1 Introduction
Massive multipleinput multipleoutput (MIMO), a.k.a. largescale antenna arrays, has become a key enabling technique for the coming fifthgeneration (5G) wireless communication systems [1][3]. In the downlink transmission of a massive MIMO system, lowcomplexity linear precoding methods [4] are shown to be nearoptimal, while nonlinear precoding approaches [5][7] are not preferred due to their high computational costs. Nevertheless, this near optimality is built on fullydigital signal processing with highresolution digitaltoanalog converters (DACs), while such a direct extension from smallscale antenna arrays to largescale ones will incur prohibitive hardware complexity. The consequent power consumption at the base station (BS) will also be huge, which does not meet the target of energyefficient transmission for future wireless communication systems. To this end, hardwareefficient largescale antenna architectures such as hybrid analogdigital structures [8][11], constantenvelope transmission [12][14], and lowresolution DACs have been proposed, where the use of lowresolution DACs, more specifically 1bit DACs, is the focus of this paper.
In the literature, there already exist some works that study the precoding design in the presence of 1bit DACs . This includes linear 1bit precoding designs in [15], [16] as well as nonlinear 1bit precoding designs [17][20], where nonlinear 1bit precoding schemes generally perform much better than linear ones. More specifically, in [17] and [18], nonlinear 1bit precoding schemes were proposed via the gradient projection algorithm based on the minimum error rate metric and minimum meansquared error (MMSE) metric, respectively. [19] proposed a 1bit precoding design via a biconvex relaxation procedure, while [20] extended the work in [19] and proposed several 1bit precoding schemes based on semidefinite relaxation (SDR), norm relaxation and sphere precoding, respectively. It should be noted that the above nonlinear 1bit precoding schemes that achieve promising error rate performances operate on a symbol level, i.e., the precoding strategy and the precoded signals are designed based on both the data symbols to be transmitted and the channel state information (CSI), as opposed to many traditional precoding designs that are dependent on CSI only [4][7].
When it comes to symbollevel precoding, there is a concept termed ‘constructive interference’ (CI) that has already received increasing research attention in recent years [21], [22]. CI is defined as the interference that pushes the received signals deeper in the decision region and farther away from the decision boundaries, which further improves the detection performance, though the MSE in this case will increase. This observation has been exploited in [23] and the references therein by symbollevel CI precoding to achieve an improved error rate performance in traditional multiuser MIMO scenarios. Inspired by this concept, [24][26] extended the idea of CI to 1bit precoding designs, and the resulting performance is shown to be promising. Moreover, while not explicitly shown, [27] also adopts the formulation of CIbased 1bit precoding, where a branchandbound (BB) based 1bit precoding algorithm that returns the optimal solution is presented. However, this 1bit design is based on the fullyBB (FBB) process, which is not practically useful in massive MIMO systems due to its unfavorable complexity.
In this paper, we design a nearoptimal 1bit precoding approach that aims to minimize the symbol error rate (SER) in the downlink transmission of a multiuser largescale antenna system, where the BB framework is leveraged. By exploiting the concept of CI and adopting the ‘symbolscaling’ metric, the SER minimization is equivalent to the maximization on the scaling coefficients, based on which we formulate the optimization problem, which is nonconvex due to the discrete 1bit constraint on the transmit signals. By relaxing the 1bit constraint and further analyzing the relaxed convex problem, we mathematically prove that the majority of the entries in the transmit signal vector obtained from solving the relaxed convex problem already comply with the 1bit constraint, i.e., only a small part of the entries need to be further normalized to meet the 1bit requirement. Building upon this observation, we introduce the proposed 1bit precoding design based on the PBB procedure to further improve the performance, where the BB process is only performed for the entries that do not satisfy the 1bit constraint. Within the BB process, we employ the ‘maxmin’ criterion to design the PBB algorithm and adopt the adaptive subdivision rule to guarantee a fast convergence speed. Compared to the traditional FBB methods whose complexity becomes prohibitive when largescale antenna arrays are considered, our proposed PBB approach makes the BB framework applicable in such scenarios with significantly reduced complexity, while still exhibiting a nearoptimal error rate performance, as validated by our numerical results.
Notations: , , and denote scalar, column vector and matrix, respectively. and denote transpose and conjugate transpose, respectively. denotes the cardinality of a set, is the sign function, and is the imaginary unit. denotes the modulus or the absolute value, and denotes the norm. and represent an matrix in the complex and real set, respectively. and represent the extraction of the real and imaginary part, respectively, and represents a identity matrix.
2 System Model
We focus on a generic downlink massive MIMO system, where the BS with each RF chain equipped with a pair of 1bit DACs communicates with multiple singleantenna users in the same timefrequency resource simultaneously. We denote the total number of transmit antennas at the BS by and the total number of users by , where . Since we focus on the effect of 1bit DACs on the data transmission, we assume ideal ADCs are adopted at each receiver and perfect CSI is available at the BS [15][20]. We denote the intended data symbol for user by , which is assumed to be drawn from a unitnorm PSK constellation, and we express the data symbol vector as . We consider a flatfading Rayleigh channel between the BS and the users, denoted by
, with each entry following a standard complex Gaussian distribution
, where is the channel vector between the BS and user . Accordingly, the transmit signal vector at the antenna port can be expressed as(1) 
where represents the unquantized transmit signal vector, and is the 1bit quantization operation. forms the unquantized signal vector based on the knowledge of and , which represents the precoding strategy adopted at the BS. In this paper, we normalize such that each entry in satisfies
(2) 
where and . The received signal vector can be expressed as
(3) 
where is the additive Gaussian noise at the receiver side with .
3 Proposed 1Bit Precoding via PBB
3.1 Problem Formulation
We present the construction of the 1bit precoding optimization problem based on the ‘symbolscaling’ CI metric in this section. The ‘symbolscaling’ CI formulation performs a signal decomposition of the data symbols as well as the noiseless received signals, where the introduced scaling coefficients are the variables to be optimized. To be more specific, we depict one quarter of an 8PSK constellation as the example in Fig. 1, where without loss of generality we denote the data symbol for user by , which is further decomposed along the two detection boundaries of into [23]
(4) 
where , and are parallel to the detection boundary and respectively, as shown in Fig. 1. We refer the interested readers to Section IV of [26] for the expression of and when a generic PSK modulation is employed, which are omitted here for brevity. We further denote the received signal excluding noise for user by , which is similarly decomposed into
(5) 
where and are the introduced scaling coefficients that jointly represent the effect of interference and 1bit quantization on . Following [23], minimizing the SER is equivalent to pushing the noiseless received signal as deep as possible in the decision region and farther away from both of the decision boundaries, which is further equivalent to maximizing the minimum value of . Accordingly, the 1bit precoding design can be formulated as
(6)  
which is a nonconvex optimization problem due to the discrete 1bit constraint , .
3.2 Analytical Study on 1bit CI Precoding
While is originally a nonconvex optimization problem and difficult to solve, by relaxing the 1bit constraints in , we arrive at a convex problem formulation:
(7)  
A suboptimal solution can then be obtained by enforcing the 1bit constraint on the signal vector obtained from solving , i.e.,
(8) 
We denote the above relaxationnormalization procedure by ‘CI 1Bit’ and the corresponding transmit signal vector by .
Based on Lagrangian and KKT conditions, in this section we further elaborate on and show that most of the entries in from solving already comply with the 1bit requirement, as a motivation for our proposed 1bit precoding approach via PBB. To be more specific, we first express and as a function of , given by
(9)  
which is obtained by comparing the real and imaginary part of both sides of (5). By expressing , , and further defining
(10) 
(9) can be expressed in a compact matrix form as
(11) 
where is given by
(12) 
Based on this transformation, is equivalent to the following optimization problem:
(13)  
where represents the th row in , is the th entry in , , and . Based on the formulation of , we derive the following important proposition, which builds the foundation of the proposed PBB procedure in the following.
Proposition: For obtained by solving , there are at least a total number of entries that already comply with the 1bit requirement.
Proof: Proving this proposition is equivalent to proving that there are at most a total number of entries in whose amplitudes are smaller than .
To begin with, by transforming into a standard minimization form:
(14)  
we express the Lagrangian of as
(15)  
where , , and are the nonnegative Lagrangian multipliers. We then construct the KKT conditions as:
rCl
∂L∂t = 1^Tβ  1 = 0 *
∂L∂~xE =  M^Tβ + μ  ν = 0
β_l( t  m_l^T~x_E )=0, β_l ≥0, ∀l ∈L
μ_m( ~x_m^E  12Nt )=0, μ_m≥0, ∀m ∈M
ν_m( ~x_m^E + 12Nt )=0, ν_m≥0, ∀m ∈M
In the following, we prove this proposition by contradiction.
Suppose that there are a total number of entries in whose amplitudes are strictly smaller than , and for notational convenience we introduce a set to include the indices of these entries, which is mathematically expressed as
(16) 
where we have based on our above assumption. According to the complementary slackness conditions (15) and (15), we obtain
(17) 
Recall (15) which can be regarded as a linear equation with as the variable, and for simplicity we introduce . Given (17), we subsequently pick the corresponding rows of whose indices belong to to formulate a sub linear equation:
(18) 
where is expressed as
(19) 
Based on that , we obtain that is fullrank. According to the linear algebra theory [28], given a fullrank coefficient matrix , a nonzero solution to (18) does not exist and there is only a trivial solution, i.e.,
(20) 
However, this solution does not comply with (15) that enforces a nonzero solution of , which causes contradiction. By following a step similar to the above, this contradiction is also observed if we assume there are a total number of entries in the obtained whose amplitudes are strictly smaller than , which completes the proof.
3.3 Proposed 1Bit Precoding via Partial BranchandBound
Based on the results in the Proposition, we propose the 1bit precoding design via PBB in this section, which essentially performs the BB process for part of the entries only, more specifically the entries in that do not comply with the 1bit requirement, as opposed to traditional BBbased schemes that perform BB process for all the entries in the transmit signal vector. This allows a considerable complexity reduction while still exhibiting a nearoptimal SER performance, as will be shown by the numerical results.
To begin with, we perform row rearrangements on to arrive at , such that it can be decomposed into
(21) 
where consists of that already satisfy the 1bit constraint and is fixed throughout the PBB procedure. consists of the residual entries in whose amplitudes are smaller than . Following the Proposition, we obtain , and . Similarly, we rearrange into such that , which is also decomposed into
(22) 
where and . The proposed PBB approach aims to further optimize with fixed, which leads to the following optimization problem:
(23)  
The subsequent BB procedure follows [27] and is omitted here due to the limited space, where we note that to guarantee a fast convergence speed, in the branching process we adopt the adaptive subdivision rule to choose the index of the entry in which is to be allocated a value in the current iteration. The corresponding index of the entry that is chosen should satisfy:
(24) 
where is the th entry in and is its corresponding index. This proposed 1bit precoding algorithm is termed ‘CI 1Bit PBB’.
4 Numerical Results
We present numerical results of the proposed 1bit precoding design in this section based on Monte Carlo simulations. In each figure, we define the transmit SNR as by assuming unit transmit power, and we compare our proposed PBB based scheme with both linear and nonlinear 1bit precoding designs in the literature.
In Fig. 2, we present the SER result for QPSK modulation in a MIMO system. Compared to existing 1bit precoding algorithms, our proposed 1bit precoding via PBB achieves a noticeable improvement in terms of the error rate and eliminates the error floor that are commonly observed for traditional 1bit precoding methods when the SNR becomes high. A similar trend is observed when we extend the modulation type to 8PSK, as depicted in Fig. 3 for a MIMO system. Both the above results validate the effectiveness of the proposed 1bit precoding approach via PBB.
5 Conclusion
In this paper, we have proposed a 1bit precoding approach via the PBB procedure, which significantly outperforms existing 1bit precoding schemes and is shown to achieve nearoptimal error rate performance. The proposed 1bit precoding scheme is built on the observation that most of the entries in the obtained transmit signal vector already satisfy the 1bit requirement by solving the relaxed 1bit precoding problem, and thus the BB process is only needed for the residual entries that do not comply with the 1bit constraint. The proposed 1bit precoding scheme also enables the BB framework to be applicable in largescale antenna arrays, which was not applicable due to the prohibitive complexity.
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