transmission strategy is speciﬁed, the corresponding outage probability for rate R (bits/s/Hz) is then Pout(SNR;R) =P[I(SNR)<R]: (19) With powerful channel codes, the probability of error when there is no outage is very small and hence the outage probability is an accurate approximation for the actual block error probability. Modern radio systems such as UMTS and LTE operate at a target error probability. Therefore, the primary performance measure is the maximum rate2at each SNR, such that the outage probability is less than , i.e., R(SNR) = max f:Pout(SNR;)g (20) C. Upper Bound For interference-free scenario, the optimal beamformers diagonalize the channel. By applying the SVD successively on all subcarriers, we retrieve the singular values associated to each subcarrier matrix. For each subcarrier, the singular values are listed in descending order and we will extract the ﬁrst Nsmodes associated to the spatial streams. Equivalently, the upper bound is given by I(SNR) =1 KK1X k=0Ns1X `=0log 1 +SNR KNs`(H[k])2 (21) 2In this work, we deﬁne the notion of rate with outage as the average data rate that is correctly received/decoded at the receiver which is equivalent to the throughput. In other standards in the literature, the rate with outage is assimilated with the transmit data rate. The only difference is if we consider rate with outage as the throughput, we account for the probability of bursts (outage) and we multiply by the term (1- ), while for the transmit data rate, the term (1- ) is not accounted anymore. 18 where`(H)is the`-th singular value of the channel matrix H. Note that the upper bound derivation follows the same rules for uplink as well as the downlink scenario. V. N UMERICAL RESULTS In this section, we discuss the numerical results of the communication performance measures obtained by Monte Carlo simulation3. Unless otherwise stated, Table II illustrates the values of the system parameters. TABLE II SYSTEM PARAMETERS [44]. Parameter mmWave Sub-6 GHz Carrier frequency 28 GHz 3.5 GHz Bandwidth 850 MHz 150 MHz Number UE antennas 4 2 Number relay/BS antennas 16TX+16RX 8TX+8RX Antenna separation =2 =2 Antenna correlation None None Number clusters 4 10 Number rays per cluster 10 20 Angular spread 22 Pathloss exponent 3 3 Number of subcarriers ( K) 16 32 Cyclic preﬁx length K=4 K=4 Number RF chains 2 2 Number spatial streams 2 2 Angle between FD arrays =2 =2 Distance between FD arrays 2 2 Signal-to-Interference Ratio -120 dB -120 dB Rician factor 5 dB 5 dB Raised cosine roll-off factor 1 1 Outer iterations ( Nouter) 10 10 A. Primary Evaluation Fig. 4 presents spectral efﬁciency results vs. average SNR. Primarily, we observe the full- digital design simulated with Nouter= 10 iterations is robust to the SI and very close to the upper bound. Unlike the full-digital case, the hybrid beamforming design experiences limitations observed mainly for the uplink, even though the downlink scheme seems to be very close to 3For all cases, 1000 realizations of the random variables were generated to perform the Monte Carlo simulation in MATLAB. 19 30 20 10 0 10051015 ) Ninner= 20Ninner= 30Ninner= 50 Uplink SNR (dB)Spectral Efﬁciency (bits/s/Hz)Upper Bound Full-Digital Downlink Fig. 4: Spectral efﬁciency results: Performance is evaluated in terms of uplink and downlink spectral efﬁciency for hybrid architecture. The uplink rate is evaluated for different number of iterations of the inner loop corresponding to alternating projections. In addition, full-digital and upper bound are evaluated and serve as benchmarking tools. MmWave conﬁguration is considered for this simulation. the full-digital performance since the downlink UE is interference-free. In the downlink, the small gap between the full-digital and hybrid beamforming is caused by the losses incurred by the hardware constraint. In addition, we observe that uplink performance improves with the number of inner alternating projections. The low performance at Ninner= 10 iterations is due to the zero-forcing constraint being approximately but not completely satisﬁed; hence, the optimal precoders and combiners are not yet obtained. At Ninner= 50 iterations, the beamformers are closely projected onto the zero-forcing null-space, which further enhances the uplink rate. The results are conﬁrmed by [51] but for machine-to-machine narrowband FD systems. B. Alternating Projections and Signal-to-Interference Ratio Fig. 5 presents the spectral efﬁciency for different numbers of inner loop alternating projections across a range of SIRs. The full-digital beamformer remains close to the upper bound while the downlink rate for hybrid architecture is still limited by the hardware constraint. In accordance with the conclusions drawn from Fig. 4, the uplink rate improves with the number of inner loop alternating projections. We further observe that the design is sensitive to high SI power (SIR between -160 and -140 dB) speciﬁcally for a few inner iterations. However, the performance can be further improved by running more iterations of alternating projections. As SIR increases to around -40 dB which corresponds to a user near the BS, the uplink performance matches the downlink (interference-free) performance regardless of the number of iterations. 20 160 140 120 100 80 60 4002468 Ninner= 10Ninner= 20Ninner= 30 Uplink) SIR (dB)Spectral Efﬁciency (bits/s/Hz)Upper Bound Full-Digital Downlink Fig. 5: Spectral efﬁciency results: Performances are evaluated at SNR = 0 dB and across a range of SIR from -160 to -40 dB which corresponds to cell-edge user (-160 dB), middle user (around -80 dB) and near user (-40 dB), respectively. The algorithm robustness is tested for various number of iterations of the alternating projections routine. Note that mmWave setting is assumed for this scenario. C. Proposed vs. Conventional Approaches Fig. 6 compares proposed and conventional approaches implemented in analog-only architec- tures. We observe that conventional designs are very sensitive to the SI because the zero-forcing and hardware conditions are not properly handled. In contrast, the proposed design is more resilient to SI and achieves higher sum spectral efﬁciency around 13 bits/s/Hz at 10 dB of SNR 30 20 10 0 10051015 ) o Angle Search[61] Beam Steering [61] SVD Approach [61] SNR (dB)Sum Spectral Efﬁciency (bits/s/Hz)Upper Bound Full-Digital Proposed Fig. 6: Sum spectral efﬁciency results: Comparisons are made between the proposed and conventional approaches. Note that the conventional techniques presented in [61] are developed for machine to machine FD systems. In this work, we changed these techniques accordingly to support the proposed system model. These performances are evaluated for analog-only architecture in accordance with [61]. We adopt the sub-6 GHz setting for this scenario. 21 whereas beam steering, SVD and angle search techniques achieve roughly 9, 9, and 7 bits/s/Hz. D. Degrees of Freedom 5 10 15 2000:20:40:60:81 16 antennas 64 antennas Zoom in for 64 antennas setting Spectral Efﬁciency (bits/s/Hz)Outage ProbabilityUpper Bound Full-Digital Uplink Downlink 1010:51111:5120:30:40:50:6 Fig. 7: Outage probability results: Comparisons are made between uplink and downlink scenarios for hybrid beamforming with full-digital as well as upper bound which serve as benchmarking tools. In addition, the comparison involves two settings with 16 and 64 antennas at the BS/relay. MmWave conﬁguration is considered in this scenario while the SNR is maintained at 0 dB while Ninner= 50 iterations. Fig. 7 illustrates the outage performance with respect to a given range of target rate re- quirements for 16 and 64 antennas at the relay/BS, respectively. We observe that for the same inner alternating projections iterations, the gap between the uplink and downlink rate for hybrid architectures changes with the number of antennas. As we conclude so far, the rate performances improves with the number of iterations of alternating projections, however, this improvement is also limited by the number of antennas. In fact, the number of antennas are shared between providing spatial multiplexing gain (sustain enough number of spatial streams) and SI cancella- tion. If the number of antennas is low, then there is not enough degree of freedom to perfectly eliminate the SI and hence the alternating projections routine (length of iterations) becomes useless, i.e., increasing Ninnerdoes not bring any further enhancement. The 64 antennas setting which provides enough degree of freedom provides not only the spectral efﬁciency (multiplexing gain) but also perfectly cancel the SI which is observed by the complete matching between the uplink and downlink rates. E. Power Consumption and RF Chains In this simulation, we will evaluate the energy efﬁciency for different variants wherein the relative parameters of each one is provided by Table III. 22 Fig. 8 illustrates the energy efﬁciency as a function of the number of RF chains at the relay/BS for different variants. We observe that for low number of RF chain, variant 1 seems to be well advocated for power-efﬁcient systems as it provides better energy efﬁciency compared to the other variants. This is explained by the fact that for a few number of RF chains, the rate at which the power consumption increases is higher that the rate of spectral efﬁciency. However, as the number of RF chains increases, the spectral efﬁciency improves at higher rate compared to the power consumption as a function of the RF chains wherein variant 4 becomes the best one since it provides hefty beamforming gain from the massive number of antennas. Furthermore, the effect of the ADC power can be easily observed by comparing variants 2 and 3 (having the same number of antennas). We notice that variant 2 is more power-efﬁcient than variant 3 since the latter requires more ADC power. We also note that variant 2 remains more power-efﬁcient than variant 3 across all the number of RF chains. TABLE III PARAMETERS OF EACH VARIANT [1], [62]. Parameter Variant 1 Variant 2 Variant 3 Variant 4 Frequency 28 GHz 39 GHz 39 GHz 73 GHz Bandwidth 850 MHz 1.4 GHz 1.6 GHz 2 GHz ADC Power 250 mW 400 mW 450 mW 550 mW BS Antennas 32 64 64 128 0 2 4 6 8 10 12 140246 Number of RF ChainsEnergy Efﬁciency (bits/Joule/Hz)Variant 1 Variant 2 Variant 3 Variant 4 Fig. 8: Energy efﬁciency performance: Results are evaluated for four variants and for different number of RF chains at the relay/BS. The SNR is ﬁxed at 0 dB. The simulations are performed for uplink scenario with Ninner= 50 iterations. 23 VI. C ONCLUSION We analyzed a single-cell, single-user MIMO scenario in which the uplink and downlink UEs are simultaneously communicating with a full-duplex relay/BS. We proposed a hybrid beam- forming design to maximize the uplink plus downlink sum capacity while bringing the SI below the noise ﬂoor. We concluded that the spectral efﬁciency improvement in the number of inner alternating projection iterations while the enhancement is also limited by the available degrees of freedom. We also showed that the proposed technique is more efﬁcient in SI cancellation compared to conventional beamsteering, SVD and angle search techniques. Although spectral efﬁciency and outage probability remain important measures, energy ef- ﬁciency is also an important concern due to the high power consumption for transceiver ar- chitectures to support mmWave bands. We evaluated energy efﬁciency for 28, 39 and 73 GHz mmwWave band variants and showed how the behavior depends on the number of RF chains. We also noticed that ADC power consumption can signiﬁcantly impact energy efﬁciency, as illustrated by variants in the 39 GHz band. An important future direction is the joint optimization of spectral and energy efﬁciency in designing the beamformers with low resolution ADCs.