Construction of One-Bit Transmit-Signal Vectors for Downlink MU-MISO Systems with PSK Signaling

Transcription

1 1 Construction of One-Bit Transit-Signal Vectors for Downlink MU-MISO Systes with PSK Signaling Gyu-Jeong Park and Song-Na Hong Ajou University, Suwon, Korea, eail: {net2616, and quantized iniu-ean squared error (QMMSE) [12], which siply applied the one-bit quantization to the outputs of the conventional linear precoding ethods. Also, a branchand-bound and a biconvex relaxation approaches were presented in [13] and [14], respectively. However, these ethods either suffer fro a severe perforance loss or require an expensive coputational coplexity (see [15] for ore details). Very recently, focusing on phase-shift-keying (PSK) constellations, a low-coplexity sybol-scaling ethod was proposed in [15], where it optiizes a transit-signal vector directly in an efficient sequential fashion as a function of an instantaneous channel atrix and users essages (called sybol-level operation). Also, it was shown that the sybolscaling ethod can yield a coparable perforance to the previous non-linear precoding ethods with a uch lower coputational coplexity. Our contributions: In this paper, we study a downlink MU-MISO syste with one-bit DACs. We first identify that antenna-selection can yield a non-trivial bit-error-rate (BER) perforance gain by alleviating error-floor probles (see Fig. 1). Clearly, this perforance gain is because antennaselection can enlarge the set of possible transit-signal vectors (i.e., search-space) copared with the previous precoding ethods in [10] [15]. However, as in conventional one-bit precoding optiization, finding an optial transit-signal vecarxiv: v1 [cs.it] 23 Jan 2019 Abstract We study a downlink ulti-user ultiple-input single-output (MU-MISO) syste in which the base station (BS) has a large nuber of antennas with cost-effective one-bit digitalto-analog converters (DACs). In this syste, we first identify that antenna-selection can yield a non-trivial sybol-error-rate (SER) perforance gain by alleviating an error-floor proble. Likewise the previous works on one-bit precoding, finding an optial transit-signal vector (encopassing precoding and antennaselection) requires exhaustive-search due to its cobinatorial nature. Motivated by this, we propose a low-coplexity twostage algorith to directly obtain such transit-signal vector. In the first stage, we obtain a feasible transit-signal vector via iterative-hard-thresholding algorith where the resulting vector ensures that each user s noiseless observation is belong to a desired decision region. In the second stage, a bit-flipping algorith is eployed to refine the feasible vector so that each user s received signal is ore robust to additive Gaussian noises. Via siulation results, we deonstrate that the proposed ethod can yield a ore elegant perforance-coplexity tradeoff than the existing one-bit precoding ethods. Index Ters Massive MIMO, one-bit DAC, precoding, antenna-selection, beaforing. I. INTRODUCTION Massive ultiple-input ultiple-output (MIMO) is one of the proising techniques to cope with the predicted wireless data traffic explosion [1] [4]. In downlink assive MIMO systes, it was shown that low-coplexity linear precoding ethods as zero-forcing (ZF) and regularized ZF (RZF) achieve an alost optial perforance [5]. In contrast, the use of a large nuber of antennas considerably increases the hardware cost and the radio-frequency (RF) circuit consuption [6]. Hybrid analog-digital precoding (a.k.a., hybrid precoding) is one of the proising ethods to address the above probles since it can reduce the nuber of RF chains [7] [9]. An alternative ethod is to ake use of low-resolution digital-to-analog converters (DACs) (e.g., 1 3 bits), which can greatly reduce the cost and power consuption per RF chain [6]. Specifically, in this case, each antenna s transit sybol is equivalent to a constant-envelop sybol, which enables the use of low-cost power aplifiers and thus reduces the hardware coplexity. Because of the potential erits of using one-bit DACs, there have been nuerous studies on the precoding ethods for such downlink assive MIMO systes [10] [15]. In [10], various non-linear precoding techniques were proposed based on seidefinite relaxation (SDR), squared l -nor relaxation, and sphere decoding. The authors proposed low-coplexity quantized precoding ethods as quantized ZF (QZF) [11] SER Exhaustive Search (with Antenna-Selection) Exhaustive Search (without Antenna-Selection) SNR [db] Fig. 1. N t = 8 and K = 2. Perforance iproveent of antenna-selection in one-bit precoding for downlink MU-MIMO systes with one-bit DACs.

2 2 tor (encopassing precoding and antenna-selection) requires exhaustive-search due to its cobinatorial nature. Motivated by this, we propose a low-coplexity algorith to solve the above proble (i.e., joint optiization of precoding and antenna-selection), which consists of the following two stages. In the first stage, we obtain a feasible transit-signal vector via iterative-hard-thresholding (IHT) algorith where the resulting vector guarantees that each user s noiseless observation is belong to a desired decision region. Naely, it can iprove the BER perforances at high-snr regies (i.e., error-floor regions) by lowering an error-floor. In the second stage, we refine the above transit-signal vector using a bit-flipping (BF) algorith so that each user s received signal is ore robust to additive Gaussian noises. In other words, it can iprove the BER perforances at low-snr regies (i.e., waterfall regions). Finally, we provide siulation results to deonstrate that the proposed ethod can iprove the perforance of the existing sybol-scaling ethod in [15] with a coparable coputational coplexity. The rest of paper is organized as follows. In Section II, we provide soe useful notations and describe the syste odel. In Section III, we propose a low-coplexity algorith to optiize a transit-signal vector directly for downlink MU-MISO systes with one-bit DACs. Siulation results are provided in IV. Section V concludes the paper. II. PRELIMINARIES In this section, we provide soe useful notations and describe the syste odel. A. Notations The lowercase and uppercase boldface letters represent colun vectors and atrices, respectively, and ( ) T denotes the conjugate transpose of a vector or atrix. For any vector x, x i represents the i-th entry of x. Let [a : b] = {a, a + 1,..., b} for non-negative integers a and b with a < b. Siilarly, let [b] = {1,..., b} for any positive integer b. Re(a) and I(a) represent its real and coplex part of a coplex vector a, respectively. Also, we define a natural apping g( ) which aps a coplex value into a real-valued vector, i.e., for each x C, we have that g(x) = [Re(x), I(x)] T. (1) The inverse apping of g is denoted as g 1. If g or g 1 is applied to a vector or a set, we assue they operate eleentwise. For exaple, we have that g([x 1, x 2 ] T ) = [Re(x 1 ), I(x 1 ), Re(x 2 ), I(x 2 )] T. (2) Also, for any coplex-value x C, we define a real-valued atrix expansion φ(x) as [ ] Re(x) I(x) φ(x) =. (3) I(x) Re(x) Finally, we let R(θ) denote a rotation atrix with a paraeter θ as [ ] cos θ sin θ R(θ) =, sin θ cos θ which rotates the following colun vector in the counterclockwise through an angle θ about the origin. R & R % c & c % R $ c $ iaginary axis b #,$ R # c # c ' c ( c ) b #,# c " R ' R ( R ) R " real axis Fig PSK constellation in coplex doain and the corresponding decision regions. R i represents the decision region of c i M. B. Syste Model We consider a single-cell downlink MU-MISO syste in which one BS with N t antennas counicates with singleantenna K N t users siultaneously in the sae tiefrequency resources. Focusing on the ipact of one-bit DACs in the transit-side operations, it is assued that the BS is equipped with one-bit DACs while each user (receiver) is with ideal analog-to-digital converters (ADCs) with infinite resolution. As in the closely related works [11], [12], [15], we also consider a noralized -PSK constellation C = {c 0, c 1,..., c 1 }, each of which constellation point is defined as c i = cos (2πi/) + j sin (2πi/) C. (4) The constellation points of 8-PSK are depicted in Fig. 2. Let µ k [0 : 1] be the user k s essage for k [K] and also let x = [ x 1,..., x Nt ] T be a transit-signal vector at the BS. Under the use of one-bit DACs and antenna-selection, each x i can be chosen with the restriction of Re(x i ) and I(x i ) { 1, 0, 1}. (5) This shows that, copared with the related works [11], [12], [15], antenna-selection can enlarge the set of possible sybols (i.e., search-space) per real (or iaginary) part of each antenna. Then, the received signal vector at the K users is given by y = ρhx + z, (6) where H C K Nt denotes the flat-fading Rayleigh channel with each entry following a coplex Gaussian distribution and z C K 1 denotes the additive Gaussian noise vector whose eleents are distributed as circularly syetric coplex Gaussian rando variables with zero-ean and unitvariance, i.e., z i CN (0, 1). Also, ρ is chosen according to the per-antenna power constraint. For siplicity, we assue the unifor power allocation for the antenna array. In this syste, our purpose is to develop a low-coplexity algorith to optiize a transit-signal vector x (i.e., joint

3 3 optiization of precoding and antenna-selection) with the assuption that the BS is aware of a perfect channel state inforation (CSI), which will be provided in Section III. Before explaining our ain result, we provide the following definition which will be used throughout the paper. Definition 1: (Decision Regions) For each c i C, a decision region R i is defined as } R i = {y C : y c i in y c j. (7) j [0: 1]:j i If the user k receives a y k R µk, then it decides the decoded essage as µ k [0 : 1]. III. THE PROPOSED TRANSMIT-SIGNAL VECTORS In this section, we derive a atheatical forulation to optiize a transit-signal vector (encopassing precoding and antenna selection) for downlink MU-MISO systes with one-bit DACs, and then present a low-coplexity algorith to solve such proble efficiently. For the ease of explanation, we first introduce the equivalent real-valued representation of the coplex input-output relationship in (6), which is given by ỹ = ρ H x + z, (8) where ỹ = g(y), x = g(x), z = g(z), and H denotes the 2K 2N t real-valued atrix which is obtained by replacing h i,j (e.g., the (i, j)-th entry of H) with the 2 2 atrix φ(h i,j ) for all i, j. For the resulting odel, we can define the realvalued constellation C = {g(c 0 ), g(c 1 ),..., g(c 1 )} where g(c i ) = [cos(2πi/), sin(2πi/)] T. (9) Fro Definition 1, the decision region in R 2 for each g(c i ) is siply obtained as R i = g(r i ). (10) Furtherore, as shown in Fig. 2, each R i can be represented as linear cobination of two basis vectors s i,1 and s i,2 as R i = {αi,1 s i,1 + α i,2 s i,2 : α i,1, α i,2 > 0}, (11) where the basis vectors are easily obtained using rotation atrices such as ( ) ( ) [ ( π( 1) l cos 2πi )] s i,l = R sin ( ) 2πi = cos π(2i+( 1) l ) ( ) sin π(2i+( 1), l ) for l = 1, 2. Using two basis vectors, we define the 2 2 [ ] real-valued atrix S i = si,1 s i,2. Since Si has full-rank, the inverse atrix of S exists and is easily coputed as ( ) ( ) S 1 1 i = sin π(2i+1) cos π(2i+1) ( ) ( ) sin(2π/) sin π(2i 1) cos π(2i 1). (12) We are now ready to explain how to optiize a transitsignal vector x efficiently. For siplicity, we let r = Hx C 2K 1 denote the noiseless received vector at the K users. Regarding our optiization proble, we first provide the following key observations: Fig. 3. α " s " (&) R " * α "," s "," +α ",) s ",) α "," s "," +α ",) s ",) The decision region R 1 in R 2 and the ipact of larger coefficients. (Feasibility condition) To ensure that all the K users recover their own essages, a transit-signal vector x should be constructed such that r k R µk (equivalently, g(r k ) R µk ), (13) for k [K]. Accordingly, g(r k ) should be represented as g(r k ) = α k,1 s µk,1 + α k,2 s µk,2, (14) for soe positive coefficients α k,1 and α k,2. This is called feasibility condition and a vector x to satisfy this condition called feasible transit-signal vector. (Noise robusteness) The condition in (13) cannot guarantee good perforances in practical SNR regies (e.g., waterfall regions) due to the ipact of additive Gaussian noises. Thus, we need to refine the above feasible transit-signal vector so that α k,1 and α k,2 are axiized. We will forulate an optiization proble atheatically which can find a transit-signal vector x to satisfy the above requireents. Fro (14), we can express the feasibility condition in a atrix for: g(r) = H x = Sα, (15) where α = [α 1,1, α 1,2,, α K,1, α K,2 ] T and S = diag(s µ1,, S µk ) denotes the block diagonal atrix having the i-th diagonal block S µi. Fro the block diagonal structure and (12), we can easily obtain the inverse atrix of S as S 1 = diag(s 1 µ 1,, S 1 µ K ). (16) Then, the feasibility condition in (15) can be rewritten as α = Λ x. (17) where Λ = S 1 H R 2K 2N t. Note that Λ is a known atrix since it is copletely deterined fro the channel atrix H and users essages (µ 1,..., µ K ). Taking the feasibility condition and noise robustness into account, our optiization proble can forulated as ax k,i : k [K], i = 1, 2} x (18) subject to α = Λ x (19) α k,1, α k,2 > 0, k [K] (20) x { 1, 0, 1} 2Nt. (21)

4 4 For the above optiization proble, the objective function ais to axiize the iniu of positive coefficients α k,i s. This is otivated by the fact that a larger value of the coefficients α k,i s yields a larger distance to the other decision regions. As an exaple, consider the decision region R 1 as illustrated in Fig. 3. Obviously, α 1,1 s 1,1 +α 1,2s 1,2 has a larger distance fro the boundary 1 than α 1,1 s 1,1 + α 1,2 s 1,2 while both have the sae distance fro the boundary 2. Likewise, if α 1,1 increases for a fixed α 1,2, then the distance fro the boundary 2 increases by keeping the distance fro the boundary 1. In order to increase the distance fro the other decision regions, therefore, it would ake sense to axiize the iniu of two coefficients α 1,1 and α 1,2. By extending this to all the K users, we can obtain the objective function in (18). Unfortunately, finding an optial solution to the above optiization proble is too coplicated due to its cobinatorial nature. We thus propose a low-coplexity two-stage algorith to solve it efficiently, which is described as follows. In the first stage, we find a feasible solution x f to satisfy the constraints (19)-(21), which is obtained by taking a solution of 1 = sign(λ x), (22) where sign(a) = 1 if a 0 and sign(a) = 1 otherwise. We solve the above non-linear inverse proble efficiently via the so-called IHT algorith. The detailed procedures are provided in Algorith 1 where the thresholding function [ ] is defined as [a] = 1 if a >, [a] = 1 if a, and [a] = 0, otherwise. In the second stage, we refine the above feasible solution by axiizing in{α k,1, α k,2 : k [K]}, which is efficiently perfored using BF algorith (see Algorith 2). Finally, fro the output of Algorith 2 (e.g., x), we can obtain the proposed transit-signal vector as x = g 1 ( x). Coputational Coplexity: Following the coplexity analysis in [15], we study the coputational coplexity of the proposed ethod with respect to the nuber of realvalued ultiplications. As benchark ethods, we consider the coputational costs of exhaustive search, exhaustive search with antenna-selection, and sybol-scaling ethod in [15], which are denoted by χ E, χ AS, and χ S, respectively. Fro [15], their coplexities are obtained as χ E = 4KN t 2 2Nt (23) χ AS = 4KN t 3 2Nt (24) χ S = 4N 2 t + 24KN t 2K. (25) Recall that the proposed ethod is coposed of the two algoriths in Algoriths 1 and 2. First, the coplexity of Algorith 1 is obtained as t 8KN t where t t ax denotes the total nuber of iterations. Also, the coplexity of Algorith 2, which is siilar to that of refined-state in sybol-scaling ethod in [15], is obtained as 8KN t. By suing the, the overall coputational coplexity of the proposed ethod is obtained as χ P = 8(t + 1)KN t. (26) Algorith 1 IHT Algorith 1: input: Λ R 2K 2Nt,, and t ax 2: initialization: x (0) = 0 3: iteration: repeat until either e (t) = 0 or t = t ax 4: output: x f x (t) Algorith 2 BF Algorith e (t) = 1 sign(λ[ x (t) ] ) x (t+1) = x (t) + Λ T e (t) 1: input: Λ R 2K 2Nt and x f R 2Nt 1 2: initialization: x = x f 3: for i = 1 : 2N t do 4: for j {1, 0, 1} do 5: β (j) = Λ x {i=j} and β (j) in = in i [2K]{β (j) i } 6: end for 7: update x i arg ax j {1,0, 1} {β (j) in } 8: end for 9: output: x R 2Nt 1 Note that for soe j {1, 0, 1}, x {i=j} is obtained fro the x by siply replacing x i with j, i.e., x {i=j} = [ x 1,..., x i 1, j, x i+1,..., x 2Nt ] T. IV. SIMULATION RESULTS In this section, we evaluate the sybol-error-rate (SER) perforances of the proposed ethod for the downlink MU- MIMO systes with one-bit DACs where N t = 128 and K = 16. For coparison, we consider QZF in [11] and the state-of-the-art sybol-scaling ethod in [15] because the forer is usually assued to be the baseline approach and the latter showed an elegant perforance-coplexity tradeoff over the other existing ethods (see [15] for ore details). Both QPSK (or 4-PSK) and 8-PSK are considered. Regarding the proposed ethod, we choose the threshold paraeter = 3 for Algorith 1, which is optiized nuerically via Monte- Carlo siulation. As entioned in Section II-B, a flat-fading Rayleigh channel is assued. Fig. 4 shows the perforance coparison of various precoding ethods when QPSK is eployed. Fro this, we observe that QZF suffers fro a serious error-floor and thus is not able to yield a satisfactory perforance. In contrast, both sybolscaling and the proposed ethods overcoe the error-floor proble, thereby outperforing QZF significantly. Moreover, it is observed that the proposed ethod can slightly iprove the perforance of QZF with uch less coputational cost (e.g., 70% coplexity reduction). Fig. 5 shows the perforance coparison of various precoding ethods when 8-PSK is eployed. It is observed that in this case, an error-floor proble is ore severe than before, which is obvious since the decision regions of 8-PSK is ore sophisticated than those of QPSK. Hence, antenna-selection can attain a ore perforance gain with a larger search-space. Accordingly, the proposed ethod can further iprove the per-

5 5 QZF [11] Sybol-Scaling [15] Proposed ethod QZF [11] Sybol-Scaling [15] Proposed ethod (6 iterations) Proposed ethod (12 iterations) Sybol Error Rate (SER) Sybol Error Rate (SER) PSK SNR [db] 8-PSK SNR Fig. 4. N t = 128, K = 16 and QPSK. Perforance coparison of QZF, sybol-scaling, and the proposed ethods for the downlink MU-MISO systes with one-bit DACs. The coputational coplexities are coputed as χ S = and χ P = Fig. 5. N t = 128, K = 16 and 8-PSK. Perforance coparison of QZF, sybol-scaling, and the proposed ethods for the downlink MU-MISO systes with one-bit DACs. The coputational coplexities are coputed as χ S = , χ P = for t = 12, and χ P = for t = 6. forance of sybol-scaling ethod by lowering an error-floor, which is verified in Fig. 5. To be specific, the proposed ethod with t = 6 outperfors the sybol-scaling ethod with an alost sae coputational cost. Furtherore, at the expense of two ties coputational cost, the proposed ethod with t can address the error-floor proble copletely. Therefore, the proposed ethod provides a ore elegant perforancecoplexity tradeoff than the state-of-the-art sybol-scaling ethod. V. CONCLUSION In this paper, we showed that the use of antenna-selection can yield a non-trivial perforance gain especially at errorfloor regions, by enlarging the set of possible transit-signal vectors. Since finding an optial transit-signal vector (encopassing precoding and antenna-selection) is too coplex, we proposed a low-coplexity two-stage ethod to directly obtain such transit-signal vector, which is based on iterativehard-thresholding and bit-flipping algoriths. Via siulation results, we deonstrated that the proposed ethod provides a ore elegant perforance-coplexity tradeoff than the stateof-the-art sybol-scaling ethod. One proising extension of this work is to devise a low-coplexity algorith which is ore suitable to our non-linear inverse proble for finding a feasible transit-signal vector. Another extension is to develop the proposed idea for one-bit DAC MU-MISO systes with quadrature-aplitude-odulation (QAM). This is ore challenging than this work focusing on PSK since soe decision regions of QAM are surrounded by the other decision regions, which akes difficult to handle. REFERENCES [1] T. L. Marzetta, Noncooperative cellular wireless with unliited nubers of base station antennas, IEEE Trans. Wireless Coun., vol. 9, no. 11, pp , Nov [2] A. Adhikary, J. Na, J.-Y. Ahn, and G. Caire, Joint spatial division and ultiplexing: The large-scale array regie, IEEE Trans. Inf. Theory, vol. 59, no. 10, pp , Oct [3] C. Masouros, M. Sellathurai, and T. Ratnarajah, Large-scale MIMO transitters in fixed physical spaces: The effect of transit correlation and utual coupling, IEEE Trans. Coun., vol. 61, no. 7, pp , Jul [4] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhin, and R. Zhang, An overview of assive MIMO: Benefits and challenges, IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp , Oct [5] C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, A vector-perturbation technique for near-capacity ultiantenna ultiuser counication-part I: Channel inversion and regularization, IEEE Trans. Coun., vol. 53, no. 1, pp , Jan [6] H. Yang and T. L. Marzetta, Total energy efficiency of cellular large scale antenna syste ultiple access obile networks, in Proc. IEEE Online Conf. Green Coun., pp , Piscataway, NJ, USA, Oct [7] A. F. Molisch, V. V. Ratna, S. Han, Z. Li, S. L. H. Nguyen, L. Li, and K. Haneda, Hybrid beaforing for assive MIMO: A survey, IEEE Coun. Mag., vol. 55, no. 9, pp , [8] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, Channel estiation and hybrid precoding for illieter wave cellular systes, IEEE J. Sel. Topics Signal Process., vol.8, no.5, pp , Oct [9] J. Mo, A. Alkhateeb, S. Abu-Surra, and R. W. Heath, Hybrid architectures with few-bit ADC receivers: Achievable rates and energy-rate tradeoffs, IEEE Trans. Wireless Coun., vol. 16, no. 4, pp , Apr [10] S. Jacobsson, G. Durisi, M. Coldrey, T. Goldstein, and C. Studer, Quantized precoding for assive MU-MIMO, IEEE Trans. Coun., vol. 65, no. 11, pp , Nov [11] A. K. Saxena, I. Fijalkow, and A. L. Swindlehurst, Analysis of one-bit quantized precoding for the ultiuser assive MIMO Downlink, IEEE Trans. Sig. Process., vol. 65, no. 17, pp , Sept [12] O. B. Usan, H. Jedda, A. Mezghani, and J. A. Nossek, MMSE precoder for assive MIMO using 1-bit quantization, in Proc. IEEE Int. Conf. Acoust. Speech Sig. Process. (ICASSP), pp , Shanghai, Mar [13] L. T. N. Landau and R. C. de Laare, Branch-and-bound precoding for ultiuser MIMO systes with 1-bit quantization, IEEE Wireless Coun. Lett., vol. 6, no. 6, pp , Dec [14] O. Castaneda, S. Jacobsson, G. Durisi, M. Coldrey, T. Goldstein, and C. Studer, 1-bit assive MU-MIMO precoding in VLSI, IEEE J. Eerging Sel. Topics Circuits and Systes, vol. 7, no. 4, pp , Dec [15] A. Li, C. Masouros, F. Liu, and A. L. Swindlehurst, Massive MIMO 1- bit DAC transission: A low-coplexity sybol scaling approach, IEEE Trans. Wireless Coun., vol. 17, no. 11, pp , Nov