sensors Beamforming Based Full-Duplex for Millimeter-Wave Communication Article

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1 sensors Artice Beamforming Based Fu-Dupex for Miimeter-Wave Communication Xiao Liu 1,2,3, Zhenyu Xiao 1,2,3, *, Lin Bai 1,2,3, Jinho Choi 4, Pengfei Xia 5 and Xiang-Gen Xia 6 1 Schoo of Eectronic and Information Engineering, Beihang University, Beijing 191, China; eeiuxiao@buaa.edu.cn X.L.);.bai@buaa.edu.cn L.B.) 2 Coaborative Innovation Center of Geospatia Technoogy, Wuhan 4379, China 3 Beijing Key Laboratory for Network-Based Cooperative Air Traffic Management, and Beijing Laboratory for Genera Aviation Technoogy, Beijing 191, China 4 Schoo of Eectrica Engineerng and Computer Science, Gwangju Institute of Science and Technoogy GIST), Gwangju 65, Korea; jchoi114@gist.ac.kr 5 Schoo of Eectronics and Information Engineering and the Key Laboratory of Embedded System and Service Computing, Tongji University, Shanghai 92, China; pengfei.xia@gmai.com 6 Department of Eectrica and Computer Engineering, University of Deaware, Newark, DE 19716, USA; xianggen@ude.edu * Correspondence: xiaozy@buaa.edu.cn; Te.: ext. 6249) Academic Editors: He Henry) Chen, Yonghui Li, Kei Sakaguchi and Yong Li Received: 26 Apri 16; Accepted: 14 Juy 16; Pubished: 21 Juy 16 Abstract: In this paper, we study beamforming based fu-dupex FD) systems in miimeter-wave mmwave) communications. A joint transmission and reception Tx/Rx) beamforming probem is formuated to maximize the achievabe rate by mitigating sef-interference SI). Since the optima soution is difficut to find due to the non-convexity of the objective function, suboptima schemes are proposed in this paper. A ow-compexity agorithm, which iterativey maximizes signa power whie suppressing SI, is proposed and its convergence is proven. Moreover, two cosed-form soutions, which do not require iterations, are aso derived under minimum-mean-square-error MMSE), zero-forcing ZF), and maximum-ratio transmission MRT) criteria. Performance evauations show that the proposed iterative scheme converges fast within ony two iterations on average) and approaches an upper-bound performance, whie the two cosed-form soutions aso achieve appeaing performances, athough there are noticeabe differences from the upper bound depending on channe conditions. Interestingy, these three schemes show different robustness against the geometry of Tx/Rx antenna arrays and channe estimation errors. Keywords: fu dupex; sef-interference canceation; beamforming; miimeter-wave; mmwave 1. Introduction Fu-dupex wireess communications FDWC) for simutaneous transmission and reception Tx/Rx) in the same frequency band [1 5] have attracted increasing attention recenty due to the potentia of doubing the spectrum efficiency. However, in order to achieve FDWC, the sef-interference SI) generated from a oca transmitter to a oca receiver must be mitigated for satisfactory performances [1,6], and this constitutes one of the critica chaenges in FDWC. There are basicay three different approaches for SI canceation. The first one is radio-frequency RF) canceation or anaog canceation), where the RF signa to be transmitted at the Tx is expoited as a reference RF signa for SI canceation in the Rx RF chain [1,3,4,7]. The second one is antenna canceation, where mutipe transmit receive) antennas are carefuy paced to generate two repicas with opposite phases [8,9] such that canceation can be achieved by just adding these two repicas. The third one is digita canceation, which is generay used together with RF or antenna canceation Sensors 16, 16, 113; doi:.339/s

2 Sensors 16, 16, of 22 to further mitigate the residua SI at baseband [1,3 5,8,9]. Bharadia et a. [] showed that fu-dupex FD) radio with a combined canceation approach is abe to cance about 1 db SI. In this paper, we consider a beamforming-based approach to mitigate SI for FDWC. A distinct advantage of this approach is that with beamforming canceation, some of the conventiona RF, antenna and baseband canceation operations may be avoided, and this greaty reduces the system compexity. Beamforming canceation is particuary meaningfu for miimeter wave mmwave) wireess communications, where arge antenna arrays are typicay required to compensate for the high pass oss in the mmwave frequency band [11 19]. Note that athough frequency re-use may be easier for mmwave communication since the pass oss is high for mmwave signa, more spectrum efficiency is amost aways a pus. To improve the spectra efficiency for mmwave communication may be aways favored for certain scenarios, e.g., mmwave backhau appications, where high capacity is required to support high data rate. Aso, in the case of mmwave ceuar with dense users, e.g., stadiums and movie theaters, FD transmission can significanty increase the muti-user capacity. On the other hand, the FD mmwave communication does not significanty increase the system compexity. To iustrate this, et us compare the compexity of an FD-mmWave node and that of a reguar mmwave node with frequency-division dupex FDD). There are a Tx RF chain and Tx antenna array, as we as a Rx RF chain and a Rx antenna array, at both the FD-mmWave node and the reguar FDD mmwave node. The ony difference is that at the FD-mmWave node the Rx needs to mitigate SI, whie at the reguar FDD mmwave node, the Rx does not need to mitigate SI. For FD-mmWave communication, SI canceation can be done by using the beamforming technoogy proposed in this paper, which ony needs to contro the antenna weight vectors and amost does not increase the system compexity. In brief, if beamforming technoogy is adopted to mitigate the SI, the compexity of an FD-mmWave node is simiar to that of a reguar FDD mmwave node. However, in order to reaize FD in mmwave communications, we face a joint Tx/Rx beamforming JTR-BF) probem to maximize the Tx/Rx achievabe rate. As this probem is non-convex, suboptima soutions are expected. The existing JTR-BF schemes proposed in mmwave communications are basicay infeasibe, because SI was not considered in these schemes [13, 26]. Athough the mitigation of oopback SI [27,28] as we as the utiization of oopback SI [29,3] are considered in mutipe-input mutipe-output MIMO) reays, these methods cannot be empoyed in our setup, as their signa modes are buit for reay systems, but in this paper a bi-directiona FD transmission system is considered. In [31], digita beamforming to cance SI in FDWC was studied from an experimenta perspective, where the JTR-BF with SI was not anayticay investigated. Athough some suboptima distributed soutions proposed for the K-user interference aignment IA) probem [32 35], e.g., the max-signa-to-interference-pus-noise-ratio Max-SINR), maximum power Max-Power), minimum eakage Min-Leakage), and minimum-mean-square-error MMSE) schemes [35], may be appicabe, they are a iterative soutions designed for genera K-user IA probems. Appying them to FD mmwave communications woud resut in high computationa compexity for arge antenna arrays. Moreover, the convergence of some of them is not yet proven in the iterature [35] to the best of our knowedge. In this paper, we propose severa suboptima soutions to the JTR-BF probem for FD mmwave communications. Firsty, an iterative agorithm, which iterativey maximizes the signa power with zero-forcing ZF) SI ZF-Max-Power), is proposed, and its convergence is proven. Next, two cosed-form soutions are derived under MMSE, ZF, and maximum-ratio transmission MRT) criteria, namey a ower bound based MMSE soution LB-MMSE) and ZF SI with MRT SI-ZF-MRT), where iterations are not required. Performance evauations show that ZF-Max-Power approaches an upper bound on the joint achievabe rate, and it needs ony two iterations on average to achieve the convergence with random initia points. These performances of ZF-MAx-Power are amost the same as those of the best baseine for the IA probem [32 35], namey Max-SINR. However, the convergence of Max-SINR is unproven yet [35] to the best of our knowedge, and the computationa compexity of ZF-Max-Power is significanty ower than that of Max-SINR since matrix inversion is not needed.

3 Sensors 16, 16, of 22 The two cosed-form soutions achieve suboptima performances to the upper bound depending on channe conditions. In addition, ZF-Max-Power and SI-ZF-MRT are robust against the geometry of Tx/Rx antenna arrays due to the operation of ZF SI, whie LB-MMSE is not. ZF-Max-Power and LB-MMSE are robust against channe estimation errors, whie SI-ZF-MRT is not. These resuts verify the feasibiity of FD mmwave communication and the effectiveness of beamforming canceation. The rest of this paper is organized as foows. In Section 2, we introduce the system mode and formuate the probem. In Section 3, we study the optimization probem, propose the ZF-Max-Power approach, and conduct the convergence anaysis and compexity comparison. In Section 4, we propose the two cosed-form beamforming schemes, namey LB-MMSE and SI-ZF-MRT. In Section 5, we present performance evauations. The concusions are drawn asty in Section 6. Notation: a, a, A, and A denote a scaar variabe, a vector, a matrix, and a set, respectivey. ), ) T and ) H denote conjugate, transpose and conjugate transpose, respectivey. In addition, [x 1, x 2,..., x M ] denotes a row vector with its eements being x i. Some other operations used in this paper are defined as foows. E{ } Expectation operation. x Absoute vaue of scaar variabe x. x 2-norm of vector x. x, y Inner product, equas to y H x. x Optima vaue of variabe x. 2. System Mode and Probem Formuation 2.1. System Mode An FD mmwave communication system consisting of two nodes, namey Node #1 and Node #2, is iustrated in Figure 1. Each node is equipped with a transmit antenna array and a receive antenna array, and supports ony one data stream [21,22,36]. We denote by n t1 and n t2 the numbers of antenna eements of the transmit arrays at Node #1 and Node #2, respectivey, whie by n r1 and n r2 those of the receive arrays at Node #1 and Node #2, respectivey. In our mode, Node #1 transmits signas to Node #2 and receives signas from Node #2 simutaneousy; thus both the nodes suffer from SI transmitted by the oca transmitters. It is noteworthy that athough we depict separate antenna arrays for the Tx/Rx chains in Figure 1 this structure is indeed common in FDWC [1,3 5,8]), the Tx/Rx chains may aso share the same antenna array [1,]. Fortunatey, our signa mode is suitabe for both cases. Note that the SI with a shared array may be even higher than with separate arrays. Node #1 Far fied Node #2 Tx Rx Near Fied Near Fied Rx Tx Figure 1. Iustration of the FD mmwave communication system.

4 Sensors 16, 16, of Channe Mode As we can see from Figure 1, there are two types of channes. The first one is the communication channe, which represents the channe for information-bearing signas exchanged between Node #1 and Node #2, i.e., H 12 and H 21, where H ij represents the channe from Node #i to Node #j. The other one is the SI channe. Ceary, H 11 and H 22 are the SI channes Communication Channe We first consider the mode of communication channe. Since the distance between these two nodes is generay much greater than the waveength of mmwave, the commony used far-fied channe mode, which has a pane wavefront, is suitabe for H 12 and H 21. According to channe measurement resuts for mmwave communication [13,37], mosty refection contributes to generating mutipath components MPCs) besides the LOS component; scattering and diffraction effects are itte due to the extremey short waveength of mmwave communication. Thus, the MPCs in mmwave communication have a feature of directivity [22 24,38,39], i.e., different MPCs have different physica anges of departure AoDs), i.e., θ m 12) and θ 21), as we as anges of arriva AoAs), e.g., φ m 12) and φ 21), as shown in Figure 1. In genera, mmwave signas have a wide band, and thus a frequency seective channe may be suitabe [4]. However, with beamforming ony a very sma number or even ony one) of strong MPCs may be searched out to form beams between Tx and Rx. As a resut, the effect of deay spread may be substantiay mitigated [41]. Due to this reason, a frequency fat channe mode is extensivey used in mmwave communication [22 24,38,39], and the narrow band) communication channes can be expressed as H 12 = n t1 n r2 M m=1 α m g 12 φ m 12) )h H 12 θ12) m ) 1) and H 21 = n t2 n r1 L =1 β g 21 φ 21) )h H 21 θ21) ) 2) where M and L are the numbers of MPCs, α m and β are the coefficients of MPCs, g 12 φ m 12) ) and g 21 φ 21) ) are receive steering vectors, h 12 θ m 12) ) and h 21 θ 21) ) are transmit steering vectors of H 12 and H 21, respectivey. For uniform inear arrays ULAs) with haf-waveength spacing, these steering vectors are defined as Equation 4), and they are a functions of the corresponding steering anges. Athough ULA is adopted in this paper, the deveoped schemes are aso feasibe for other types of arrays, ike uniform panar array UPA) or circuar array, because different types of arrays affect ony the channe matrices. For convenience, we have the foowing normaization: M E{ α m 2 L } = E{ β 2 } = 1. 3) m=1 =1 In the case of Tx/Rx sharing the same antenna array at a node, we have n r1 = n t1, n r2 = n t2, L = M, θm 12 = φm 21, θm 21 = φm 12, and α m = β m, m = 1, 2,..., M, i.e., parameters of H 12 are the same as those of H 21. However, since Tx/Rx have different RF chains [1,], the beamforming and combining vectors are basicay different. Note that for mmwave communications there may be other modes. For instance, in [42] a custered channe mode was adopted, where the channe incudes severa custers, and a custer consists of many MPCs with sma ange differences. Different modes are suitabe for different communication circumstances. As our schemes do not expoit the specific feature of the communication channe, they can be used for different modes.

5 Sensors 16, 16, of 22 g 12 φ m 12) [ ) = exp jπ cosφ m 12) ) ), exp jπ1 cosφ m 12) ) ),..., exp g 21 φ 21) [ ) = exp jπ cosφ 21) ) ), exp jπ1 cosφ 21) ) ),..., exp h 12 θ m 12) [ ) = exp jπ cosθ m 12) ) ), exp jπ1 cosθ m 12) ) ),..., exp h 21 θ 21) [ ) = exp jπ cosθ 21) ) ), exp jπ1 cosθ 21) ) ),..., exp jπn r2 1) cosφ 12) m ))] T/ nr2 jπn r1 1) cosφ 21) ))] T/ nr1 jπn t1 1) cosθ 12) m ))] T/ nt1 jπn t2 1) cosθ 21) ))] T/ nt2 4) SI Channe Next, we consider the strength of SI and the SI channe. Note that even in mmwave band, where the center frequency is high and the signa strength attenuates rapidy, SI may be sti much more significant than the background noise, because Tx/Rx antennas ocate cose to each other in FD mmwave communication. For instance, if the waveength of the carrier frequency is 1 mm, the signa bandwidth is 1 GHz, and the transmission power is dbm, according to the Friis formua, the SI at a position cm λ) away from a transmit antenna is og 4π ) = 42 dbm, which is much greater than a typica noise power σ 2 = og κtb) = og ) = dbm, where κ, T, B are the Botzmann constant, ambient temperature and bandwidth, respectivey. Hence, SI shoud be taken into account in FD mmwave communication. As the distance between the transmit and receive arrays at each node is short in portabe devices, the far-fied range condition, i.e., R 2D 2 /λ [43], may not hod for SI channe, where D is the diameter of the antenna aperture, λ is the waveength of the carrier. For instance, considering a haf-waveength spaced ULA with 64 eements, the far-fied range shoud satisfy R 232λ) 2 /λ = 48λ, which is basicay too arge for sma-size devices ike mobie phones or aptops even at the mmwave band. Thus, the SI channes may empoy the near-fied mode, which has a spherica wavefront [43 47]. In such a case, the SI channes highy depend on the pacement of the transmit and receive arrays as we as the circumstances. In this paper we consider an antenna pacement as shown in Figure 2. The distance between the first eements of the two arrays is d, and the ange between these two ULAs is ω. Athough this pacement together with the two parameters, i.e., d and ω, cannot cover a the possibe antenna pacements, it aows us to perform tractabe anaysis. Moreover, it can refect the effects of two typica factors of antenna pacement, i.e., the distance and ange between these two arrays, on the beamforming performance. With this antenna pacement, the coefficient corresponding to the jth row and ith coumn of H 11 or H 22 is [43 47]. [H 11 or H 22 ] ij = h ij = ρ exp j2π r ) ij r ij λ where r ij is the distance between the i-th eement of the transmit array and the j-th eement of the receive array, and ρ is a constant for power normaization such that trh 11 H H 11 ) = n t1n r1 and trh 22 H H 22 ) = n t2n r2. The expression for r ij is shown in Equation 6). ) 2 ) 2 ) ) r ij = d tanω) + j 1) λ 2 + d sinω) + i 1) λ 2 2 d tanω) + j 1) λ d 2 sinω) + i 1) λ 2 cosω) 6) In the case of Tx/Rx sharing the same antenna array, we have d = and ω =. The SI channe mode adopted in this paper is a simpified one, which is typica [46,47] but not necessariy accurate. In practice, the SI channe can be very compicated, incuding signa refection, scattering and the couping effects between adjacent antennas. Hence, it is very difficut, if not impossibe, to eaborate an accurate SI channe. On the other hand, the proposed schemes can be used 5)

6 Sensors 16, 16, of 22 for arbitrary SI channes, incuding the competey accurate one. Hence, we adopt the simpe and typica mode shown in Equation 6) here, which can refect the robustness of the proposed schemes against the geometry of the Tx/Rx antenna arrays within a node. Transmit Array Receive Array Figure 2. The transmit and receive antenna arrays of a node Probem Formuation With the above system and channe modes, the received signas at Node #1 and Node #2 are written as y 1 = w H r1 ε 21 H 21 w t2 s 2 + ε 11 H 11 w t1 s 1 + n 1 ) 7) and y 2 = w H r2 ε 12 H 12 w t1 s 1 + ε 22 H 22 w t2 s 2 + n 2 ) 8) respectivey, where s 1 and s 2 are the transmitted symbos with unit power at Node #1 and Node #2, respectivey, ε 21 and ε 12 are the average powers of the desired received signas, ε 11 and ε 22 are the average powers of the SI, n 1 and n 2 are the Gaussian white noise vectors with E{n 1 n H 1 } = I n r1 and E{n 2 n H 2 } = I n r2, respectivey, w t1 and w t2 are the transmit antenna weight vectors AWVs), and w r1 and w r2 are the receive AWVs. The two-norms of a these AWVs are normaized to 1. With the transmit and receive AWVs, the joint achievabe rate JAR) can be expressed as R =og ε 21 w H r1 H ) 21w t ε 11 w H r1 H + 11w t1 2 og2 1 + ε 12 w H r2 H ) 12w t ε 22 w H r2 H 22w t2 2 where ε 11 w H r1 H 11w t1 2 and ε 22 w H r2 H 22w t2 2 are the average powers of SI at Nodes #1 and #2, respectivey. The JTR-BF probem is formuated as 9) maximize w t1,w r1,w t2,w r2 R subject to w t1 2 = w r1 2 = 1 w t2 2 = w r2 2 = 1 ) As the main purpose of this paper is to investigate the feasibiity of FD mmwave communications and evauate the performance of beamforming canceation, the channe matrices and powers in Equation ) are assumed known a-priori. In practice, these parameters can be estimated provided that the channe does not change too fast. For instance, as an mmwave communication channe has the feature of directivity and is sparse in the ange domain, schemes ike AoD/AoA estimation [48], beam searching [26,39,49], iterative training [22,5], and even compressed sensing [51,52] can be adopted for the communication channe estimation. The estimation of SI channe can be more straightforward, e.g., one can estimate a scaar channe coefficient between a singe Tx/Rx antenna pair i.e., the i-th transmit antenna and the j-th receive antenna) once a time. After n ri n ti i = 1, 2) measurements, the SI channe matrix can be estimated. Athough n ri n ti is arge in mmwave communication, each measurement may

7 Sensors 16, 16, of 22 require ony one symbo thanks to the high strength of SI, rather than a ong training sequence ike those in [22,26,39,49 51]. Thus, the time cost of the SI channe estimation is yet affordabe. In addition, in our mode both ampitudes and phases of the AWVs are controabe. In fact, a constant-ampitude CA) array with ony phases controabe has ower compexity [22,26,39,49,51]. An N-eement CA array needs N phase shifters, which are not difficut to impement. In contrast, an N-eement array with both ampitudes and phases controabe requires N phase shifters and N variabe gain ampifier. Hence, a CA array has a ower compexity than an ampitude-and-phase-controabe array, especiay when N is arge, and thus is favored in mmwave communications, where N is arge in genera. However, the array structure without the CA constraint cannot be definitey rued out, because it aso arouses particuar attention in both agorithm design [23,24,5,53,54] and impementation [55]. Moreover, the performance achieved with non-ca-constraint arrays can be seen as a bound achieved with the CA arrays, if the same scheme is adopted. Therefore, in this paper we adopt the arrays without the CA constraint to investigate the feasibiity of FD mmwave communication and evauate the beamforming performance, etting the probem with CA constraint be a further work. 3. The ZF-Max-Power Approach It is cear that the JAR in Equation ) is not a concave function, and the equaity constraints are not affine. Thus, Equation ) is not a convex/concave probem, and its gobay optima soution is hard to find. Consequenty, we first give an upper bound JAR, and then propose the ZF-Max-Power approach in this section Upper Bound of the JAR According to Equation 9), since ε 11 w H r1 H 11w t1 2 and ε 22 w H r2 H 22w t2 2, an upper bound function on the JAR, denoted by R ub, in the presence of SI can be easiy obtained as R og ε 21 w H r1 H 21w t2 2) + og ε 12 w H r2 H 12w t1 2) R ub 11) The corresponding optima AWVs for R ub are w t1 = RpSingVectH 12 ), w t2 = RpSingVectH 21 ) w r1 = LpSingVectH 21 ), w r2 = LpSingVectH 11 ) 12) where LpSingVectX) and RpSingVectX) represent the eft and right principa singuar vectors of X, respectivey. Thus, we have R R ub R ub < 13) where R ub is the maximum of the upper bound R ub, and thus it is an upper bound on R The ZF-Max-Power Approach As there are mutipe couped variabes in Equation ), we consider using the aternating optimization AO) approach [56] to obtain a suboptima soution of Equation ). The basic idea of AO is to aternatey optimize a few parameters, and assume the other parameters fixed and known [56]. In each round, a sub-probem with a few parameters are formuated and soved. Generay, this approach requires that the optima soution to each sub-probem can be found. However, for the probem in Equation ), the AO approach cannot be directy used, because, as we can see, even given w r1 and w r2, optima w t1 and w t2 sti cannot be easiy found. To make the AO approach feasibe, we propose the ZF-Max-Power scheme in this paper. The motivation of this scheme is as foows. Since the SI is usuay significant in FD mmwave communication, we can force the SI to zero and maximize the signa power. In particuar, we add

8 Sensors 16, 16, of 22 a constraint that the SI is competey mitigated. probem becomes maximize R ub w t1,w r1,w t2,w r2 subject to w t1 2 = w r1 2 = 1 In such a case, we have R = R ub, and the w t2 2 = w r2 2 = 1 w H r1 H 11w t1 = w H r2 H 22w t2 = Since we have added a new constraint to the origina probem in Equation ), the soution of the new probem in Equation 14) is suboptima to the origina probem. To sove the new probem in Equation 14), we need the foowing resut. Lemma 1. Given an arbitrary set of ineary independent vectors {a i C L 1 } i=1,2,...,n; N<L and an arbitrary vector a C L 1, the vector b C L 1 that maximizes b, a 2 with the constraint that b, a i = i=1,2,...,n and b = 1 is before normaization) b = a N i=1 a, b i b i, where b 1 = a 1 / a 1 and Proof. See Appendix A. b i = 14) a i i 1 j=1 a i, b j b j a i i 1 j=1 a i, b j b j, i > 1 ) Specificay, when N = 1 the optima b in Lemma 1 before normaization is b = a a, a 1 a 1 a 1 a 1 16) and when N = 2 the optima b in Lemma 1 before normaization is where b = a a, b 2 = a 2 a 2, a 2 a 2, a 1 a 1 a 1 a 1 a, b 2 b 2 17) a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 18) Let us go back to the ZF-Max-Power scheme. According to Equation 16), given fixed w t1 and w t2, the optima w r1 and w r2 for Equation 14) before normaization) are w r1 = H 21 w t2 H 21 w t2, H 11 w t1 H 11 w t1 H 11w t1 H 11 w t1 19) and w r2 = H 12 w t1 H 12 w t1, H 22 w t2 H 22 w t2 H 22w t2 H 22 w t2 respectivey. Simiary, given fixed w r1 and w r2, the optima w t1 and w t2 for Equation 14) before normaization) are w t1 = H H 12 w r2 H H 12 w H H 11 r2, w r1 H H 11 w r1 HH 11 w r1 H H 11 w 21) r1 and w t2 = H H 21 w r1 H H 21 w r1, H H 22 w r2 H H 22 w r2 HH 22 w r2 H H 22 w r2 respectivey. Finay, the ZF-Max-Power scheme can be summarized as in Agorithm 1. ) 22)

9 Sensors 16, 16, of 22 Agorithm 1 The ZF-Max-Power Scheme. 1) Initiaize: Initiaize the transmit AWVs as w t1 = RpSingVectH 12 ), w t2 = RpSingVectH 21 ). 2) Iteration: Iterate the foowing process ρ times, then stop. Compute w r1 and w r2 according to Equations 19) and ), respectivey; then normaize them. Compute w t1 and w t2 according to Equations 21) and 22), respectivey; then normaize them. 3) Resut: w t1 and w t2 are the transmit AWVs, and w r1 and w r2 are the receive AWVs. It is noted that there are various stopping rues for the ZF-Max-Power scheme. A simpe one is to stop after a certain number of iterations, e.g., ρ iterations used in Agorithm 1. Another one is to compute R ub after the n-th iteration and get R n) ub according to Equation 9). When Rn) ub /Rn 1) ub < µ, stop the iteration, where µ is a predefined threshod sighty greater than 1, e.g., µ = Convergence Anaysis and Compexity Comparison To prove the convergence of the ZF-Max-Power scheme, we need the foowing theorem. Theorem 1. Let R n) denote the vaue of R after the n-th iteration. Then {R n) n = 1, 2,...} is a non-descending sequence, i.e., R n+1) R n). Proof. See Appendix B. According to Equation 13), R R ub <. Thus, {Rn) n = 1, 2,...} converges to a suboptima vaue, which guarantees the convergence of ZF-Max-Power. On the other hand, the computationa compexity of ZF-Max-Power is much ower than that of Max-SINR [35]. For simpicity, suppose n t1 = n t2 = n r1 = n r2 = N. According to Agorithm 1, in each iteration the main compexity of ZF-Max-Power ies in mutipications between a channe matrix and an AWV, as shown in Equations 19) 22). Hence, the computationa compexity of ZF-Max-Power is roughy ON 2 ), because there are about N 2 scaar mutipications for a mutipication of a channe matrix and an AWV. In contrast, according to [35] matrix inversion is required in each iteration of Max-SINR. Hence, Max-SINR has a computationa compexity of ON 3 ), which is significanty higher than that of ZF-Max-Power, especiay in FD mmwave communication where N is arge. 4. Cosed-Form Soutions In this section, we consider different criteria for the JTR-BF probem that provide us with cosed-form soutions LB-MMSE It is natura to perform receive and transmit beamforming separatey when considering cosed-form soutions. According to Equation 9), an optima soution can be achieved for receive beamforming by using the MMSE approach which is equivaent to maximizing the SINR. However, for transmit beamforming, the optima soution is hard to find. Thus, we propose to optimize the ower bound on the JAR with MMSE, and the method is referred to as the ower bound MMSE LB-MMSE) approach.

10 Sensors 16, 16, 113 of 22 According to Equation 9), we have ε 21 w H r1 R og H 21w t2 2 ε 12 w H r2 H ) 12w t1 2 2 w H r1 w r1 + ε 11 w H r1 H 11w t1 2 w H r2 w r2 + ε 22 w H r2 H 22w t2 2 ε 21 w H r1 = og H 21w t2 w H t2 HH 21 w r1 ε 12 w H 2 w H r1 I + ε11 H 11 w t1 w H ) r2 H 12w t1 w H t1 HH 12 w ) 23) r2 t1 HH 11 wr1 w H r2 I + ε22 H 22 w t2 w H ) R 1 t2 HH 22 wr2 First, we find w r1 and w r2 to maximize the ower bound R 1 by temporay treating w t1 and w t2 as fixed parameters. This subprobem is actuay to maximize SINR or minimize MSE) at the two nodes. According to Lemma 2 in Appendix C, the optima w r1 and w r2 before normaization) can be respectivey found as 1H21 w r1 = I + ε 11 H 11 w t1 w H t1 11) HH w t2 ) 1H12 24) w r2 = I + ε 22 H 22 w t2 w H t2 HH 22 w t1 With these two receive AWVs, we further have Equation ), where inequaities a) and b) are based on Lemmas 3 and 4, respectivey, in Appendix C. ) ) 1H21 ) )) 1H12 R 1 = og 2 ε 12 ε 21 w H t2 HH 21 I + ε 11 H 11 w t1 w H t1 HH 11 w t2 w H t1 HH 12 I + ε 22 H 22 w t2 w H t2 HH 22 w t1 a) og 2 b) og 2 = og 2 w H ε 12 ε t2 H H 21 H ) 2 21w t2 21 w H t2 HH 21 I + ε11 H 11 w t1 w H t1 HH 11 ε 12 ε 21 ε 12 ε 21 w H t1 w H t2 w H t2 HH 21 H 21w t2 w H I + ε11 H H 11 H ) t1 HH 12 H 12w t1 11 wt1 I + ε22 H H 22 H 22 w H t2 w H ) t1 H H 12 H ) 2 ) 12w t1 H21 w t2 w H t1 HH 12 I + ε22 H 22 w t2 w H ) t2 HH 22 H12 w t1 ) ) wt2 w H t2 HH 21 H 21w t2 w H I + ε22 H H 22 H ) t1 HH 12 H ) 12w t1 22 wt2 I + ε11 H H 11 H ) R 2 11 wt1 w H t1 ) Next, et us find w t1 and w t2 to maximize the ower bound R 2. This subprobem is equivaent to the foowing two optimization probems: arg w t1 max w H t1 w H t1 HH 12 H 12w t1 I + ε11 H H 11 H 26) 11) wt1 and arg w t2 max w H t2 These are generaized Rayeigh quotient probems, before normaization) are w H t2 HH 21 H 21w t2 I + ε22 H H 22 H 27) 22) wt2 ) ) 1H w t1 = peigvect I + ε 11H H 11 H H H 12 and the optima transmit AWVs 28) and ) ) 1H w t2 = peigvect I + ε 22H H 22 H H H 21 29) respectivey, where peigvectx) denotes the principa eigenvector of X. In brief, by expoiting the proposed LB-MMSE, the transmit AWVs are computed as Equations 28) and 29), respectivey. Based on the transmit AWVs, the receive AWVs before normaization) are found as Equation 24).

11 Sensors 16, 16, of SI-ZF-MRT In this scheme, we first consider transmit beamforming by adopting MRT, and then use ZF to suppress the SI for receive beamforming. This approach is referred to as SI-ZF-MRT. By expoiting MRT for transmit beamforming, we have To suppress the SI, we have w t1 = H H 12 w r2; w t2 = H H 21 w r1 3) w H r1 H 11H H 12 w r2 = = w H r2 H 22H H 21 w r1 31) There are many soutions of w r1 and w r2 for Equation 31). Among those, we want to find the receive AWVs to maximize JAR whie satisfying Equation 31). With Equations 3) and 31), the JAR becomes R = og ε 21 w H r1 H 21H H 21 w r1 2) + og ε 12 w H r2 H 12H H 12 w r2 2) 32) There are two options to design w r1 and w r2. One is to first derive w r1 that maximizes Equation 32) without considering the constraint Equation 31); then derive w r2 to optimize Equation 32) with Equation 31) satisfied. With this option, we have and ) w r1 = peigvect H 21 H H 21 w r2 = a a, where a = peigvect H 12 H H 12), a1 = H 12 H H 11 w r1, and b 2 = H 22H H 21 w r1 H 22 H H 21 w r1, H 22 H H 21 w r1 H 22 H H 21 w r1, 33) a 1 a 1 a 1 a 1 a, b 2 b 2 34) a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 35) The other one is simiar to the first one but with the positions of w r1 and w r2 exchanged. With this option, we have ) w r2 = peigvect H 12 H H 12 36) and w r1 = a a, where a = peigvect H 21 H H 21), a1 = H 11 H H 12 w r2, and b 2 = H 21H H 22 w r2 H 21 H H 22 w r2, H 21 H H 22 w r2 H 21 H H 22 w r2, a 1 a 1 a 1 a 1 a, b 2 b 2 37) a 1 a 1 a 1 a 1 a 1 a 1 a 1 a 1 38) In brief, for SI-ZF-MRT, the two options are i) to find the receive AWVs according to Equations 33) and 34), and then obtain the transmit AWVs according to Equation 3); ii) to find the receive AWVs according to Equations36) and 37), and then obtain the transmit AWVs according to Equation 3). Therefore, the one which has a higher JAR can be seected as the soution for SI-ZF-MRT. Simiar to SI-ZF-MRT, SI-ZF-maximum-ratio combining MRC) is aso appicabe to the joint beamforming probem. With SI-ZF-MRC, receive beamforming is firsty performed by using MRC. Afterwards, transmit beamforming is carried out to maximize the JAR with the SI forced to zero. Since SI-ZF-MRC is simiar to SI-ZF-MRT in formuation and performance, we do not present the detais here.

12 Sensors 16, 16, of Steering Beamforming The conventiona SBF in mmwave communication can aso be introduced here to compare with the aternatives. SBF does not require fu channe information. Instead, it ony requires the knowedge of the transmit and steering vectors for the most significant MPC of the communication channe, and does not consider the SI. Suppose the m-th and -th mutipath components are the most significant ones from Node #1 to Node #2 and from Node #2 to Node #1, respectivey. By using SBF, the AWVs become w t2 = h 21 θ 21) ), w t1 = h 12 θ m 12) ) w r1 = g 21 φ 21) ), w r2 = g 12 φ 12) m ) By comparing the performance of SBF with those of the proposed schemes, we can see whether or not it is infeasibe not to consider the SI in FD mmwave communication, and how much performance degradation it causes if we do not consider the SI in beamforming. 5. Simuation Resuts In this section, we present the performances of a the invoved schemes through numerica simuations, where the communication channe mode and the SI channe mode introduced in Section 2 are adopted. In a the evauations, the near-fied SI channes are deterministic, and are decided by Equation 5); whie the far-fied signa channes are random. Both LOS and non-los NLOS) channes are considered for the communication channes. For NLOS channe, the transmit and receive steering anges are randomy generated within [, 2π), and the coefficients obey circuary symmetric compex Gaussian distribution with the same average power. For LOS channe, the LOS component has a fixed coefficient and fixed transmit and receive steering anges, whie the other NLOS components have random steering anges and coefficients with average power db ower than the LOS component. The tota power of a generated channe obeys Equation 3), and the tota number of MPCs is 3 We ve aso simuated with other numbers of MPCs, and simiar resuts were obtained.). For each curve in a the figures in this section, we have generated reaizations with the LOS or NLOS channe modes, and computed the average JAR based on these reaizations. Moreover, we have considered both types of array settings in the evauations, namey Tx/Rx have separate antenna arrays and the same antenna array at a node. In a the simuations, n t1 = n r1 = n t2 = n r2 = 32. Firsty, we consider the JAR and convergence performances of the ZF-Max-Power scheme with random initia transmit AWVs, which are shown in Figure 3 with reevant parameters isted in the caption. Both cases of separate arrays and the same array are incuded under LOS/NLOS channes. As ZF-Max-Power is an iterative method, we compare it with Max-SINR, which achieves the best performance within the typica soutions for the IA probem [35]. It is observed that ZF-Max-Power achieves a suboptima performance cose to the upper bound after convergence, under both LOS and NLOS channes, especiay with separate arrays. The sight superiority of the case with separate arrays is due to that the different channe parameters of H 12 and H 21 provide more degrees of spatia freedoms for beamforming than the case of the same array, where the channe parameters are the same. Moreover, the convergence speed of ZF-Max-Power is fast in a the cases. Basicay, ony two iterations are required to achieve convergence. Interestingy, under the considered scenario, ZF-Max-Power achieves amost the same JAR and convergence performances as Max-SINR. However, it is noteworthy that ZF-Max-Power is a centraized iterative approach, where the iteration is performed at a certain node, and does not expoit channe reciprocity. By contrast, Max-SINR is a distributed iterative approach, where the iteration is performed at a distributed nodes by expoiting channe reciprocity. As a consequence, the convergence of ZF-Max-Power can be proven, whie that of Max-SINR is sti unproven in the iterature [35] to the best of our knowedge. Moreover, ZF-Max-Power has a significanty ower computationa compexity than Max-SINR. 39)

13 Sensors 16, 16, of JAR Bps/Hz) JAR Bps/Hz) 5 LOS, Upper Bound LOS, Sep. Arrays, ZF Max Power LOS, Same Array, ZF Max Power LOS, Sep. Arrays, Max SINR LOS, Same Array, Max SINR 5 NLOS, Upper Bound NLOS, Sep. Arrays, ZF Max Power NLOS, Same Array, ZF Max Power NLOS, Sep. Arrays, Max SINR NLOS, Same Array, Max SINR ρ ρ Figure 3. JAR and convergence performances of ZF-Max-Power with random initia transmit AWVs Left: LOS channe, Right: NLOS channe). ε 11 = ε 22 = 4 db. For LOS channe, ε 12 = ε 21 = db, whie for NLOS channe, ε 12 = ε 21 = db. For the case of separate arrays, d/λ = 1 and ω = π/6 rad, whie for the case of sharing the same array, d/λ = and ω = rad. Next, et us see the JAR performances of the proposed schemes with respect to varying ω, d, i.e., the geometry of the arrays, when separate arrays are expoited at a node. Figure 4 shows the JAR performances with respect to varying ω under LOS and NLOS channes, respectivey, where SI is fixed. The eft hand side figure of Figure 5 shows the JAR performances with respect to d under LOS channe. Simiar resuts can be observed under NLOS channe. As we can see, SI is assumed fixed in the eft hand side figure of Figure 5, which may be not practicay reasonabe, because in practice d significanty affects SI. However, in order to adequatey evauate the effects of d on the JAR performance, we assume a fixed SI in the eft hand side figure of Figure 5. For rigorousness, we aso adopt varying SI in the right hand side figure of Figure 5, which is in accordance with the practice, where SI deteriorates with d 2. It is noteworthy that the strength of SI is typicay much higher than that of the desired signa, because SI comes from the oca transmitter at the same node, whie the designed signa comes from the remote transmitter at the other node. Reevant parameters for these figures are isted in the corresponding captions. JAR Bps/Hz) Upper Bound ZF Max Power LB MMSE SI ZF MRT SBF LB MMSE with d/λ= JAR Bps/Hz) Upper Bound ZF Max Power LB MMSE SI ZF MRT SBF LB MMSE with d/λ= ω π rad) ω π rad) Figure 4. JAR performance of the invoved schemes with respect to varying ω under LOS Left) and NLOS Right) channes in the case of separate arrays. ε 11 = ε 22 = 4 db, ε 12 = ε 21 = db, d/λ = 2. For LB-MMSE, the JAR with d/λ = is aso potted.

14 Sensors 16, 16, of 22 3 JAR Bps/Hz) 5 Upper Bound ZF Max Power LB MMSE SI ZF MRT SBF JAR Bps/Hz) 5 Upper Bound ZF Max Power LB MMSE SI ZF MRT SBF d/λ d/λ Figure 5. JAR performance of the invoved schemes with respect to varying d under LOS channe in the case of separate arrays. ω = π rad. In the Left) hand figure SI is assumed fixed, i.e., ε 11 = ε 22 = 4 db, ε 12 = ε 21 = db; whie in the Right) hand figure SI varies with d/λ, i.e., ε 11 = ε 22 = 6 og d/λ) db, ε 12 = ε 21 = db. By comparing these figures with each other, it can be observed that: i) ii) iii) ZF-Max-Power is robust against ω, d and SI, and approaches the upper bound in a these cases. This is because ZF-Max-Power not ony forces SI to zero, but aso iterativey maximizes signa power. Thus, it achieves compeing performance that is insensitive to the geometry of the Tx/Rx arrays and SI. SI-ZF-MRT is aso robust against ω, d and SI, thanks to its zero-forcing fitering to SI. In addition, it aso achieves an acceptabe performance, which is cose to the upper bound. It is noted that the JAR gap between SI-ZF-MRT and the upper bound is greater under LOS channe than that under NLOS channe. This phenomenon can be expained by referring to Equation 34), where w r2 is in fact set within an n r2 2)-dimension subspace, due to the two zero forcing equations shown in Equation 31). Ceary if a in Equation 34), which represents the dimension with the argest power of the channe, has ess energy projected on the n r2 2)-dimension subspace, the JAR performance wi be poorer. Under LOS channe, the majority of the channe energy concentrates on a singe path, or a singe dimension. Once this dimension has a sma projection on the subspace, the performance wi be poor. In contrast, under NLOS channe the channe energy eveny disperses on mutipe paths. Ony when a of these paths have a sma projection on the subspace, the performance wi be poor. In other words, the probabiity of a poor performance is ower under NLOS channe than that under LOS channe. Hence, on average the JAR gap between SI-ZF-MRT and the upper bound is greater under LOS channe than that under NLOS channe. LB-MMSE is sensitive to ω and d. From Figure 4 we observe that the performance of LB-MMSE fuctuates as ω changes, and the fuctuation is different for different d. From Figure 5 we observe that the performance of LB-MMSE has a -shape as d increases, but behaves stabe when d is arge. To understand these, we need to go back to Equations 28) and 29). From these two equations we can see that the transmit AWVs are decided to maximize the SINR rather than minimize SI based on the oca information. Taking Equation 28) for iustration, since usuay ε 11 is big, when H 11 has a ow rank, the eigenvector of H H 12 H 12 has a high probabiity to ocate within the nu space of H 11. In such a case, a high signa power can be achieved whie itte SI ocates within the signa subspace; thus good performance is achieved. Note that this statement is just for iustration. In practice, H 11 is generay with fu rank except when ω = or π. However, when most energy of H 11 ocates at a ow-dimensiona subspace, the situation wi be simiar to the statement that H 11 has a ow rank. In comparison, when H 11 has a high or even fu rank, SI wi amost unavoidaby ocate within the signa subspace and affects the received SINR, and thus the performance wi be

15 Sensors 16, 16, 113 of 22 iv) poor. When d is sma, the energy dispersion of H 11 is sensitive to ω and d according to the SI channe mode, and thus the JAR performance is aso sensitive to ω and d. However, when d is arge, the SI channe amost reduces to a directiona channe with rank 1, and thus SI has a ow probabiity to ocate within the signa subspace. In such a case, LB-MMSE can staby achieve a near-optima performance. SBF is aso sensitive to ω, d and SI. This is because SBF does not even consider SI in the beamforming design. Meanwhie, from Figure 5 it is found that SBF becomes improved as d increases. In the right hand side of Figure 5 the improving speed of SBF is faster than that in Figure 5, because SI is reduced as d increases. This phenomenon suggests that when the near-fied SI channe graduay reduces to a directiona channe, the conventiona beamforming schemes that to simpy steer towards each other may aso achieve good performance, because usuay the communication channe and SI channe have difference steering anges. However, in practica FD mmwave communication where d is generay sma, the SI channe does not have the feature of directivity; thus SBF is much poorer than the other candidates, and the performance of SBF does not show monotonicity with ω, as shown in Figure 4. Thus, SBF may not be a good choice for FD mmwave communication, where SI must be taken into account. Then, we compare the JAR performances of the discussed schemes with separate arrays and the same array. Figure 6 shows the comparison resuts with respect to SI under LOS and NLOS channes, respectivey, where reevant parameters are isted in the captions. From these two figures we observe that the schemes with separate arrays basicay achieve better performance than those with the same array. This advantage is aso due to that the different channe parameters of H 12 and H 21 when using separate arrays provide arger degrees of spatia freedom for beamforming than the case of using the same array, where the channe parameters are the same. Moreover, both ZF-Max-Power and SI-ZF-MRT are insensitive to the increase of SI, thanks to the operation of ZF SI, whie the performance of SBF becomes poorer as the increase of SI, due to no operation of ZF SI. Interestingy, the JAR of LB-MMSE with separate arrays sowy decreases as the increase of SI whereas that with the same array changes itte, which shows that LB-MMSE with the same array is more robust against the SI. To expain this, et us ook at the third and fourth ines of Equation ). In third ine, and w H t1 HH 12 H 12w t1 w H t2i+ε 22 H H 22 H 22)w t2 w H t2 HH 21 H 21w t2 w H t1i+ε 11 H H 11 H 11)w t1 can be roughy seen as the receive SINRs at Node #1 and Node #2 without considering the receive AWVs, respectivey. However, in order to obtain cosed-form expressions of the transmission AWVs, the denominators or numerators) of these two components are exchanged and optimized respectivey, as shown in Equations 26) and 27). This means that the optimizations in Equations 26) and 27) are not to directy optimize the receive SINRs at Node #1 and Node #2. Hence, in genera LB-MMSE is not so robust against the SI. However, in the case with the same array, the ink from Node #1 to Node #2 is symmetric; thus the denominators or numerators) of the two components in the third ine of Equation ) can be seen equa or at east proportiona to each other. In such a case, Equations 26) and 27) are in fact to optimize the receive SINRs at Node #1 and Node #2 without considering the receive AWVs. Hence, LB-MMSE with the same array is reativey more robust against the SI. Finay, we evauate the effects of channe estimation errors on the proposed schemes. For the SI channe, there exists Gaussian error; whie for the communication channe, it is possibe to miss some MPCs during the beam search process. Figure 7 shows the effects of these estimation errors on the proposed schemes with separate arrays the resuts are simiar with the same array) under LOS and NLOS channes, respectivey, where the parameters are specified in the captions. From these two figures we can observe that both ZF-Max-Power and LB-MMSE are reativey robust against the channe estimation error. Even ony one MPC is acquired, they can achieve promising performance, especiay under LOS channe. However, SI-ZF-MRT is not robust against the estimation error of the communication channe, i.e., if ony one MPC is acquired, the performance of SI-ZF-MRT

16 Sensors 16, 16, of 22 becomes rather poor, this is because the fu channe information is invoved in the SI ZF operation according to Equation 31). JAR Bps/Hz) 5 Upper Bound Separate, ZF Max Power Separate, LB MMSE Separate, SI ZF MRT Separate, SBF Same, ZF Max Power Same, LB MMSE Same, SI ZF MRT Same, SBF JAR Bps/Hz) 5 Upper Bound Separate, ZF Max Power Separate, LB MMSE Separate, SI ZF MRT Separate, SBF Same, ZF Max Power Same, LB MMSE Same, SI ZF MRT Same, SBF SI ε 11 and ε 22 db) SI ε 11 and ε 22 db) Figure 6. JAR comparison between different array settings separate arrays versus the same array) under LOS Left) and NLOS Right) channes with varying SI. ε 12 = ε 21 = db. For the case of separate arrays, ω =.6π rad, d/λ = 1. 3 JAR Bps/Hz) 5 A MPCs, ZF Max Power A MPCs, LB MMSE A MPCs, SI ZF MRT 1 MPC, ZF Max Power 1 MPC, LB MMSE 1 MPC, SI ZF MRT JAR Bps/Hz) 5 A MPCs, ZF Max Power A MPCs, LB MMSE A MPCs, SI ZF MRT 1 MPC, ZF Max Power 1 MPC, LB MMSE 1 MPC, SI ZF MRT MSE of SI Channe Estimation db) MSE of SI Channe Estimation db) Figure 7. Effects of channe estimation errors on the proposed schemes with separate arrays under LOS Left) and NLOS Right) channes. ε 12 = ε 21 = db, ε 11 = ε 22 = 4 db, ω = π rad, d/λ = 1. It is noteworthy that circuit imperfections are not taken into consideration in our system mode, i.e., in Equations 7) and 8). In a practica FD system, there are aways Tx/Rx hardware and impementation imperfections, incuding ow-noise ampifier LNA) noise figure, phase noise, in-phase and quadrature IQ) mismatch, noninear distortion of power ampifier, etc. These imperfections may be more severe in mmwave communication systems than in ow-frequency systems, because of the higher carrier frequency and arger bandwidth. Since SI is strong, the Tx imperfections, which are carried by the transmitted signas s 1 and s 2 in Equations 7) and 8), wi aso arrive at the Rx. When beamforming canceation is adopted to force the SI to zero, the performance wi be affected itte by the Tx circuit imperfections provided that the AWV contro is perfect, because a the SI, incuding the imperfections, can be fitered out by beamforming. However, in practice, the AWV contro may have error. In such a case the SI cannot be competey fitered out by beamforming, and the residua SI, which contains Tx imperfections, wi degrade the system performance. Athough baseband BB) canceation can be used to dea with the residua SI after beamforming canceation, it basicay cannot effectivey cance the residua Tx imperfections.

17 Sensors 16, 16, of 22 3 JAR Bps/Hz) No EVM Error, With BB Canceation No EVM Error, Without BB Canceation 5 EVM Error db with BB Canceation EVM Error db with BB Canceation EVM Error 5 db with BB Canceation AWV Error db) JAR Bps/Hz) No EVM Error, With BB Canceation 5 No EVM Error, Without BB Canceation EVM Error db with BB Canceation EVM Error db with BB Canceation EVM Error 5 db with BB Canceation AWV Error db) Figure 8. Effects of AWV error and EVM error on the JAR performance of ZF-Max-Power under LOS Left) and NLOS Right) channes. ε 12 = ε 21 = db, ε 11 = ε 22 = 5 db, ω =.8π rad, d/λ = 1. To further iustrate the effect of the circuit imperfections on the system performance, we mode a the typica Tx imperfections as a zero-mean Gaussian distributed error vector magnitude EVM) noise [57], and its average power can be measured in db with respect to the transmission power. On the other hand, we aso need to consider AWV contro error, which can aso be modeed as a zero-mean Gaussian variabe, and its average power can aso be measured in db with respect to the 2-norm of its corresponding weight. By expoiting this mode, we can evauate the effects of AWV error and EVM error on the JAR performance of ZF-Max-Power as shown in Figure 8, where reevant parameters are isted in the caption. The effects are simiar to the performances of the other schemes. From this figure we can find that when AWV contro is perfect or AWV error is sma enough, the SI as we as the EVM noise can be mitigated successfuy by beamforming. However, when there is significant AWV error, the system performance deteriorates as the AWV error becomes greater. In such a case, BB canceation is needed to cance the residua SI. From the figure we observe that with BB canceation, the performance is greaty improved and does not depend on the AWV error. On the other hand, BB canceation can hardy mitigate the residua EVM noise, because it is difficut to estimate reevant parameters of the EVM noise [57]. Hence, we can observe that even when BB canceation is adopted, if EVM noise exists, the performance sti deteriorates as the AWV error becomes greater. 6. Concusions In this paper, we investigate FD mmwave communications, where we empoy beamforming to cance SI, and study a JTR-BF probem in the presence of significant SI. As the probem of finding the optima beamforming vectors to maximize the JAR is non-convex, severa suboptima soutions are proposed. Firsty, ZF-Max-Power, which restricts the origina probem by ZF SI and aternativey optimizes the desired power, is proposed, and its convergence is proven. It is shown that the computationa compexity of ZF-Max-Power is ower than that of Max-SINR by one order of magnitude. Next, two cosed-form soutions, namey LB-MMSE and SI-ZF-MRT are proposed, by jointy using MMSE, ZF and MRT criteria. Performance evauations show that ZF-Max-Power approaches an upper bound on the JAR, and it needs ony 2 iterations on average to achieve the convergence with random initia points. LB-MMSE and SI-ZF-MRT achieve suboptima performances. In addition, we find that ZF-Max-Power and SI-ZF-MRT are robust against the geometry of Tx/Rx antenna arrays due to the operation of ZF SI, whie LB-MMSE is not. ZF-Max-Power and LB-MMSE are robust against channe estimation error, whie SI-ZF-MRT is not. Furthermore, these schemes basicay achieve better JAR performance with a separate-array setting than those sharing the same array. The resuts demonstrate the feasibiity of FD mmwave communications and the effectiveness of beamforming canceation.

18 Sensors 16, 16, of 22 Acknowedgments: The authors woud ike to thank the funding support from the Nationa Natura Science Foundation of China NSFC) under grant Nos. 67, 9384, , and , Nationa Key R&D Program under grant No. 16YFB, Nationa Basic Research Program of China under grant No.11CB77, and Foundation for Innovative Research Groups of the Nationa Natura Science Foundation of China under grant No Author Contributions: Z.X., J.C., P.X. and X.-G.X. conceived and designed the idea and simuations; X.L. and L.B. performed the simuations; X.L. and Z.X. anayzed the performance; a the authors contributed to writing and improving the paper. Conficts of Interest: The authors decare no confict of interest. Appendix A. Proof of Lemma 1 Since {a i C L 1 } i=1,2,...,n; N<L is a group of inear independent vectors, they span a inear subspace denoted by S S{a i i=1,2,...,n }. Suppose {v i } i=1,2,...,n is an orthogona basis of the inear subspace S. Then constraint a, a i = i=1,2,...,n is equivaent to a, v i = i=1,2,...,n. Remark The vector b C L 1 to maximize b, a 2 with the constraint that b, v i = i=1,2,...,n and b = 1 is before normaization) b = a N i=1 a, v i v i. According to the above remark, the next thing we need to prove is that {b i i=1,2,...,n } shown in Equation ) is an orthogona basis of {b i i=1,2,...,n }. This proposition hods because {b i i=1,2,...,n } is obtained according to the standard Gram-Schmidt process from {a i i=1,2,...,n }. Therefore, Lemma 1 hods. Appendix B. Proof of Theorem 1 Under the constraints in Equation 14), the JAR becomes Rw t1, w t2, w r1, w r2 ) = og ε 21 w H r1 H 21w t2 2) + og ε 12 w H r2 H 12w t1 2) R ub B1) which shows that R is a function of w t1, w t2, w r1, and w r2. Let w n) t1 and w n) t2 the receive AWVs after the n-th iteration. Then we have R n) R n+1) = Rw n+1) denote the transmit AWVs after the n-th iteration, and w n) r1 t1, w n+1) t2, w n+1) r1, w n+1) r2 ). According to Agorithms 1, as we as Equations 19) and ), we have and w n) r2 = Rw n) t1, wn) t2, wn) r1, wn) r2 ) and Rw n) t1, wn) t2, wn+1) r1, w n+1) r2 ) Rw n) t1, wn) t2, wn) r1, wn) r2 ) = Rn) B2) Aso, according to Agorithms 1, as we as Equations 21) and 22), we have Rw n+1) t1, w n+1) t2, w n+1) r1 Therefore, we have R n+1) R n). Appendix C. Lemmas 2, 3, 4, w n+1) r2 ) = R n+1) Rw n) t1, wn) t2, wn+1) r1, w n+1) r2 ) B3) Lemma 2. Given R C M M a positive define Hermitian matrix, the soution to maximize xh aa H x x H Rx x = R 1 a. is Proof. Since R is a positive define Hermitian matrix, R 1/2 is aso an invertibe Hermitian matrix. Thus, we have x H aa H x x H Rx = x H aa H x R 1/2 x ) H R 1/2 x ) C1)

19 Sensors 16, 16, of 22 Let x = R 1/2 x, the probem becomes to maximize xh R H/2 a)r H/2 a) H x. Therefore, the soution is x H x x = R H/2 a = R 1/2 a, i.e., x = R 1 a. Lemma 3. Given R C M M a positive define Hermitian matrix, a H R 1 a ah a) 2 a H Ra. Proof. Since R C M M is a positive define Hermitian matrix, it can be expressed as R = M i=1 λ iv i v H i, where λ i > and {v i } i=1,2,...,m constitutes an orthogona base in C M M. Thus, a can be expressed as a = M i=1 α iv i. Besides, we have R 1 = M i=1 1 λ i v i v H i. Thus, = = ) ) a H R 1 a a H Ra M i=1 M i=1 M i=1 =a H a) 2 α i 2 λ i α i 4 + α i ) M ) α i 2 λ i i=1 M i=1 i 1 j=1 M i 1 i=1 j=1 ) λj α i 2 α j 2 + λ i λ i λ j α i 2 α j 2 C2) Lemma 4. Given R C M M a positive define Hermitian matrix with the formuation R = I + εbb H, a H a a H Ra 1 1+εb H b. Proof. Let v = a/ a. We have a H a a H Ra = a H a a H I + εbb H ) a = 1 v H I + εbb H ) v 1 = 1 + ε b H v ε = 1 bh b b εb H b C3) References 1. Sabharwa, A.; Schniter, P.; Guo, D.; Biss, D.; Rangarajan, S.; Wichman, R. In-band fu-dupex wireess: Chaenges and opportunities. IEEE J. Se. Areas Commun. 14, 32, Ahmed, E.; Etawi, A.; Sabharwa, A. Simutaneous transmit and sense for cognitive radios using fu-dupex: A first study. In Proceedings of the IEEE Antennas and Propagation Society Internationa Symposium APSURSI), Chicago, IL, USA, 8 14 Juy 12; IEEE: Chicago, IL, USA, 12; pp Duarte, M.; Dick, C.; Sabharwa, A. Experiment-driven characterization of fu-dupex wireess systems. IEEE Trans. Wire. Commun. 11, 11, Duarte, M.; Sabharwa, A.; Aggarwa, V.; Jana, R.; Ramakrishnan, K.; Rice, C.; Shankaranarayanan, N. Design and characterization of a fu-dupex mutiantenna system for WiFi networks. IEEE Trans. Veh. Techno. 14, 63, Jain, M.; Choi, J.I.; Kim, T.; Bharadia, D.; Seth, S.; Srinivasan, K.; Levis, P.; Katti, S.; Sinha, P. Practica, rea-time, fu dupex wireess. In Proceedings of the 17th Annua Internationa Conference on Mobie Computing and Networking, Las Vegas, NV, USA, September 11; ACM: New York, NY, USA, 11; pp

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