I Introduction
Massive multipleinput multipleoutput (MIMO) systems have been recognized as a promising technology for the fifth generation (5G) and future wireless communications [1]. With channel state information (CSI) at the base station (BS), fullydigital massive MIMO systems can greatly improve the spectral efficiency by employing simple linear precoding schemes, where it has been shown that zeroforcing (ZF)based precoding approaches are nearoptimal when favorable propagation conditions exist [1]. Nevertheless, with a fullydigital largescale antenna array at the BS, the corresponding large number of radio frequency (RF) chains and highresolution digitaltoanalog converters (DACs) lead to greatly increased hardware complexity, cost, and power consumption at the BS, which hinders the implementation of massive MIMO in practical wireless systems. One popular technique that achieves a performancecost compromise is to reduce the number of RF chains by employing hybrid structures [2][6], where the precoding is divided into the analog domain and digital domain. With a reduced number of RF chains, the hardware costs and power consumption at the BS can be reduced accordingly.
In addition to the hybrid structures, another promising technique, which is the focus of this paper, is to reduce the power consumption per RF chain by employing lowresolution DACs. It is known that the power consumption of DACs grows exponentially with the resolution and linearly with the bandwidth [7]. Therefore, employing lowresolution DACs, especially 1bit DACs, can greatly alleviate the hardware cost and the power consumption at the BS in the downlink. Due to the benefits that 1bit DACs can provide, in the recent literature there are an increasing number of contributions that propose 1bit transmit precoding designs for the massive MIMO downlink [8][17]. In [8], the 1bit quantization is directly applied to the linear ZF precoding and its performance is analytically studied. In [9][10], linear precoding schemes based on the minimummeansquared error (MMSE) criterion are proposed. However, the above quantized linear precoding methods are shown to suffer severe performance losses in the mediumtohigh signaltonoise ratio (SNR) regime due to the 1bit quantization.
To further improve the system performance in the presence of 1bit DACs, nonlinear precoding approaches that directly design the transmit signals based on the data symbols are further proposed in [11][16], where a symbol perturbation scheme for QPSK modulation is introduced in [11], [12] proposes a nonlinear precoding method based on the gradient descend method (GDM), while other nonlinear precoding methods are proposed in [13][14] based on biconvex relaxation. More computationally intensive nonlinear methods have been proposed in [15][17] that achieve better performance, but with a complexity that scales at least as the square of the number of antennas. While such approaches offer better performance compared with the lowcomplexity quantized linear precoding methods, their complexity may be too high for practical implementation.
To improve the performance of the more practical lowcomplexity quantized linear precoding methods, we propose a lowcomplexity refinement method based on the constructive interference (CI) formulation, which is applicable to any existing 1bit precoders for PSK modulation. By decomposing the data symbols and received signals along the corresponding detection thresholds, and using an appropriate coordinate transformation, we first construct a simple symbolscaling formulation as our performance metric based on CI. Based on this metric, we further introduce a refinement that improves the symbolscaling metric of existing quantized precoding schemes by modifying the transmit signal one antenna at a time. Our numerical results validate the effectiveness of the proposed refinement method on a number of existing quantized precoding approaches, and reveal the fact that the performance improvements are more significant for the quantized linear methods.
: , , and
denote scalar, vector and matrix, respectively.
, , and denote transposition, conjugate transposition, and trace of a matrix respectively. denotes the modulus of a complex number, and represents the set of complex matrices. and denote the real and imaginary part of a complex number, respectively.Ii System Model
We consider a multiuser massive MISO downlink with 1bit DACs employed at the BS, where the BS with transmit antennas communicates with singleantenna users simultaneously in the same timefrequency resource and , as depicted in Fig. 1. We focus on transmit precoding designs with 1bit DACs, and ideal analogtodigital converters (ADCs) with infinite precision are assumed for each receiver. Following the closelyrelated literature [8][12], the data symbols are assumed to be from a normalized PSK modulation, denoted as . We denote the quantized transmit signal vector as , and can be expressed as
(1) 
In (1), denotes a generic precoding strategy, where for a linear precoding scheme is equivalent to multiplication by a precoding matrix, while for nonlinear precoding approaches denotes the mapping strategy to form the unquantized signals based on . denotes the elementwise 1bit quantization operation for both the real and imaginary part, and in this paper we normalize each entry in as
(2) 
where . The above normalization guarantees that . Subsequently, we express the received signal for the th user as
(3)  
where denotes the flatfading Rayleigh fading channel vector for user
with each entry following a standard Gaussian distribution,
is the additive Gaussian noise at the receiver with zero mean and variance
, and denotes the total available power per antenna, where for simplicity we have assumed a uniform power allocation.Iii Proposed Refinement Method based on CI
In this section, the symbollevel performance metric based on constructive interference is first introduced, followed by the proposed refinement process.
Iiia Constructive Interference
We define CI as interference that pushes the received symbols away from the detection thresholds of the modulation constellation [18][20]. The exploitation of CI was first introduced in [18] and further extended in [20] with the introduction of the constructive region. These papers showed that, as long as the resulting interfered signals are located in the constructive region, the distance to the decision thresholds is increased, and the introduction of the constructive region brings additional performance gains [20]. While we focus on PSK modulation in this paper, the extension to QAM modulations are applicable, and we refer the interested readers to [21][22] for a detailed description.
IiiB The SymbolScaling Performance Metric
Based on the above description for CI, we propose to consider an alternative symbolscaling metric for the 1bit quantized massive MIMO downlink based on the CI formulation, which characterizes the effect of the interference on each data symbol in a simple way. To be specific, for a generic PSK modulation, we first express each data symbol as
(4) 
where denotes the th constellation point which starts from and follows an anticlockwise direction, as depicted in Fig. 2 where we employ 8PSK as an example. We then decompose each data symbol along the corresponding detection thresholds, where without loss of generality we assume , which leads to
(5) 
Based on the geometry, we can obtain the following expression for the decomposed signal and
(6)  
where and denote the coordinates of the bases and in the realimaginary plane, respectively, and is a scaling factor to ensure the equality of (5):
(7) 
For example, for the constellation point considered in Fig. 2, we obtain the expressions
(8)  
Subsequently, we decompose the noiseless received signal of each user along the two detection thresholds of :
(9) 
where
(10) 
are real scaling factors for and , respectively. A larger value for or represents a larger distance to the detection threshold, and accordingly the performance of 1bit quantized MIMO systems is dominated by the minimum value of and . Based on this fact, we employ the minimum value in the real scaling factors and as the performance metric.
IiiC Proposed Refinement Method
Before we introduce the refinement method, based on (6) and (9), we first obtain and as a function of the quantized transmit signal vector . We expand (9) by its real and imaginary part, expressed as
(11)  
where for simplicity we have employed the following notations
(12) 
Accordingly, we obtain [23]
(13)  
By introducing
(14)  
we simplify (13) into
(15) 
By further introducing
(16) 
and
(17)  
(15) can be further expressed in a compact form as
(18) 
where is given by
(19) 
In (18), we denote as the th entry in and we omit the notations and . Accordingly, the symbolscaling performance metric is the minimum value of in , which we aim to improve via the proposed refinement process.
In the following we introduce the proposed refinement method based on the above formulation, which can be applied to any existing precoding methods for the 1bit massive MIMO downlink to further improve performance. The central idea of the refinement scheme aims to improve the minimum value of by modifying the transmit signals on the antennas. If we consider all the possible signal combinations, the refinement is equivalent to an exhaustive search, which is too computationally costly to implement. Therefore, to consider a more practical approach and keep the computational cost as low as possible, we propose to only modify the transmit signal on one antenna at a time. To be specific, we first calculate the scaling vector based on obtained from some existing algorithm via (5)(18), and further obtain the minimum value of in , which is denoted as . Subsequently, we perform an iterative method, where within the th iteration, we change the sign of an entry in , and calculate the updated scaling vector . If the minimum value of in the updated scaling vector, which is denoted as , is larger than , we update accordingly. In the case that the minimum value of becomes smaller, we keep the th entry in unaltered and move to the next iteration. While the above approach is not guaranteed to converge to the global optimum, it will be shown that the performance improvement is indeed significant, especially for the quantized linear ZF methods.
Based on the above description, we summarize the refinement algorithm in Algorithm 1.
Iv Numerical Results
To evaluate the performance of the proposed symbol scaling approach, in this section we present numerical results in terms of the bit error rate (BER) based on Monte Carlo simulations, where in each plot the transmit SNR is defined as . Both QPSK and 8PSK modulations are considered in the simulations. The refinement method is simulated together with the following existing 1bit quantized precoding methods:

‘ZFUnquantized’: the conventional ZF precoding with infiniteprecision DACs (only as reference);

‘SP (R)’: the 1bit symbol perturbation precoding technique in [11] for QPSK;

‘GDM (R)’: GDMbased 1bit precoding method in [12];

‘Constructive (R)’: the 1bit precoding method based on CI in [23].
The inclusion of ‘R’ in the above abbreviations denotes the cases where the proposed refinement method is applied to the original 1bit precoding techniques.
In Fig. 3, we present numerical results for a smallscale MIMO system with transmit antennas and users. When the proposed refinement method is not applied, we observe that quantized nonlinear precoding approaches generally offer a better BER performance than the quantized linear methods, and the quantized linear ZF scheme has the worst BER performance. Specifically, an error floor is observed for all existing precoding approaches in the high SNR regime, which is due to the 1bit quantization. When the refinement process is applied to the existing algorithms, we observe significant improvements in the BER performance for all techniques, and the error floor vanishes for most of the approaches. Specifically, the proposed refinement process offers the highest performance improvement for the lowcomplexity quantized linear ZF method.
In the following, we show the numerical results for massive MIMO systems. In Fig. 4, we present the BER performance for QPSK modulation with transmit antennas and users. An improved BER performance is observed compared to the case of smallscale MIMO systems in Fig. 3, due to the increase in the ratio . Again, when the refinement method is not introduced, the lowcomplexity quantized linear ZF scheme achieves the worst BER performance and an error floor is observed. The more complicated nonlinear methods achieve significantly better BER performance. When the refinement process is further introduced, the BER performance of all techniques is further improved, and the scheme with the worst BER performance improves the most. Specifically, we observe that, with refinement, the quantized ZF scheme achieves a comparable performance to the nonlinear quantized precoding schemes, where an SNR loss of only 1.5dB is observed compared to the best BER performance achieved by ‘Constructive, R’.
In Fig. 5, the BER performance with 8PSK modulation for a massive MIMO system with transmit antennas and users is depicted. A similar trend to Fig. 3 and Fig. 4 is observed, where the quantized linear ZF method without refinement achieves the worst BER performance, while the refinement process offers the best performance improvement for the quantized ZF scheme. Specifically, there is an SNR loss of only 1dB for ‘ZF 1Bit, R’ compared to the approach ‘Constructive, R’ that returns the best BER performance.
Considering the complexity, the quantized linear ZF method with the refinement process achieves the best performancecomplexity tradeoff, and is therefore the most promising technique in a practical 1bit massive MIMO system.
V Conclusion
In this paper, a lowcomplexity refinement method is proposed for the massive MIMO downlink with 1bit DACs, which is applicable to any existing quantized precoding methods with PSK modulations. By first formulating the symbolscaling performance metric based on constructive interference, the refinement process modifies the transmit signal on one antenna at a time and further improves this performance metric with a low computational cost. Numerical results have shown that the refinement offers additional performance improvements for existing schemes, especially for the lowcomplexity quantized linear ZF method.
Acknowledgment
This work was supported by the Royal Academy of Engineering, U.K., the Engineering and Physical Sciences Research Council (EPSRC) project EP/M014150/1, the China Scholarship Council (CSC), and the U.S. National Science Foundation under grant CCF1703635.
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