Robust Hybrid Precoding for Beam Misalignment in Millimeter-Wave Communications

In this paper, we focus on the phenomenon of beam misalignment in Millimeter-wave (mmWave) multi-receiver communication systems, and propose robust hybrid precoding designs that alleviate the performance loss caused by this effect. We consider two distinct design methodologies: I) the synthesis of a `flat mainlobe' beam model which maximizes the minimum effective array gain over the beam misalignment range, and II) the inclusion of the `error statistics' into the design, where the array response incorporating the distribution of the misalignment error is derived. For both design methodologies, we propose a hybrid precoding design that approximates the robust fully-digital precoder, which is obtained via alternating optimization based on the gradient projection (GP) method. We also propose a low-complexity alternative to the GP algorithm based on the least square projection (LSP), and we further deploy a second-stage digital precoder to mitigate any residual inter-receiver interference after the hybrid analog-digital precoding. Numerical results show that the robust hybrid precoding designs can effectively alleviate the performance degradation incurred by beam misalignment.



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I Introduction

The availability of rich spectrum in the millimeter-wave (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 sub-6 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 large-scale 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 fully-digital precoder in traditional sub-6GHz bands, it is unfortunately not promising to consider fully-digital 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 cost-efficiently and energy-efficiently, the concept of hybrid analog-digital structure has been introduced in [4], which provides a promising trade-off 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 low-dimension digital precoder, followed by the processing of a high-dimension analog precoder [10, 11, 12]. For the analog precoding, low-cost 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 fully-digital 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 large-scale 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 two-dimension 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 signal-to-interference-plus-noise ratio (SINR) was derived. For a mmWave ad-hoc network, the authors in

[24] have derived a closed-form 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 small-cell scenarios, and propose a beam alignment method based on adaptive subspace sampling and hierarchical beam codebooks. Nevertheless, only single-user transmission with analog-only processing was considered. Furthermore, the frequent beam re-alignment for a large-scale antenna array may not be favorable for delay-sensitive applications in mmWave communications. In [25], a hybrid precoding scheme is proposed to resist the AoA estimation errors based on the null-space projection in the analog domain and diagonal-loading method in the digital domain, respectively. However, this scheme is only applicable to a single-user mmWave communication system with the partially-connected 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’ 111A 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 single-receiver analog beamforming and the relaxation of its optimization problem does not guarantee element-wise 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 multi-receiver hybrid precoding design in mmWave communications.

Motivated by this, in this paper we propose robust hybrid precoding schemes for a generic multi-receiver mmWave communication system. We consider the hybrid precoding design that approximates the robust fully-digital precoder by minimizing their Euclidean distance. To suppress the residual inter-receiver interference, we further introduce a second-stage digital precoder based on zero-forcing. 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 fully-digital precoders (DPs) based on the ‘flat mainlobe’ model and the ‘error statistics’ for a multi-receiver 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 max-min non-convex optimization is solved in two steps: we firstly formulate an equivalent min-max problem with the zero-forcing principle that fully cancels the inter-receiver interference, which is further transformed into a second-order 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 closed-form robust DP that maximizes the array gain subject to zero inter-receiver interference.

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

  • We further design a second-stage digital precoder to fully mitigate the residual inter-receiver interference that is incurred due to the approximation process involved in the precoding design. The second-stage 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 non-robust 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 fully-digital precoder design and its approximation by the hybrid precoding are presented. Section IV introduces the second-stage digital precoder that cancels the inter-receiver interference. Numerical results are presented in Section V, and we conclude the paper in Section VI.

Notations: Bold upper-case letters , bold lower-case 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 Moore-Penrose pseudo inverse of ; denotes a block diagonal matrix with matrices on the block-diagonal; 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

Figure 1: Block diagram for a multi-receiver mmWave MIMO system

Ii-a System Model

We consider a multi-receiver 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 single-stream 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


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 element-wise 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


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.

Ii-B 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]:


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


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


where , and are the channel gain, AoD and AoA of the strongest path between the BS and the -th RU, respectively.

Ii-C Error Model for Beam Misalignment

We define the beam misalignment error in AoA/AoD as


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


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]


Iii Hybrid Precoding Design by Approximating the Fully-Digital Precoder

In this section, we present the hybrid precoding design by minimizing the difference between the hybrid precoder and the optimal robust fully-digital precoder. For the robust fully-digital precoder, we consider designs based on both the ‘flat mainlobe’ model and the ‘error statistics’ in the beam alignment.

Iii-a Robust Fully-Digital Precoder Design based on ‘Flat Mainlobe’ (DP-FM)

To alleviate the loss in the array again resulting from beam misalignment, the ‘flat-mainlobe’ model aims to design the robust fully-digital precoder that maximizes the minimal array gain for each RU over the expected range of misalignment. Let be the fully-digital 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


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 fully-digital precoder subject to zero inter-receiver interference and transmit power limit, formulated as


where is the set of RUs, cancels the inter-receiver interference and is the average transmit power constraint at the BS. The above problem can be equivalently reformulated into the following min-max form:


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


where denotes the maximum matching error between and for all the RUs. The left-hand side of can be equivalently expressed as [26]


where . We further define the interference matrix given by


which includes the array response for the other RUs in the set . With , we obtain

and express the singular value decomposition (SVD) of



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


where , each represents the weight for the corresponding singular vector and . Using the expression of in (16) and defining , can be formulated into


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


Based on the above, is solved using an alternating optimization process, as shown in Algorithm 1. In Algorithm 1, the objective function is positive and minimized within each iteration at Step 4 and Step 5. Accordingly, the algorithm converges to a locally optimal solution [26, 32, 33].

1:  Input: , ,
2:  Initialize , ;
3:  while  and  do
4:     For a given , obtain by solving ;
5:     Update ;
6:     Update ; .
7:  end while
8:  Output:
Algorithm 1 Alternating Optimization for the Robust Fully-Digital Precoder in

Iii-B Robust Fully-Digital Precoder Design based on ‘Error Statistics’ (DP-ES)

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 ‘flat-mainlobe’ metric introduced in Section III-A 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


where , and is computed as


where , . Since is small, using Maclaurin series, we have


Using the Maclaurian series for the exponential function, we can further express




Hence, (20) can now be simplified as


By defining and with some algebraic manipulations, the above expression can be recast as


We denote the robust fully-digital precoder based on ‘error statistics’ as , where is the precoder for the -th RU. Using the principle of zero-forcing [34], we propose to maximize the array gain for the -th RU subject to zero inter-receiver interference, and the optimization problem on the fully-digital precoder in the case of imperfect beam alignment is given by


where is the transmit array gain towards the -th RU, , is the estimated AoD at the BS for the -th RU and cancels the inter-receiver interference. The interference matrix is given by


which includes the array response for the other RUs. As , we have and express the singular value decomposition (SVD) of as


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


where , each represents the weight for the corresponding singular vector, and . Using the expression of in (29), can be formulated into


which has the following optimal solution:


Accordingly, the robust fully-digital precoder for RUs is given by


To satisfy the transmit power constraint at the BS, i.e. , we scale to obtain the optimal robust fully-digital precoder based on the ‘error statistic’ metric, given by


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 closed-form expression for the robust fully-digital precoder.

Iii-C Hybrid Precoding Approximation (HPA)

Based on the obtained robust fully-digital 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 non-negative matrix factorization [39]. Similar to the work in [10], we design the hybrid precoding by formulating the following optimization:


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 fully-digital 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 least-square sub-problem:


which leads to


where in is satisfied by normalizing by the factor [10, Lemma 1].

1:  Input: , ,
2:  Initialize , , ;
3:  Initialize using random phase;
4:  while  and  do
5:     For a given , calculate ;
6:     Update ;
7:     ;
8:     ; .
9:  end while
10:  .
11:  Output:
Algorithm 2 Hybrid Precoding by Approximating the Fully-Digital Precoder

The analog precoder for a given is designed by solving the following sub-problem:


By defining , and , is reformulated as the following constant modulus least-square (CMLS) problem:


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.

Iii-C1 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


where for the unconstrained objective of , is given by


To evaluate the objective function on the manifold, the retraction operation of a tangent vector at the point is defined as [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


While the MO algorithm can provide a near-optimal performance, the update of the analog precoder involves a line-search 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).

Iii-C2 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 element-wise 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


onto the feasible region defined in , while moving along the decent direction , with the step size given by


The projection onto can be viewed as a phase-extraction operation and has a closed-form solution, given by


Algorithm 3 summarizes the GP method, which employs the Polak-Ribiere 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 element-wise constant modulus constraint set does not increase the objective, as the solution sequence converges to a KKT point.

1:  Input: ,
2:  Initialize with random phase, , , , ;
3:  while  and  do
4:     ;