I Introduction
MUMIMO is one of the key techniques in the future communication system, in which the base station (BS) is equipped with large number of antennas [1]. Scaling up the number of transmit antennas brings up numerous advantages: enhanced system throughput, improved radiated energy efficiency and simplified algorithms of signal processing [2]. On the other hand, gigantic increase in energy consumption, a key challenge to the use of massive MIMO, is brought by the increasing antenna quantity.
Recently, coarsely quantized MUMIMO has been extensively investigated and recognized as an effective way to decrease the energy cost [3]. The authors in [4] show that, compared with massive MIMO systems with ideal DACs, the sum rate loss in 1bit massive MUMIMO systems can be compensated by disposing approximately times more antennas at the BS. Besides, the recently advanced nonlinear precoding algorithms, such as the semidefinite relaxation based precoder [5], the finitealphabet (FA) precoder [6] and the alternating direction method of multipliers (ADMM) based precoder [7], can improve the bit error rate (BER) performance. Generally speaking, equipping the BS with lowresolution DACs can greatly reduce the power consumption without introducing significant performance penalties by assuming perfect knowledge of CSI.
In practical situations, however, the CSI available to the BS is imperfect due to limited feedback and finiteenergy training, resulting in receivers performance degradation. Therefore, robust precoding design that takes into consideration the channel uncertainties is of great importance.
In this paper, a random matrix with finite secondorder statistics is used to model the imperfection in CSI. We formulate the robust quantized precoding problem as minimization of inter user interference subject to two constraints: a discrete set (outputs of multibit DACs) constraint and a bounded CSI error constraint. By exploiting the overlap between the multibit DACs outputs and the singlebit ones, we reformulate the original multivariable discrete optimization problem into a binary optimization problem. Furthermore, based on the observation that the bounded CSI error constraint involves a quadratic form of some random variables, using the Sprocedure lemma
[8], the bounded CSI error constraint is recast into a tractable formulation with convex constraints. In this manner, we obtain the relaxed version of the original precoding problem, which is finally tackled by the standard semidefinite relaxation (SDR) method [9]. Lastly, the effectiveness of the proposed precoding algorithm is verified by numerical simulations.Notations: Throughout this paper, vectors and matrices are given in lower and uppercase boldface letters, e.g.,
and , respectively. We use to denote the element at the th row and th column. The symbols , , , , and denote the expectation operator, the trace operator, the columnwise vectorization, the transpose, the conjugate transpose of , respectively. For a vector , , , and are respectively used to represent the real part, the imaginary part and norm of . The symbols andare respectively referred to an identity matrix and a zeros matrix with proper size.
Ii System Model
We consider a singlecell MUMIMO downlink system of one BS serving singleantenna user terminals (UTs). We assume the BS is equipped with transmit antennas and ignore the RF impairments. Besides, we follow [10] for the model of realtime downlink channel through a GaussMarkov uncertainty of the form
(1) 
where is the imperfect observation of the channel available to the BS, and denotes error matrix whose elements are independently sampled from a bounded set:
(2) 
The parameter indicates the degree of uncertainty of the related channel measurement . Specifically, means perfect CSI, the values of correspond to partial CSI and accounts for no CSI.
Iia InputOutput Relationship for MUMIMO Downlink
Let be the dimensional UTs intended constellation points. With the knowledge of CSI , the BS precodes into a dimensional vector , satisfying the average power constraints . Assuming perfect synchronization, the received signal at UEs can be expressed as
(3) 
where is a complex vector with element being complex addictive Gaussian noise distributed as .
IiB Quantization
Let be the precoding matrix and the precoded vector of the unquantized system. For the quantized MUMIMO downlink system, each precoded signal component is quantized separately into a finite set of prescribed labels by a bit symmetric uniform quantizer . It is assumed that the real and imaginary parts of precoded signals are quantized separately. The resulting quantized signals read
(4) 
Specially, the realvalued quantizer maps realvalued input (real parts or imaginary parts of the precoded signals) to a set of labels , which are determined by the set of thresholds , such that . For a bit DAC with step size , the thresholds and quantization labels (outputs) are respectively given by
(5) 
and
(6) 
In the case of 1 bit DACs, the output set reduces to . For any output drawn from bit uniform quantifier, can be represented by
(7) 
where , , and are constant coefficients satisfying The relation in (7) indicates that the multiple DACs outputs can be represented by the linear combination of several independent singlebit DACs outputs.
Iii Robust Quantized Precoding Design
Iiia Problem Formulation
Similar to [6], we concentrate on a performance metric that minimizes inter user interference,
under the average power constraint. is the unknown precoding factor taking into consideration the power constraint. Specially, in the case of channel uncertainties, the quantized precoding problem can be formulated as
(8) 
where . The complexvalued optimization problem in (8) can be equivalently rewritten as a realvalued problem:
(9) 
where, with a slight abuse of notations, we define
The nonlinear precoding problem in (9) can be tackled by the naive exhaustive search with complexity of order . The unendurable computational complexities of these methods impede their application in massive MUMIMO.
IiiB Robust Precoding Design
In order to simultaneously achieve robust and efficient precoding, we reformulate the problem in (9) as a binary optimization problem by leveraging the favorable relation shown in (7). Define an auxiliary matrix and denote
(10) 
(11) 
it follows from (7) that
(12) 
Specially, we have
(13) 
and
(14) 
for the 2bit case and the 3bit case ^{1}^{1}1For the Bbit case, we can have Here, we only employ 13 bit DACs as the performance gap between MUMIMO system with ideal DACs and the one with Bbit DACs is negligible, which would be confirmed in Section IV., respectively. With expressions in (10) and (12), one can rewrite the precoding problem in (9) in the following equivalent form:
(15) 
where , for .
By introducing a slack variable , we can equivalently rewrite the problem in (15) as
(16a)  
(16b)  
(16c)  
(16d)  
(16e)  
(16f) 
where
and means is a nonnegative definite matrix.
In the sequel, by applying the SProcedure lemma, we rewrite the constraints in (2) and (16b) that involve quadratic inequalities in error vectors in the linear matrix inequality constraints as stated in the following.
Theorem III.1.
Given the bounded channel uncertainty in (2), the condition in (16b) can be relaxed to
(17a)  
(17b) 
where is an auxiliary matrix and is a slack variable.
Proof.
: We start by restating the SProcedure lemma.
Lemma 1 (SProcedure lemma, [8]).
Let , for , be defined as
where , and . Suppose there exists an with . Then, for any , the equalities
hold if and only if there exists such that
With Theorem III.1, the optimization problem in (15) can be reformulated as
(24) 
The precoding problem in (24) is still NPhard due to the nonconvex rank constraint in (16f). We follow the standard strategy [10] that relaxes the problem in (24) by omitting the nonconvex constraint (16f). Then the remaining convex problem in (24) can be efficiently solved by numerical algorithms, e.g., the interior point method [13]. Finally, let denote the solution of (24), using the rounding strategy as described in [9], we can obtain the desired realvalued precoded vector by
(25) 
Then, the complexvalued precoded vector can be determined as
(26) 
We refer to the proposed method as RSDR. The complexity analysis of RSDR is presented in the following.
IiiC Complexity Analysis
The quantized precoding problem (8) can be directly tackled by the naive exhaustive search (NES) [14] with complexity of order or by the sphere decoding (SD) method [15, 16] of exponential complexity. In this paper, the original quantized precoding problem is recast into a tractable formulation (24) and finally solved by the interior point method.
Since the proposed RSDR has a similar form to the standard SDP in [17], we can borrow the analytical result therein. According to [17], the worstcase complexity of solving the SDP is , where is the number of constraints in (24) and is the solution precision. In practice, it has been shown in [17] that scales more typically with , and hence the number of operations is bounded by .
Iv Simulations
Iva System setup
This section evaluates the performance of the proposed precoder via numerical simulations in comparison with the ADMM precoder and the FA precoder. Each provided result is an average over 1000 independent runs.
We use the Lloyd quantization method [18] to determine the quantization outputs (6) and the least square regression method [19] to calculate the coefficients in (13) and (14). Specially, for the 2bit case, we have . For the 3bit case, we have and . We assume that the entries of
are zeromean Gaussian distributed, i.e.,
. The Water Filling (WF) precoder [20] with infiniteresolution DACs and perfect CSI is regarded as the benchmark.IvB Performance Evaluation
In Fig. 1, we consider a 128antenna MUMIMO system with 10 UTs. The CSIrelated parameter is . It can be observed that the performance of all the compared precoders, in terms of BER, improve as a result of a increased bits of DACs.
One can also observe from Fig. 1 that the proposed RSDR precoder outperforms the ADMM precoder and the FA precoder. For instance, with same average transmit power and 3bit DACs, the proposed precoding scheme can attain 3 dB of target BER whenever , whereas the FA precoder and the ADMM precoder respectively require higher SNR conditions ( and ), to support 3 dB of BER target at the same CSI condition. Besides, the performance gap between the ideal ZF precoder (infinite DACs) with perfect CSI and the proposed precoder with 3bit DACs is negligible, indicating the robustness of the the proposed precoding scheme.
In Fig. 2, we investigate the performance of the proposed precoder for different modulation schemes, QPSK, 8PSK and 16QAM. The BS is equipped with 128 antennas and serves 16 UTs with different channel uncertainties, e.g., . It can be seen that the proposed RSDR precoder shows robustness for all the considered modulation schemes under channel uncertainties. Specially, compared to the ideal case (), the performance gaps are about 0.2 dB, 1 dB, and 3 dB, for QPSK, 8PSK and 16QAM with a target BER of , respectively.
Fig. 3 compares the average execution time (in seconds) of RSDR in the case of different channel conditions. Compared to the SDR precoder [5] that has perfect access to CSI, the proposed RSDR precoder only incurs two added constraints. The algorithms are performed on a desktop PC with an Intel Core I77700K CPU at 4.2 GHz with 32 GB RAM. It can be seen that the average execution time of RSDR increases as the number of the BS antennas increases. Compared to the perfect CSI case, the RSDR can obtain robust performance at the expense of slightly more numerical computations, indicating the effectiveness of the new precoder.
In summary, the proposed RSDR enables robust precoding against channel uncertainty in polynomial time [17]. We note here that the efficiency of the proposed precoder needs to be improved to meet the demand in 5G or 6G communication systems, in which the BS is foreseen to be equipped with hundreds, even thousands of antennas.
V Conclusions
In this paper, we have investigated the quantized precoding problem for the MUMIMO in the presence of imperfect CSI. By exploiting the bridging relation between the multibit DACs outputs and the singlebit ones, and taking advantaging of the Sprocedure lemma, the original precoding problem is reformulated into a tractable formulation and solved by the SDR method. Simulation results have verified that the proposed precoder is robust to various channel uncertainties and can support the MUMIMO system with higherorder modulations.
Vi Acknowledgements
The authors would like to thank for technical experts: Tian Pan, Guangyi Yang, Yingzhe Li,and Rui Gong, all from Huawei Technologies for their fruitful discussions.
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