I Introduction
Massive multipleinput multipleoutput (MIMO) is one of the promising techniques to cope with the predicted wireless data traffic explosion [1, 2, 3, 4]. In downlink massive MIMO systems, it was shown that lowcomplexity linear precoding methods as zeroforcing (ZF) and regularized ZF (RZF) achieve an almost optimal performance [5]. In contrast, the use of a large number of antennas considerably increases the hardware cost and the radiofrequency (RF) circuit consumption [6]. Hybrid analogdigital precoding (a.k.a., hybrid precoding) is one of the promising methods to address the above problems since it can reduce the number of RF chains [7, 8, 9]. An alternative method is to make use of lowresolution digitaltoanalog converters (DACs) (e.g., 13 bits), which can greatly reduce the cost and power consumption per RF chain [6]. Specifically, in this case, each antenna’s transmit symbol is equivalent to a constantenvelop symbol, which enables the use of lowcost power amplifiers and thus reduces the hardware complexity.
Because of the potential merits of using onebit DACs, there have been numerous studies on the precoding methods for such downlink massive MIMO systems [10, 11, 12, 13, 14, 15]. In [10], various nonlinear precoding techniques were proposed based on semidefinite relaxation (SDR), squared norm relaxation, and sphere decoding. The authors proposed lowcomplexity quantized precoding methods as quantized ZF (QZF) [11] and quantized minimummean squared error (QMMSE) [12], which simply applied the onebit quantization to the outputs of the conventional linear precoding methods. Also, a branchandbound and a biconvex relaxation approaches were presented in [13] and [14], respectively. However, these methods either suffer from a severe performance loss or require an expensive computational complexity (see [15] for more details). Very recently, focusing on phaseshiftkeying (PSK) constellations, a lowcomplexity symbolscaling method was proposed in [15], where it optimizes a transmitsignal vector directly in an efficient sequential fashion as a function of an instantaneous channel matrix and users’ messages (called symbollevel operation). Also, it was shown that the symbolscaling method can yield a comparable performance to the previous nonlinear precoding methods with a much lower computational complexity.
Our contributions: In this paper, we study a downlink MUMISO system with onebit DACs. We first identify that antennaselection can yield a nontrivial biterrorrate (BER) performance gain by alleviating errorfloor problems (see Fig. 1). Clearly, this performance gain is because antennaselection can enlarge the set of possible transmitsignal vectors (i.e., searchspace) compared with the previous precoding methods in [10, 11, 12, 13, 14, 15]. However, as in conventional onebit precoding optimization, finding an optimal transmitsignal vector (encompassing precoding and antennaselection) requires exhaustivesearch due to its combinatorial nature. Motivated by this, we propose a lowcomplexity algorithm to solve the above problem (i.e., joint optimization of precoding and antennaselection), which consists of the following two stages. In the first stage, we obtain a feasible transmitsignal vector via iterativehardthresholding (IHT) algorithm where the resulting vector guarantees that each user’s noiseless observation is belong to a desired decision region. Namely, it can improve the BER performances at highSNR regimes (i.e., errorfloor regions) by lowering an errorfloor. In the second stage, we refine the above transitsignal vector using a bitflipping (BF) algorithm so that each user’s received signal is more robust to additive Gaussian noises. In other words, it can improve the BER performances at lowSNR regimes (i.e., waterfall regions). Finally, we provide simulation results to demonstrate that the proposed method can improve the performance of the existing symbolscaling method in [15] with a comparable computational complexity.
The rest of paper is organized as follows. In Section II, we provide some useful notations and describe the system model. In Section III, we propose a lowcomplexity algorithm to optimize a transmitsignal vector directly for downlink MUMISO systems with onebit DACs. Simulation results are provided in IV. Section V concludes the paper.
Ii Preliminaries
In this section, we provide some useful notations and describe the system model.
Iia Notations
The lowercase and uppercase boldface letters represent column vectors and matrices, respectively, and denotes the conjugate transpose of a vector or matrix. For any vector , represents the th entry of . Let for nonnegative integers and with . Similarly, let for any positive integer . and represent its real and complex part of a complex vector , respectively. Also, we define a natural mapping which maps a complex value into a realvalued vector, i.e., for each , we have that
(1) 
The inverse mapping of is denoted as . If or is applied to a vector or a set, we assume they operate elementwise. For example, we have that
(2) 
Also, for any complexvalue , we define a realvalued matrix expansion as
(3) 
Finally, we let denote a rotation matrix with a parameter as
which rotates the following column vector in the counterclockwise through an angle about the origin.
IiB System Model
We consider a singlecell downlink MUMISO system in which one BS with antennas communicates with singleantenna users simultaneously in the same timefrequency resources. Focusing on the impact of onebit DACs in the transmitside operations, it is assumed that the BS is equipped with onebit DACs while each user (receiver) is with ideal analogtodigital converters (ADCs) with infinite resolution. As in the closely related works [11, 12, 15], we also consider a normalized PSK constellation , each of which constellation point is defined as
(4) 
The constellation points of PSK are depicted in Fig. 2. Let be the user ’s message for and also let be a transmitsignal vector at the BS. Under the use of onebit DACs and antennaselection, each can be chosen with the restriction of
(5) 
This shows that, compared with the related works [11, 12, 15], antennaselection can enlarge the set of possible symbols (i.e., searchspace) per real (or imaginary) part of each antenna. Then, the received signal vector at the users is given by
(6) 
where
denotes the flatfading Rayleigh channel with each entry following a complex Gaussian distribution and
denotes the additive Gaussian noise vector whose elements are distributed as circularly symmetric complex Gaussian random variables with zeromean and unitvariance, i.e.,
. Also, is chosen according to the perantenna power constraint. For simplicity, we assume the uniform power allocation for the antenna array.In this system, our purpose is to develop a lowcomplexity algorithm to optimize a transmitsignal vector (i.e., joint optimization of precoding and antennaselection) with the assumption that the BS is aware of a perfect channel state information (CSI), which will be provided in Section III. Before explaining our main result, we provide the following definition which will be used throughout the paper.
Definition 1
(Decision Regions) For each , a decision region is defined as
(7) 
If the user receives a , then it decides the decoded message as .
Iii The Proposed TransmitSignal Vectors
In this section, we derive a mathematical formulation to optimize a transmitsignal vector (encompassing precoding and antenna selection) for downlink MUMISO systems with onebit DACs, and then present a lowcomplexity algorithm to solve such problem efficiently.
For the ease of explanation, we first introduce the equivalent realvalued representation of the complex inputoutput relationship in (6), which is given by
(8) 
where , , , and denotes the realvalued matrix which is obtained by replacing (e.g., the th entry of ) with the matrix for all . For the resulting model, we can define the realvalued constellation where
(9) 
From Definition 1, the decision region in for each is simply obtained as
(10) 
Furthermore, as shown in Fig. 2, each can be represented as linear combination of two basis vectors and as
(11) 
where the basis vectors are easily obtained using rotation matrices such as
for . Using two basis vectors, we define the realvalued matrix . Since has fullrank, the inverse matrix of exists and is easily computed as
(12) 
We are now ready to explain how to optimize a transmitsignal vector efficiently. For simplicity, we let denote the noiseless received vector at the users. Regarding our optimization problem, we first provide the following key observations:

(Feasibility condition) To ensure that all the users recover their own messages, a transmitsignal vector should be constructed such that
(13) for . Accordingly, should be represented as
(14) for some positive coefficients and . This is called feasibility condition and a vector to satisfy this condition called feasible transmitsignal vector.

(Noise robusteness) The condition in (13) cannot guarantee good performances in practical SNR regimes (e.g., waterfall regions) due to the impact of additive Gaussian noises. Thus, we need to refine the above feasible transmitsignal vector so that and are maximized.
We will formulate an optimization problem mathematically which can find a transmitsignal vector to satisfy the above requirements. From (14), we can express the feasibility condition in a matrix form:
(15) 
where and denotes the block diagonal matrix having the th diagonal block . From the block diagonal structure and (12), we can easily obtain the inverse matrix of as
(16) 
Then, the feasibility condition in (15) can be rewritten as
(17) 
where . Note that is a known matrix since it is completely determined from the channel matrix and users’ messages . Taking the feasibility condition and noise robustness into account, our optimization problem can formulated as
(18)  
subject to  (19)  
(20)  
(21) 
For the above optimization problem, the objective function aims to maximize the minimum of positive coefficients ’s. This is motivated by the fact that a larger value of the coefficients ’s yields a larger distance to the other decision regions. As an example, consider the decision region as illustrated in Fig. 3. Obviously, has a larger distance from the boundary 1 than while both have the same distance from the boundary 2. Likewise, if increases for a fixed , then the distance from the boundary 2 increases by keeping the distance from the boundary 1. In order to increase the distance from the other decision regions, therefore, it would make sense to maximize the minimum of two coefficients and . By extending this to all the users, we can obtain the objective function in (18).
Unfortunately, finding an optimal solution to the above optimization problem is too complicated due to its combinatorial nature. We thus propose a lowcomplexity twostage algorithm to solve it efficiently, which is described as follows.

In the first stage, we find a feasible solution to satisfy the constraints (19)(21), which is obtained by taking a solution of
(22) where if and otherwise. We solve the above nonlinear inverse problem efficiently via the socalled IHT algorithm. The detailed procedures are provided in Algorithm 1 where the thresholding function is defined as if , if , and , otherwise.

In the second stage, we refine the above feasible solution by maximizing , which is efficiently performed using BF algorithm (see Algorithm 2). Finally, from the output of Algorithm 2 (e.g., ), we can obtain the proposed transmitsignal vector as .
Computational Complexity: Following the complexity analysis in [15], we study the computational complexity of the proposed method with respect to the number of realvalued multiplications. As benchmark methods, we consider the computational costs of exhaustive search, exhaustive search with antennaselection, and symbolscaling method in [15], which are denoted by , , and , respectively. From [15], their complexities are obtained as
(23)  
(24)  
(25) 
Recall that the proposed method is composed of the two algorithms in Algorithms 1 and 2. First, the complexity of Algorithm 1 is obtained as where denotes the total number of iterations. Also, the complexity of Algorithm 2, which is similar to that of refinedstate in symbolscaling method in [15], is obtained as . By summing them, the overall computational complexity of the proposed method is obtained as
(26) 
Note that for some , is obtained from the by simply replacing with , i.e.,
Iv Simulation Results
In this section, we evaluate the symbolerrorrate (SER) performances of the proposed method for the downlink MUMIMO systems with onebit DACs where and . For comparison, we consider QZF in [11] and the stateoftheart symbolscaling method in [15] because the former is usually assumed to be the baseline approach and the latter showed an elegant performancecomplexity tradeoff over the other existing methods (see [15] for more details). Both QPSK (or 4PSK) and 8PSK are considered. Regarding the proposed method, we choose the threshold parameter for Algorithm 1, which is optimized numerically via MonteCarlo simulation. As mentioned in Section IIB, a flatfading Rayleigh channel is assumed.
Fig. 4 shows the performance comparison of various precoding methods when QPSK is employed. From this, we observe that QZF suffers from a serious errorfloor and thus is not able to yield a satisfactory performance. In contrast, both symbolscaling and the proposed methods overcome the errorfloor problem, thereby outperforming QZF significantly. Moreover, it is observed that the proposed method can slightly improve the performance of QZF with much less computational cost (e.g., complexity reduction).
Fig. 5 shows the performance comparison of various precoding methods when 8PSK is employed. It is observed that in this case, an errorfloor problem is more severe than before, which is obvious since the decision regions of 8PSK is more sophisticated than those of QPSK. Hence, antennaselection can attain a more performance gain with a larger searchspace. Accordingly, the proposed method can further improve the performance of symbolscaling method by lowering an errorfloor, which is verified in Fig. 5. To be specific, the proposed method with outperforms the symbolscaling method with an almost same computational cost. Furthermore, at the expense of two times computational cost, the proposed method with can address the errorfloor problem completely. Therefore, the proposed method provides a more elegant performancecomplexity tradeoff than the stateoftheart symbolscaling method.
V Conclusion
In this paper, we showed that the use of antennaselection can yield a nontrivial performance gain especially at errorfloor regions, by enlarging the set of possible transmitsignal vectors. Since finding an optimal transmitsignal vector (encompassing precoding and antennaselection) is too complex, we proposed a lowcomplexity twostage method to directly obtain such transmitsignal vector, which is based on iterativehardthresholding and bitflipping algorithms. Via simulation results, we demonstrated that the proposed method provides a more elegant performancecomplexity tradeoff than the stateoftheart symbolscaling method. One promising extension of this work is to devise a lowcomplexity algorithm which is more suitable to our nonlinear inverse problem for finding a feasible transmitsignal vector. Another extension is to develop the proposed idea for onebit DAC MUMISO systems with quadratureamplitudemodulation (QAM). This is more challenging than this work focusing on PSK since some decision regions of QAM are surrounded by the other decision regions, which makes difficult to handle.
References
 [1] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 35903600, Nov. 2010.
 [2] A. Adhikary, J. Nam, J.Y. Ahn, and G. Caire, “Joint spatial division and multiplexing: The largescale array regime,” IEEE Trans. Inf. Theory, vol. 59, no. 10, pp. 64416463, Oct. 2013.
 [3] C. Masouros, M. Sellathurai, and T. Ratnarajah, “Largescale MIMO transmitters in fixed physical spaces: The effect of transmit correlation and mutual coupling,” IEEE Trans. Commun., vol. 61, no. 7, pp. 27942804, Jul. 2013.
 [4] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An overview of massive MIMO: Benefits and challenges,” IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 742758, Oct. 2014.
 [5] C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, “A vectorperturbation technique for nearcapacity multiantenna multiuser communicationpart I: Channel inversion and regularization,” IEEE Trans. Commun., vol. 53, no. 1, pp. 195202, Jan. 2005.
 [6] H. Yang and T. L. Marzetta, “Total energy efficiency of cellular large scale antenna system multiple access mobile networks,” in Proc. IEEE Online Conf. Green Commun., pp. 2732, Piscataway, NJ, USA, Oct. 2013.
 [7] A. F. Molisch, V. V. Ratnam, S. Han, Z. Li, S. L. H. Nguyen, L. Li, and K. Haneda, “Hybrid beamforming for massive MIMO: A survey,” IEEE Commun. Mag., vol. 55, no. 9, pp. 134141, 2017.

[8]
A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,”
IEEE J. Sel. Topics Signal Process., vol.8, no.5, pp.831846, Oct. 2014.  [9] J. Mo, A. Alkhateeb, S. AbuSurra, and R. W. Heath, “Hybrid architectures with fewbit ADC receivers: Achievable rates and energyrate tradeoffs,” IEEE Trans. Wireless Commun., vol. 16, no. 4, pp. 22742287, Apr. 2017.
 [10] S. Jacobsson, G. Durisi, M. Coldrey, T. Goldstein, and C. Studer, “Quantized precoding for massive MUMIMO,” IEEE Trans. Commun., vol. 65, no. 11, pp. 46704684, Nov. 2017.
 [11] A. K. Saxena, I. Fijalkow, and A. L. Swindlehurst, “Analysis of onebit quantized precoding for the multiuser massive MIMO Downlink,” IEEE Trans. Sig. Process., vol. 65, no. 17, pp. 46244634, Sept. 2017.
 [12] O. B. Usman, H. Jedda, A. Mezghani, and J. A. Nossek, “MMSE precoder for massive MIMO using 1bit quantization,” in Proc. IEEE Int. Conf. Acoust. Speech Sig. Process. (ICASSP), pp. 33813385, Shanghai, Mar. 2016.
 [13] L. T. N. Landau and R. C. de Lamare, “Branchandbound precoding for multiuser MIMO systems with 1bit quantization,” IEEE Wireless Commun. Lett., vol. 6, no. 6, pp. 770773, Dec. 2017.
 [14] O. Castaneda, S. Jacobsson, G. Durisi, M. Coldrey, T. Goldstein, and C. Studer, “1bit massive MUMIMO precoding in VLSI,” IEEE J. Emerging Sel. Topics Circuits and Systems, vol. 7, no. 4, pp. 508522, Dec. 2017.
 [15] A. Li, C. Masouros, F. Liu, and A. L. Swindlehurst, “Massive MIMO 1bit DAC transmission: A lowcomplexity symbol scaling approach,” IEEE Trans. Wireless Commun., vol. 17, no. 11, pp. 75597575, Nov. 2018.
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