Construction of One-Bit Transmit-Signal Vectors for Downlink MU-MISO Systems with PSK Signaling
11 Construction of One-Bit Transmit-Signal Vectors forDownlink MU-MISO Systems with PSK Signaling
Gyu-Jeong Park and Song-Nam HongAjou University, Suwon, Korea,email: { net2616, snhong } @ajou.ac.kr Abstract —We study a downlink multi-user multiple-inputsingle-output (MU-MISO) system in which the base station (BS)has a large number of antennas with cost-effective one-bit digital-to-analog converters (DACs). In this system, we first identify thatantenna-selection can yield a non-trivial symbol-error-rate (SER)performance gain by alleviating an error-floor problem. Likewisethe previous works on one-bit precoding, finding an optimaltransmit-signal vector (encompassing precoding and antenna-selection) requires exhaustive-search due to its combinatorialnature. Motivated by this, we propose a low-complexity two-stage algorithm to directly obtain such transmit-signal vector.In the first stage, we obtain a feasible transmit-signal vectorvia iterative-hard-thresholding algorithm where the resultingvector ensures that each user’s noiseless observation is belongto a desired decision region. In the second stage, a bit-flippingalgorithm is employed to refine the feasible vector so that eachuser’s received signal is more robust to additive Gaussian noises.Via simulation results, we demonstrate that the proposed methodcan yield a more elegant performance-complexity tradeoff thanthe existing one-bit precoding methods.
Index Terms —Massive MIMO, one-bit DAC, precoding,antenna-selection, beamforming.
I. I
NTRODUCTION
Massive multiple-input multiple-output (MIMO) is one ofthe promising techniques to cope with the predicted wirelessdata traffic explosion [1]–[4]. In downlink massive MIMO sys-tems, it was shown that low-complexity linear precoding meth-ods as zero-forcing (ZF) and regularized ZF (RZF) achieve analmost optimal performance [5]. In contrast, the use of a largenumber of antennas considerably increases the hardware costand the radio-frequency (RF) circuit consumption [6]. Hybridanalog-digital precoding (a.k.a., hybrid precoding) is one ofthe promising methods to address the above problems since itcan reduce the number of RF chains [7]–[9]. An alternativemethod is to make use of low-resolution digital-to-analogconverters (DACs) (e.g., 1 ∼ (cid:96) ∞ -norm relaxation,and sphere decoding. The authors proposed low-complexity quantized precoding methods as quantized ZF (QZF) [11] -30 -20 -10 0 10 20 30 40 50 SNR [dB] -4 -3 -2 -1 S E R Exhaustive Search(with Antenna-Selection) Exhaustive Search(without Antenna-Selection)
Fig. 1. N t = 8 and K = 2 . Performance improvement of antenna-selectionin one-bit precoding for downlink MU-MIMO systems with one-bit DACs. and quantized minimum-mean squared error (QMMSE) [12],which simply applied the one-bit quantization to the outputsof the conventional linear precoding methods. Also, a branch-and-bound and a biconvex relaxation approaches were pre-sented in [13] and [14], respectively. However, these meth-ods either suffer from a severe performance loss or requirean expensive computational complexity (see [15] for moredetails). Very recently, focusing on phase-shift-keying (PSK)constellations, a low-complexity symbol-scaling method wasproposed in [15], where it optimizes a transmit-signal vectordirectly in an efficient sequential fashion as a function ofan instantaneous channel matrix and users’ messages (calledsymbol-level operation). Also, it was shown that the symbol-scaling method can yield a comparable performance to theprevious non-linear precoding methods with a much lowercomputational complexity. Our contributions:
In this paper, we study a downlinkMU-MISO system with one-bit DACs. We first identify thatantenna-selection can yield a non-trivial bit-error-rate (BER)performance gain by alleviating error-floor problems (seeFig. 1). Clearly, this performance gain is because antenna-selection can enlarge the set of possible transmit-signal vectors(i.e., search-space) compared with the previous precodingmethods in [10]–[15]. However, as in conventional one-bit pre-coding optimization, finding an optimal transmit-signal vec- a r X i v : . [ c s . I T ] J a n tor (encompassing precoding and antenna-selection) requiresexhaustive-search due to its combinatorial nature. Motivatedby this, we propose a low-complexity algorithm to solvethe above problem (i.e., joint optimization of precoding andantenna-selection), which consists of the following two stages.In the first stage, we obtain a feasible transmit-signal vectorvia iterative-hard-thresholding (IHT) algorithm where the re-sulting vector guarantees that each user’s noiseless observationis belong to a desired decision region. Namely, it can improvethe BER performances at high-SNR regimes (i.e., error-floorregions) by lowering an error-floor. In the second stage, werefine the above transit-signal vector using a bit-flipping (BF)algorithm so that each user’s received signal is more robustto additive Gaussian noises. In other words, it can improvethe BER performances at low-SNR regimes (i.e., waterfallregions). Finally, we provide simulation results to demonstratethat the proposed method can improve the performance ofthe existing symbol-scaling method in [15] with a comparablecomputational complexity.The rest of paper is organized as follows. In Section II,we provide some useful notations and describe the systemmodel. In Section III, we propose a low-complexity algorithmto optimize a transmit-signal vector directly for downlinkMU-MISO systems with one-bit DACs. Simulation results areprovided in IV. Section V concludes the paper.II. P RELIMINARIES
In this section, we provide some useful notations anddescribe the system model.
A. Notations
The lowercase and uppercase boldface letters representcolumn vectors and matrices, respectively, and ( · ) T denotesthe conjugate transpose of a vector or matrix. 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 . Similarly, let [ b ] = { , ..., b } for any positive integer b . Re( a ) and Im( a ) represent its real and complex part of a complex vector a ,respectively. Also, we define a natural mapping g ( · ) whichmaps a complex value into a real-valued vector, i.e., for each x ∈ C , we have that g ( x ) = [Re( x ) , Im( x )] T . (1)The inverse mapping of g is denoted as g − . If g or g − isapplied to a vector or a set, we assume they operate element-wise. For example, we have that g ([ x , x ] T ) = [Re( x ) , Im( x ) , Re( x ) , Im( x )] T . (2)Also, for any complex-value x ∈ C , we define a real-valuedmatrix expansion φ ( x ) as φ ( x ) = (cid:20) Re( x ) − Im( x )Im( x ) Re( x ) (cid:21) . (3)Finally, we let R ( θ ) denote a rotation matrix with a parameter θ as R ( θ ) = (cid:20) cos θ − sin θ sin θ cos θ (cid:21) , which rotates the following column vector in the counterclock-wise through an angle θ about the origin. 𝑐 " 𝑐 𝑐 $ 𝑐 % 𝑐 & 𝑐 ’ 𝑐 ( 𝑐 ) ℛ ℛ $ ℛ % ℛ & ℛ ’ ℛ ( ℛ ) ℛ " real axisimaginary axis 𝑏 𝑏 Fig. 2. 8-PSK constellation in complex domain and the correspondingdecision regions. R i represents the decision region of c i ∈ M . B. System Model
We consider a single-cell downlink MU-MISO system inwhich one BS with N t antennas communicates with single-antenna K (cid:28) N t users simultaneously in the same time-frequency resources. Focusing on the impact of one-bit DACsin the transmit-side operations, it is assumed that the BS isequipped with one-bit DACs while each user (receiver) iswith ideal analog-to-digital converters (ADCs) with infiniteresolution. As in the closely related works [11], [12], [15],we also consider a normalized m -PSK constellation C = { c , c , ..., c m − } , each of which constellation point is definedas c i = cos (2 πi/m ) + j sin (2 πi/m ) ∈ C . (4)The constellation points of -PSK are depicted in Fig. 2. Let µ k ∈ [0 : m − be the user k ’s message for k ∈ [ K ] and alsolet x = (cid:2) x , ..., x N t (cid:3) T be a transmit-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 Im( x i ) ∈ {− , , } . (5)This shows that, compared with the related works [11], [12],[15], antenna-selection can enlarge the set of possible sym-bols (i.e., search-space) per real (or imaginary) part of eachantenna. Then, the received signal vector at the K users isgiven by y = √ ρ Hx + z , (6)where H ∈ C K × N t denotes the flat-fading Rayleigh channelwith each entry following a complex Gaussian distributionand z ∈ C K × denotes the additive Gaussian noise vec-tor whose elements are distributed as circularly symmetriccomplex Gaussian random variables with zero-mean and unit-variance, i.e., z i ∼ CN (0 , . Also, ρ is chosen according tothe per-antenna power constraint. For simplicity, we assumethe uniform power allocation for the antenna array.In this system, our purpose is to develop a low-complexityalgorithm to optimize a transmit-signal vector x (i.e., joint optimization of precoding and antenna-selection) with theassumption that the BS is aware of a perfect channel stateinformation (CSI), which will be provided in Section III.Before explaining our main result, we provide the followingdefinition which will be used throughout the paper. Definition 1: (Decision Regions)
For each c i ∈ C , a deci-sion region R i is defined as R i ∆ = (cid:26) y ∈ C : | y − c i | ≤ min j ∈ [0: m − j (cid:54) = i | y − c j | (cid:27) . (7)If the user k receives a y k ∈ R µ k , then it decides the decodedmessage as µ k ∈ [0 : m − . (cid:4) III. T HE P ROPOSED T RANSMIT -S IGNAL V ECTORS
In this section, we derive a mathematical formulation tooptimize a transmit-signal vector (encompassing precodingand antenna selection) for downlink MU-MISO systems withone-bit DACs, and then present a low-complexity algorithmto solve such problem efficiently.For the ease of explanation, we first introduce the equiv-alent real-valued representation of the complex input-outputrelationship in (6), which is given by ˜ y = √ ρ ˜ H ˜ x + ˜ z , (8)where ˜ y = g ( y ) , ˜ x = g ( x ) , ˜ z = g ( z ) , and ˜ H denotes the K × N t real-valued matrix which is obtained by replacing h i,j (e.g., the ( i, j ) -th entry of H ) with the × matrix φ ( h i,j ) for all i, j . For the resulting model, we can define the real-valued constellation ˜ C = { g ( c ) , g ( c ) , ..., g ( c m − ) } where g ( c i ) = [cos(2 πi/m ) , sin(2 πi/m )] T . (9)From Definition 1, the decision region in R for each g ( c i ) issimply obtained as ˜ R i = g ( R i ) . (10)Furthermore, as shown in Fig. 2, each ˜ R i can be representedas linear combination of two basis vectors s i, and s i, as ˜ R i ∆ = { α i, s i, + α i, s i, : α i, , α i, > } , (11)where the basis vectors are easily obtained using rotationmatrices such as s i,(cid:96) = R (cid:18) π ( − (cid:96) m (cid:19) (cid:20) cos (cid:0) πim (cid:1) sin (cid:0) πim (cid:1)(cid:21) = cos (cid:16) π (2 i +( − (cid:96) ) m (cid:17) sin (cid:16) π (2 i +( − (cid:96) ) m (cid:17) , for (cid:96) = 1 , . Using two basis vectors, we define the × real-valued matrix S i ∆ = (cid:2) s i, s i, (cid:3) . Since S i has full-rank, theinverse matrix of S exists and is easily computed as S − i = 1sin(2 π/m ) sin (cid:16) π (2 i +1) m (cid:17) − cos (cid:16) π (2 i +1) m (cid:17) − sin (cid:16) π (2 i − m (cid:17) cos (cid:16) π (2 i − m (cid:17) . (12)We are now ready to explain how to optimize a transmit-signal vector x efficiently. For simplicity, we let r = Hx ∈ C K × denote the noiseless received vector at the K users.Regarding our optimization problem, we first provide thefollowing key observations: ℛ " 𝛼 " 𝒔 "(&) 𝛼 "," 𝒔 "," + 𝛼 ",) 𝒔 ",) 𝛼 "," 𝒔 "," + 𝛼 ",)* 𝒔 ",) Fig. 3. The decision region R in R and the impact of larger coefficients. • (Feasibility condition) To ensure that all the K usersrecover their own messages, a transmit-signal vector x should be constructed such that r k ∈ R µ k ( equivalently , g ( r k ) ∈ ˜ R µ k ) , (13)for k ∈ [ K ] . Accordingly, g ( r k ) should be represented as g ( r k ) = α k, s µ k , + α k, s µ k , , (14)for some positive coefficients α k, and α k, . This iscalled feasibility condition and a vector x to satisfy thiscondition called feasible transmit-signal vector. • (Noise robusteness) The condition in (13) cannot guar-antee good performances in practical SNR regimes (e.g.,waterfall regions) due to the impact of additive Gaus-sian noises. Thus, we need to refine the above feasibletransmit-signal vector so that α k, and α k, are maxi-mized.We will formulate an optimization problem mathematicallywhich can find a transmit-signal vector x to satisfy theabove requirements. From (14), we can express the feasibilitycondition in a matrix form: g ( r ) = ˜ H ˜ x = S α , (15)where α = [ α , , α , , · · · , α K, , α K, ] T and S = diag ( S µ , · · · , S µ K ) denotes the block diagonal matrix havingthe i -th diagonal block S µ i . From the block diagonal structureand (12), we can easily obtain the inverse matrix of S as S − = diag ( S − µ , · · · , S − µ K ) . (16)Then, the feasibility condition in (15) can be rewritten as α = Λ ˜ x . (17)where Λ ∆ = S − ˜ H ∈ R K × N t . Note that Λ is a known matrixsince it is completely determined from the channel matrix H and users’ messages ( µ , ..., µ K ) . Taking the feasibilitycondition and noise robustness into account, our optimizationproblem can formulated as max ˜ x min { α k,i : k ∈ [ K ] , i = 1 , } (18)subject to α = Λ ˜ x (19) α k, , α k, > , k ∈ [ K ] (20) ˜ x ∈ {− , , } N t . (21) For the above optimization problem, the objective functionaims to maximize the minimum of positive coefficients α k,i ’s.This is motivated by the fact that a larger value of thecoefficients α k,i ’s yields a larger distance to the other decisionregions. As an example, consider the decision region R asillustrated in Fig. 3. Obviously, α , s , + α (cid:48) , s , has a largerdistance from the boundary 1 than α , s , + α , s , whileboth have the same distance from the boundary 2. Likewise,if α , increases for a fixed α , , then the distance fromthe boundary 2 increases by keeping the distance from theboundary 1. In order to increase the distance from the otherdecision regions, therefore, it would make sense to maximizethe minimum of two coefficients α , and α , . By extendingthis to all the K users, we can obtain the objective functionin (18).Unfortunately, finding an optimal solution to the above op-timization problem is too complicated due to its combinatorialnature. We thus propose a low-complexity two-stage algorithmto solve it efficiently, which is described as follows. • In the first stage, we find a feasible solution x f to satisfythe constraints (19)-(21), which is obtained by taking asolution of = sign ( Λ ˜ x ) , (22)where sign ( a ) = 1 if a ≥ and sign ( a ) = − otherwise.We solve the above non-linear inverse problem efficientlyvia the so-called IHT algorithm. The detailed proceduresare provided in Algorithm 1 where the thresholdingfunction [ · ] ∆ is defined as [ a ] ∆ = 1 if a > ∆ , [ a ] ∆ = − if a ≤ − ∆ , and [ a ] ∆ = 0 , otherwise. • In the second stage, we refine the above feasible solutionby maximizing min { α k, , α k, : k ∈ [ K ] } , which isefficiently performed using BF algorithm (see Algorithm2). Finally, from the output of Algorithm 2 (e.g., ˜ x ),we can obtain the proposed transmit-signal vector as x = g − (˜ x ) . Computational Complexity:
Following the complexityanalysis in [15], we study the computational complexity ofthe proposed method with respect to the number of real-valued multiplications. As benchmark methods, we considerthe computational costs of exhaustive search, exhaustive searchwith antenna-selection, and symbol-scaling method in [15],which are denoted by χ E , χ AS , and χ S , respectively. From[15], their complexities are obtained as χ E = 4 KN t × N t (23) χ AS = 4 KN t × N t (24) χ S = 4 N + 24 KN t − K. (25)Recall that the proposed method is composed of the twoalgorithms in Algorithms 1 and 2. First, the complexity ofAlgorithm 1 is obtained as t (cid:63) × KN t where t (cid:63) ≤ t max denotes the total number of iterations. Also, the complexityof Algorithm 2, which is similar to that of refined-state insymbol-scaling method in [15], is obtained as KN t . Bysumming them, the overall computational complexity of theproposed method is obtained as χ P = 8( t (cid:63) + 1) KN t . (26) Algorithm 1
IHT Algorithm input: Λ ∈ R K × N t , ∆ , and t max initialization: ˜ x (0) = iteration: repeat until either e ( t ) = or t = t max e ( t ) = − sign ( Λ [˜ x ( t ) ] ∆ )˜ x ( t +1) = ˜ x ( t ) + Λ T e ( t ) output: ˜ x f ← ˜ x ( t ) Algorithm 2
BF Algorithm input: Λ ∈ R K × N t and x f ∈ R N t × initialization: ˜ x = ˜ x f for i = 1 : 2 N t do for j ∈ { , , − } do β ( j ) = Λ ˜ x { i = j } and β ( j )min = min i ∈ [2 K ] { β ( j ) i } end for update ˜ x i ← arg max j ∈{ , , − } { β ( j )min } end for output: ˜ x ∈ R N t × ♦ Note that for some j ∈ { , , − } , ˜ x { i = j } is obtained fromthe ˜ x by simply replacing ˜ x i with j , i.e., ˜ x { i = j } = [˜ x , ..., ˜ x i − , j, ˜ x i +1 , ..., ˜ x N t ] T . IV. S
IMULATION R ESULTS
In this section, we evaluate the symbol-error-rate (SER)performances of the proposed method for the downlink MU-MIMO systems with one-bit DACs where N t = 128 and K = 16 . For comparison, we consider QZF in [11] andthe state-of-the-art symbol-scaling method in [15] because theformer is usually assumed to be the baseline approach and thelatter showed an elegant performance-complexity tradeoff overthe other existing methods (see [15] for more details). BothQPSK (or 4-PSK) and 8-PSK are considered. Regarding theproposed method, we choose the threshold parameter ∆ = 3 for Algorithm 1, which is optimized numerically via Monte-Carlo simulation. As mentioned in Section II-B, a flat-fadingRayleigh channel is assumed.Fig. 4 shows the performance comparison of various precod-ing methods when QPSK is employed. From this, we observethat QZF suffers from a serious error-floor and thus is not ableto yield a satisfactory performance. In contrast, both symbol-scaling and the proposed methods overcome the error-floorproblem, thereby outperforming QZF significantly. Moreover,it is observed that the proposed method can slightly improvethe performance of QZF with much less computational cost(e.g., complexity reduction).Fig. 5 shows the performance comparison of various pre-coding methods when 8-PSK is employed. It is observed thatin this case, an error-floor problem is more severe than before,which is obvious since the decision regions of 8-PSK is moresophisticated than those of QPSK. Hence, antenna-selectioncan attain a more performance gain with a larger search-space.Accordingly, the proposed method can further improve the per- -20 -15 -10 -5 0 5 10 15 20 25 30 35 SNR [dB] -6 -5 -4 -3 -2 -1 S y m bo l E rr o r R a t e ( S E R ) QZF [11]Symbol-Scaling [15]Proposed method
Fig. 4. N t = 128 , K = 16 and QPSK. Performance comparison ofQZF, symbol-scaling, and the proposed methods for the downlink MU-MISOsystems with one-bit DACs. The computational complexities are computed as χ S = 114656 and χ P = 36044 . formance of symbol-scaling method by lowering an error-floor,which is verified in Fig. 5. To be specific, the proposed methodwith t (cid:63) = 6 outperforms the symbol-scaling method with analmost same computational cost. Furthermore, at the expenseof two times computational cost, the proposed method with t (cid:63) can address the error-floor problem completely. Therefore,the proposed method provides a more elegant performance-complexity tradeoff than the state-of-the-art symbol-scalingmethod. V. C ONCLUSION
In this paper, we showed that the use of antenna-selectioncan yield a non-trivial performance gain especially at error-floor regions, by enlarging the set of possible transmit-signalvectors. Since finding an optimal transmit-signal vector (en-compassing precoding and antenna-selection) is too complex,we proposed a low-complexity two-stage method to directlyobtain such transmit-signal vector, which is based on iterative-hard-thresholding and bit-flipping algorithms. Via simulationresults, we demonstrated that the proposed method provides amore elegant performance-complexity tradeoff than the state-of-the-art symbol-scaling method. One promising extension ofthis work is to devise a low-complexity algorithm which ismore suitable to our non-linear inverse problem for finding afeasible transmit-signal vector. Another extension is to developthe proposed idea for one-bit DAC MU-MISO systems withquadrature-amplitude-modulation (QAM). This is more chal-lenging than this work focusing on PSK since some decisionregions of QAM are surrounded by the other decision regions,which makes difficult to handle.R
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