I Introduction
The availability of rich spectrum in the millimeterwave (mmWave) frequency bands makes mmWave communication one of the most promising candidates for future wireless communication systems to address the current challenge of bandwidth shortage [1, 2, 3, 4, 5, 6, 7, 8]. Specifically, the bands from 30 GHz to 300 GHz have been considered as the primary contender for the future 5G network [8]. While mmWave signals are vulnerable to path loss, penetration loss and rain fading compared to the signals in the sub6 GHz bands [2], the short wavelength at mmWave frequencies allows large antenna arrays to be packed at the radio frequency (RF) front end for mmWave transceivers. The deployment of a largescale antenna array enables the exploitation of highly directional beamforming to combat the attenuation from the environment [3, 9, 4, 10].
While it is possible to employ a fullydigital precoder in traditional sub6GHz bands, it is unfortunately not promising to consider fullydigital processing for mmWave communications, due to the prohibitive cost and power consumption of the hardware components working at mmWave bands. To address this problem and implement mmWave communications both costefficiently and energyefficiently, the concept of hybrid analogdigital structure has been introduced in [4], which provides a promising tradeoff between cost, complexity, and capacity of the mmWave network. The underlying principle behind the hybrid structure is to employ a reduced number of RF chains at the transceivers and divide the signal processing into an analog part and a digital part. Accordingly, the data streams at the mmWave transceivers are firstly processed by a lowdimension digital precoder, followed by the processing of a highdimension analog precoder [10, 11, 12]. For the analog precoding, lowcost phase shifters are commonly used [9], which imposes a constant modulus requirement on the analog precoding matrix. Due to this constraint, the performance of hybrid precoder is usually inferior to the fullydigital precoder. In addition, analog precoding based on switches has also been considered in [13, 14] as an alternative to constant modulus phase shifters.
There have been many recent works on hybrid precoding design in mmWave systems [15, 10, 16, 13, 14, 11, 12, 17, 18, 19, angVirtual2017751]. A commonality in these works is the attempt to maximize the overall spectral efficiency of the network with the assumption of perfect channel state information (CSI), which implicitly assumes perfect alignment between the transmitting and receiving beams. However, in practical mmWave scenarios where perfect CSI is usually not available [20, 21]
, the estimation errors in the angle of arrival (AoA) or angle of departure (AoD) result in beam misalignment. In addition to the channel estimation errors, the imperfection in the antenna array, which includes array perturbation and mutual coupling
[10, 22, 23], also contributes towards the imperfect alignment of beams. Besides, environmental vibrations such as wind, moving vehicles, etc, can also be potential sources of beam misalignment [9]. The deployment of a largescale antenna array that generates narrow beams for mmWave communications also makes the system highly sensitive to beam misalignment.To investigate the effect of imperfect alignment between the transmitting and receiving beams, existing studies in [22, 24, 7, 6] focus on analyzing the performance loss in terms of ergodic capacity. The works in [7, 6] evaluate the coverage performance of mmWave cellular networks with imperfect beam alignment, where [7] adopts an enhanced antenna model that is able to express the mainlobe beamwidth and array gain as a function of the number of antennas. With the 3GPP twodimension directional antenna model, the impact of beam misalignment on the performance of a 60GHz wireless network was studied in [22]
, where the probability distribution of the signaltointerferenceplusnoise ratio (SINR) was derived. For a mmWave adhoc network, the authors in
[24] have derived a closedform expression for the ergodic capacity per receiver to quantify the performance loss due to the alignment error between the transmitting and receiving beams.While there are already works that investigate the performance loss of beam misalignment, there are only a limited number of studies that consider the robust hybrid precoding design in the presence of beam misalignment [9, 25]. In [9], the authors consider the beam misalignment for backhaul links in smallcell scenarios, and propose a beam alignment method based on adaptive subspace sampling and hierarchical beam codebooks. Nevertheless, only singleuser transmission with analogonly processing was considered. Furthermore, the frequent beam realignment for a largescale antenna array may not be favorable for delaysensitive applications in mmWave communications. In [25], a hybrid precoding scheme is proposed to resist the AoA estimation errors based on the nullspace projection in the analog domain and diagonalloading method in the digital domain, respectively. However, this scheme is only applicable to a singleuser mmWave communication system with the partiallyconnected structure, where each RF chain is connected to a subset of antennas. For the robust design against beam misalignment, the analysis in [7, 22] and [24] has established that the ideal ‘flat mainlobe’ ^{1}^{1}1A constant large antenna gain within the narrow mainlobe and zero elsewhere. model is robust to the loss in array gain, especially in the case of extremely narrow beams. Accordingly, the ideal ‘flat mainlobe’ model is theoretically conducive to alleviate the loss caused by beam misalignment. While there has been a previous attempt to synthesize a realistic ‘flat mainlobe’ beampattern in [26], it considers only a singlereceiver analog beamforming and the relaxation of its optimization problem does not guarantee elementwise constant modulus for the analog precoder, which is required for mmWave transceivers that employ phase shifters. In addition, the statistics of the AoD/AoA estimation errors has been studied in [24] and [27], and it is shown in [25] that the inclusion of the ‘error statistics’ into the precoding design can also lead to an improved performance for the case of beam misalignment. However, the above two concepts have not been well explored for robust multireceiver hybrid precoding design in mmWave communications.
Motivated by this, in this paper we propose robust hybrid precoding schemes for a generic multireceiver mmWave communication system. We consider the hybrid precoding design that approximates the robust fullydigital precoder by minimizing their Euclidean distance. To suppress the residual interreceiver interference, we further introduce a secondstage digital precoder based on zeroforcing. For the precoding design, we consider two distinct methodologies to incorporate the beam misalignment error: a) the robust design based on the ‘flat mainlobe’ model, and b) the robust design based on the prior knowledge of the ‘error statistics’ in beam alignment.
The main contributions of the paper are summarized as follows:

We develop two robust fullydigital precoders (DPs) based on the ‘flat mainlobe’ model and the ‘error statistics’ for a multireceiver mmWave system to alleviate the performance loss owing to beam misalignment. The DP based on ‘flat mainlobe’ model aims to maximize the minimal array gain for the receivers over the expected beam misalignment range. The resulting maxmin nonconvex optimization is solved in two steps: we firstly formulate an equivalent minmax problem with the zeroforcing principle that fully cancels the interreceiver interference, which is further transformed into a secondorder cone programming (SOCP). For the robust DP based on ‘error statistics’, we analytically derive the expected array response for the transmitter incorporating the beam alignment error distribution, which we utilize to obtain the closedform robust DP that maximizes the array gain subject to zero interreceiver interference.

Based on the obtained robust fullydigital precoder, we introduce the hybrid precoding design by minimizing the Euclidean distance between the hybrid precoder and the fullydigital precoder. The resulting optimization is decoupled into two subproblems and solved via alternating optimization, where the digital precoder and the analog precoder are obtained using the leastsquare approximation and the gradient projection (GP) method, respectively. A lowcomplexity scheme based on least square projection (LSP) is also introduced with a closedform expression of the analog precoder for further complexity reduction.

We further design a secondstage digital precoder to fully mitigate the residual interreceiver interference that is incurred due to the approximation process involved in the precoding design. The secondstage digital precoder applies a channel inversion on the effective channel of each receiver.

Our complexity analysis for the analog precoding design reveals that the computational cost of the LSP method is significantly lower than the GP method, while the GP method requires only approximately to of the computational cost compared to the scheme based on manifold optimization (MO) proposed in [10]. Moreover, numerical results show that significant performance gains can be observed for the proposed robust designs compared to the nonrobust hybrid precoders in the presence of imperfect beam alignment.
The rest of the paper is organized as follows. Section II describes the system model, channel model and beam alignment error model. In Section III, the robust fullydigital precoder design and its approximation by the hybrid precoding are presented. Section IV introduces the secondstage digital precoder that cancels the interreceiver interference. Numerical results are presented in Section V, and we conclude the paper in Section VI.
Notations: Bold uppercase letters , bold lowercase letters and letters
denote matrices, vectors and scalars respectively;
is the entry on the th row and th column of ; Conjugate, transpose and conjugate transpose of are represented by , and ; denotes the Frobenius norm of ; is the MoorePenrose pseudo inverse of ; denotes a block diagonal matrix with matrices on the blockdiagonal; indicates vectorization; is the norm of the vector ; Expectation of a complex variable is noted by ; and denote the Hadamard and Kronecker product of two matrices;is the identity matrix;
returns the absolute value of a complex number; denotes the argument of a complex number; denotes the real part of a complex number.Ii System Model
Iia System Model
We consider a multireceiver mmWave system in the downlink, as shown in Fig.1 where a base station (BS) with antennas and RF chains is communicating with receiver units (RUs), where is assumed at the BS to support simultaneous transmission with RUs. Each RU is equipped with antennas and a single RF chain, i.e., a singlestream transmission is assumed for each RU. During transmission, the BS applies a digital precoder followed by an analog precoder , and the signal vector to be transmitted is therefore
(1) 
where , is the symbol transmitted to the th RU and . is the total transmit power at the BS, and in this work we have assumed uniform power allocation among different RUs. Since is implemented with analog phase shifters, its entries should satisfy the elementwise constant modulus constraint, i.e., . The total power constraint is enforced by normalizing such that .
Based on the above, the received signal for the th RU is obtained as
(2) 
where is the analog combiner for RU , is the mmWave channel matrix between the BS and the th RU, and is the additive Gaussian noise at each RU. Similar to its counterpart at the BS, each entry in satisfies the constant modulus constraint.
IiB Channel Model
MmWave channels are expected to be sparse with a limited number of propagation paths, and accordingly the channel between the BS and the th RU is given by [4]:
(3) 
where is the number of propagation paths between the BS and the th RU, and is the complex gain of the th path following . and are the AoD and AoA along the th path, respectively, with and being the corresponding antenna array response vectors of the BS and the th RU, respectively. For uniform linear arrays (ULAs) considered in this paper, for an element antenna array is given by
(4) 
where and are the antenna spacing and signal wavelength, respectively. Since transmission and reception at the mmWave system are done through highly directional beams and with single stream per RU, when multiple paths are available, it is reasonable to steer the beam toward the strongest path [28, 29]. Therefore, in this work, we adopt a single path channel model as in [29], where the channel described in (3) is reduced to
(5) 
where , and are the channel gain, AoD and AoA of the strongest path between the BS and the th RU, respectively.
IiC Error Model for Beam Misalignment
We define the beam misalignment error in AoA/AoD as
(6) 
where is the actual AoA/AoD and is the estimated AoA/AoD. Following [25, 27], the beam alignment error
is characterized by a random variable following a uniform distribution, given by
(7) 
where and
represents the standard deviation of the beam alignment error. We assume the random misalignment error
is bounded as , where is the mainlobe beamwidth of the transceiver units. Beam deviation exceeding is treated as alignment failure, and appropriate methods based on the work in [30] can be used for realignment. In this paper, as we focus on the precoding design at the BS side, we employ the analog combiner that maximizes the array gain at each RU, given by [29](8) 
Iii Hybrid Precoding Design by Approximating the FullyDigital Precoder
In this section, we present the hybrid precoding design by minimizing the difference between the hybrid precoder and the optimal robust fullydigital precoder. For the robust fullydigital precoder, we consider designs based on both the ‘flat mainlobe’ model and the ‘error statistics’ in the beam alignment.
Iiia Robust FullyDigital Precoder Design based on ‘Flat Mainlobe’ (DPFM)
To alleviate the loss in the array again resulting from beam misalignment, the ‘flatmainlobe’ model aims to design the robust fullydigital precoder that maximizes the minimal array gain for each RU over the expected range of misalignment. Let be the fullydigital precoder matrix, where is the precoding vector for the th RU, and then the array gain of the BS corresponding to the th RU is given by
(9) 
where is the set of angular range covering the expected misalignment such that contains samples that are uniformly distributed in the range of . Let be the matrix containing the array response of the BS in the range of , and with the goal of constructing a ‘flat mainlobe’ beampattern for each RUs, we propose to maximize the minimum array gain over by optimizing the robust fullydigital precoder subject to zero interreceiver interference and transmit power limit, formulated as
(10)  
s.t  
where is the set of RUs, cancels the interreceiver interference and is the average transmit power constraint at the BS. The above problem can be equivalently reformulated into the following minmax form:
(11)  
s.t.  
where the vector is to approximate a flat array gain over . The unit gain is attained when the beam is perfectly aligned in the direction of , which is the upper bound for the array gain in the direction . Hence, and are ‘equivalent’ in the sense that the minimization of will maximize . Converting into the epigraph form, we further obtain
(12)  
s.t.  
where denotes the maximum matching error between and for all the RUs. The lefthand side of can be equivalently expressed as [26]
(13) 
where . We further define the interference matrix given by
(14) 
which includes the array response for the other RUs in the set . With , we obtain
and express the singular value decomposition (SVD) of
as(15) 
where is the matrix that consists of the right singular vectors. In accordance with the equality constraint in , we obtain that the precoding vector is in the null space of , and can be expressed as a linear combination of the right singular vectors that correspond to zero singular values, given by
(16) 
where , each represents the weight for the corresponding singular vector and . Using the expression of in (16) and defining , can be formulated into
(17)  
s.t.  
where . This problem is not jointly convex with respect to w.r.t. and because of the coupling of these variables in the constraint . Nevertheless, for a fixed , we note that is convex w.r.t. and is a SOCP, which can be efficiently solved by convex optimization tools [31]. From (13) and (17), for a given , we update following
IiiB Robust FullyDigital Precoder Design based on ‘Error Statistics’ (DPES)
In this section, the ‘error statistics’ metric is incorporated in the DP design to optimize the array gain in the presence of beam misalignment which compared to the ‘flatmainlobe’ metric introduced in Section IIIA can be expressed in a closed form, as detailed below. Based on (4), the array response in the presence of beam alignment error is given by
(19) 
where , and is computed as
(20) 
where , . Since is small, using Maclaurin series, we have
(21) 
Using the Maclaurian series for the exponential function, we can further express
(22) 
where
(23) 
Hence, (20) can now be simplified as
(24) 
By defining and with some algebraic manipulations, the above expression can be recast as
(25) 
We denote the robust fullydigital precoder based on ‘error statistics’ as , where is the precoder for the th RU. Using the principle of zeroforcing [34], we propose to maximize the array gain for the th RU subject to zero interreceiver interference, and the optimization problem on the fullydigital precoder in the case of imperfect beam alignment is given by
(26)  
s.t. 
where is the transmit array gain towards the th RU, , is the estimated AoD at the BS for the th RU and cancels the interreceiver interference. The interference matrix is given by
(27) 
which includes the array response for the other RUs. As , we have and express the singular value decomposition (SVD) of as
(28) 
where is the matrix that consists of the right singular vectors. Similar to the previous section, the precoding vector can be expressed as a linear combination of the right singular vectors of that correspond to zero singular values, given by
(29) 
where , each represents the weight for the corresponding singular vector, and . Using the expression of in (29), can be formulated into
(30) 
which has the following optimal solution:
(31) 
Accordingly, the robust fullydigital precoder for RUs is given by
(32) 
To satisfy the transmit power constraint at the BS, i.e. , we scale to obtain the optimal robust fullydigital precoder based on the ‘error statistic’ metric, given by
(33) 
Discussion: The major difference between the precoding design based on ‘flat mainlobe’ model and ‘error statistics’ model is that the former design maximizes the array gain over an angular range around for each RU, while the latter maximizes the array gain only along by using the expected array response in the presence of the beam alignment error. Accordingly, while the ‘flat mainlobe’ model aims to maximize the effective array gain, the ‘error statistics’ metric maximizes the average array gain over the misalignment error range for all the RUs and enjoys a closedform expression for the robust fullydigital precoder.
IiiC Hybrid Precoding Approximation (HPA)
Based on the obtained robust fullydigital precoder, in this section we introduce the design of the hybrid precoder and based on matrix factorization and alternating optimization. Alternating optimization is widely employed in optimization problems involving different subsets of variables, which also finds its applications in matrix completion [35], [36], image reconstruction [37], blind deconvolution [38] and nonnegative matrix factorization [39]. Similar to the work in [10], we design the hybrid precoding by formulating the following optimization:
(34)  
s.t  
which is a matrix factorization problem and is solved by alternately optimizing and . Algorithm 2 describes the framework to obtain a feasible hybrid precoder by approximating the fullydigital precoder based on the principle of alternating optimization, as shown below. To be more specific, the digital precoder is designed based on a fixed analog precoder by solving the following leastsquare subproblem:
(35)  
s.t 
which leads to
(36) 
where in is satisfied by normalizing by the factor [10, Lemma 1].
The analog precoder for a given is designed by solving the following subproblem:
(37)  
s.t. 
By defining , and , is reformulated as the following constant modulus leastsquare (CMLS) problem:
(38)  
s.t. 
In the following, we first briefly review an existing algorithm for based on manifold optimization, followed by the description for our proposed algorithms based on GP and LSP.
IiiC1 Analog Precoding Design  Manifold Optimization (MO) [10]
For the MO as a benchmark in this paper, the constraint in is defined as a Riemannian manifold [40]. Endowing the complex plane with the Euclidean metric , the manifold where is introduced, which is the search space of over Riemannian submanifold of .
Defining the tangent space for a point as , the Riemannian gradient is obtained by the orthogonal projection of the Euclidean gradient onto , given as
(39) 
where for the unconstrained objective of , is given by
(40) 
To evaluate the objective function on the manifold, the retraction operation of a tangent vector at the point is defined as [41]
(41) 
The counterpart of the classical conjugate gradient method (CGM) for the defined manifold is used to search for the optimal analog precoder in [10], where it is observed that updating the Riemannian gradient and the descent direction requires the operation between two vectors in different tangent spaces and . Subsequently, the tangent vector needs to be mapped from to , which is given by
(42) 
While the MO algorithm can provide a nearoptimal performance, the update of the analog precoder involves a linesearch algorithm, which within each iteration involves a) an orthogonal projection of Euclidean gradient onto the tangent space defined in (39), b) a retraction of the tangent vector on the manifold defined in (41), and c) the construction of a transport from one tangent space to another, as defined in (42).
IiiC2 Proposed Analog Precoding Design  Gradient Projection (GP)
In this section, we propose an iterative analog precoding design based on the GP method, which only requires the projection of the solution sequence onto the elementwise constant modulus constraint set. The GP method is in nature a revamped version of the Conjugate Gradient Method (CGM), which searches for the optimal solution by projecting each subsequent point
(43) 
onto the feasible region defined in , while moving along the decent direction , with the step size given by
(44) 
The projection onto can be viewed as a phaseextraction operation and has a closedform solution, given by
(45) 
Algorithm 3 summarizes the GP method, which employs the PolakRibiere parameter [33] to update the decent direction in each iteration, as shown in Step 9 of Algorithm 3. [32] has demonstrated that the projection onto the elementwise constant modulus constraint set does not increase the objective, as the solution sequence converges to a KKT point.
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