Symbol-level precoding is symbol-perturbed ZF when energy Efficiency is sought
aa r X i v : . [ c s . I T ] M a r SYMBOL-LEVEL PRECODING IS SYMBOL-PERTURBED ZF WHEN ENERGYEFFICIENCY IS SOUGHT
Yatao Liu and Wing-Kin Ma
Department of Electronic Engineering, The Chinese University of Hong Kong, Hong Kong SAR, ChinaE-mail: [email protected], [email protected]
ABSTRACT
This paper considers symbol-level precoding (SLP) for multiusermultiple-input single-output (MISO) downlink. SLP is a nonlinearprecoding scheme that utilizes symbol constellation structures. Ithas been shown that SLP can outperform the popular linear beam-forming scheme. In this work we reveal a hidden connection be-tween SLP and linear beamforming. We show that under an en-ergy minimization design, SLP is equivalent to a zero-forcing (ZF)beamforming scheme with perturbations on symbols. This iden-tity gives new insights and they are discussed in the paper. As aside contribution, this work also develops a symbol error probability(SEP)-constrained SLP design formulation under quadrature ampli-tude modulation (QAM) constellations.
Index Terms — multiuser MISO, symbol-level precoding, en-ergy efficiency, symbol error probability.
1. INTRODUCTION
In multiuser MIMO downlink scenarios, linear precoding or beam-forming is arguably the most widely used physical-layer transceiverscheme. Linear beamforming is simple in terms of transceiver struc-tures, and it has been found in numerous studies that linear beam-forming is effective in improving system performance such as totalthroughput. Recently there has been interest in another class of pre-coding schemes called constructive interference, directional modu-lation, or symbol-level precoding (SLP) [1–9]. For convenience, wewill use the name SLP when we refer to such schemes. SLP lever-ages on the fact that transmitted symbols are drawn from a constel-lation, such as quadrature amplitude modulation (QAM) and M -aryphase shift keying (MPSK), in real world. By contrast, in linearbeamforming, we usually take a level of abstraction from the sym-bol level. To be specific, when we design linear beamformers, itis common to adopt quality-of-service (QoS) performance metricssuch as the signal-to-interference-and-noise ratio (SINR), achievablerate, and symbol mean square error (MSE). The use of such metricsfrees us from symbol level details and allows us to directly workon the higher level problem of beamforming optimization. On theother hand, such an abstraction also precludes utilization of symbolconstellation structures that appear in all practical digital communi-cation systems.The early idea of SLP dates back to the early 2010 under thename of constructive interference [1–3]. There, the rationale is toconsider a symbol-dependent linear beamforming scheme in whichinterference is beneficially aligned at the symbol level. This requiresexploitation of the underlying constellation structures, and an SLPdesign for one constellation (e.g., QAM) can be different from thatof another (e.g., MPSK). In recent studies, this rationale is graduallyshifting toward a more general philosophy, where SLP is regarded as a generally nonlinear precoder that takes an optimization form.It is worthwhile to mention that, coincidently, the same precodingphilosophy is also seen in the concurrent developments of constantenvelope precoding [10, 11] and one-bit MIMO precoding [12].The principles of SLP and linear beamforming are, in essence,different. In this paper, we draw a connection between the two. Sim-ply speaking, we show that if we seek to minimize the total transmis-sion energy in the SLP design, the resulting SLP is equivalent to azero-forcing (ZF) beamforming scheme with perturbations on sym-bols. This result gives new insights into SLP. We derive the aboveresult based on a symbol error probability (SEP)-constrained SLPdesign formulation for QAM constellations, which, as a side contri-bution, is developed in this paper. After the submission of this work,it has come to our attention that the symbol-perturbed ZF scheme wementioned above was independently developed in [13]. However,the work in [13] neither noticed nor showed the equivalent relation-ship of SLP and symbol-perturbed ZF. In our work we also studya less explored issue in the existing SLP literature, which is aboutblock-level optimization of symbol gains at the user side and will bediscussed in Section 5. Incorporating this issue into the design givesrise to interesting insights as we will illustrate through simulations.
2. BACKGROUND
We consider a multiuser multiple-input single-output (MISO) down-link scenario where a multi-antenna base station (BS) serves K single-antenna users. The channels from the BS to the users areassumed to be frequency-flat block faded. Under such settings, thereceived signals of the users over one transmission block may bemodeled as y i,t = h Hi x t + v i,t , i = 1 , . . . , K, t = 1 , . . . , T. (1)Here, y i,t is the received signal of user i at symbol time t ; x t ∈ C N is the multi-antenna transmitted signal from the BS at symbol time t ,with N being the number of transmit antennas at the BS; h i ∈ C N represents the MISO channel from the BS to user i ; T is the trans-mission block length; v i,t is noise and we assume v i,t ∼ CN (0 , σ v ) where σ v denotes the noise variance. Assuming perfect channel stateinformation (CSI) at the BS, the task is to send symbol streams, onedesignated for one user, via a pertinent MIMO precoding scheme.Let us briefly recall how the above task is done in conventionallinear beamforming. Let { s i,t } Tt =1 be a symbol stream of user i . Thetransmitted signal x t of linear beamforming takes the form x t = K X i =1 w i s i,t , (2)where w i ∈ C N is the beamformer associated with the i th sym-bol stream. There are numerous ways to design the beamformers14–18], although the QoS performance metrics used often fall intoseveral types. In particular, it is common to adopt the SINR SINR i , ρ i | h Hi w i | P j = i ρ j | h Hi w j | + σ v , where ρ i = E [ | s i,t | ] , as the QoS performance metric. The SINRtakes a level of abstraction from the symbol level: it evaluates in-terference by means of average power, and consequently the under-lying constellation structures are not exploited. A popular beam-forming formulation under the SINR metric is the following SINR-constrained design: min w ,..., w K E [ k x t k ] = P Ki =1 ρ i k w i k s . t . SINR i ≥ γ i , i = 1 , . . . , K, (3)where γ i > , i = 1 , . . . , K, are pre-specified SINR requirements;see the literature [14, 17] for further description.
3. SEP-CONSTRAINED SYMBOL-LEVEL PRECODING
In this section we consider SLP. Unlike linear beamforming, whichrestricts the transmitted signal x t to take the linear form (2), SLPallows x t to be any vector (in C N ). It aims at finding an appropriate x t such that desired symbols are shaped at the user side. To be morespecific, we intend to achieve, as accurately as possible, h Hi x t ≈ d i s i,t , for all i, t , (4)for some given signal gain factors d , . . . , d K > ; s i,t ’s are againthe symbols. In doing so, we also incorporate other design consid-erations such as energy efficiency. Several design formulations forSLP have been proposed in previous works [3–9], and in this workwe are interested in an SEP-constrained formulation. In the SEP-constrained formulation, we seek to to minimize the total transmis-sion energy in an instantaneous sense, and, at the same time, we mustguarantee the SEP of every user to be no worse than a pre-specifiedvalue. Mathematically, this is formulated as an optimization problem min x t k x t k s . t . SEP i,t ≤ ε i , i = 1 , . . . , K, (5)where SEP i,t denotes the symbol error probability of s i,t given s i,t ,which we will define and characterize later, and ε i ’s are pre-specifiedSEP requirements. We should mention that SLP requires solvingoptimization problems on a per-symbol basis, whilst linear beam-forming usually solves an optimization problem once for the wholetransmission block (cf. (2)–(3)).Let us characterize the SEP, which depends on the constellationand the detection process at the user side. We assume that the symbolstream { s i,t } Tt =1 of user i is drawn from a QAM constellation S i = { s R + j s I | s R , s I ∈ {± , ± , . . . , ± (2 L i − }} , where j = √− , and L i is a positive integer; note that the con-stellation size is L i . Also, we assume that every user has accessto its corresponding signal gain factor d i . In practice, this can bemade possible by designing the training phase such that users are Note that in arriving at the SINR expression, we have made two mild as-sumptions, namely, that i) every stream { s i,t } Tt =1 is independent and identi-cally distributed (i.i.d.) with mean zero and variance ρ i = E [ | s i,t | ] , and ii)one stream is statistically independent of another stream. able to acquire d i ’s from the training signals. With knowledge of d i ’s, the users detect their symbol streams by a simple detection pro-cess ˆ s i,t = dec i ( y i,t /d i ) , where dec i denotes the decision functioncorresponding to S i . To get some insight with what we will see inthe SEP derivations, observe from the signal model (1) that y i,t d i = s i,t + b i,t + v i,t d i , where b i,t = h Hi x t − d i s i,t denotes a residual term of the approximation in (4). Now, define SEP i,t = Pr(ˆ s i,t = s i,t | s i,t ) . as the (conditional) SEP in (5). Also, define SEP
Ri,t = Pr( ℜ (ˆ s i,t ) = ℜ ( s i,t ) | s i,t ) , SEP
Ii,t = Pr( ℑ (ˆ s i,t ) = ℑ ( s i,t ) | s i,t ) , as the conditional SEPs of the real and imaginary parts of s i,t , re-spectively. As a standard SEP analysis result, one can show that SEP
Ri,t = Q (cid:18) √ σ v ( d i − ℜ ( b i,t )) (cid:19) + Q (cid:18) √ σ v ( d i + ℜ ( b i,t )) (cid:19) ≤ Q (cid:18) √ σ v ( d i − |ℜ ( b i,t ) | ) (cid:19) , |ℜ ( s i,t ) | < L i − , SEP
Ri,t = Q (cid:18) √ σ v ( d i + ℜ ( b i,t )) (cid:19) , ℜ ( s i,t ) = 2 L i − , SEP
Ri,t = Q (cid:18) √ σ v ( d i − ℜ ( b i,t )) (cid:19) , ℜ ( s i,t ) = − L i + 1 . (6)Also, the same result applies to SEP
Ii,t if we replace “ R ” with “ I ”and “ ℜ ” with “ ℑ ”.Our next task is to turn the SEP constraints in (5) to a formsuitable for optimization. Let ¯ ε i = 1 − √ − ε i , and observe that SEP
Ri,t ≤ ¯ ε i , SEP
Ii,t ≤ ¯ ε i = ⇒ SEP i,t ≤ ε i . (7)Also, it is shown from (6) that SEP
Ri,t ≤ ¯ ε i ⇐ = − d i + a Ri,t ≤ ℜ ( b i,t ) ≤ d i − c Ri,t , (8)where a Ri,t = α i , |ℜ ( s i,t ) | < L i − β i , ℜ ( s i,t ) = 2 L i − −∞ , ℜ ( s i,t ) = − L i + 1 c Ri,t = α i , |ℜ ( s i,t ) | < L i − −∞ , ℜ ( s i,t ) = 2 L i − β i , ℜ ( s i,t ) = − L i + 1 with α i = σ v √ Q − (cid:16) ¯ ε i (cid:17) , β i = σ v √ Q − (¯ ε i ) , and that the same result applies to SEP
Ii,t if we replace “ R ” with “ I ”and “ ℜ ” with “ ℑ ”. Using (7)–(8), we obtain the implication − d i + a Ri,t ≤ ℜ ( b i,t ) ≤ d i − c Ri,t , − d i + a Ii,t ≤ ℑ ( b i,t ) ≤ d i − c Ii,t = ⇒ SEP i,t ≤ ε i . By plugging the above implication into the constraints of Prob-lem (5), we obtain a tractable SLP design problem. Let us summa-rize the result. act 1
The SEP-constrained SLP design problem (5) can be han-dled, in a restrictive sense, by the following problem min x t k x t k s . t . − d + a Rt ≤ ℜ ( Hx t − Ds t ) ≤ d − c Rt , − d + a It ≤ ℑ ( Hx t − Ds t ) ≤ d − c It , (9) where H = [ h , . . . , h K ] H , d = [ d , . . . , d K ] T , D =Diag( d , . . . , d K ) , a Rt = [ a R ,t , . . . , a RK,t ] T , a It = [ a I ,t , . . . , a IK,t ] T , c Rt = [ c R ,t , . . . , c RK,t ] T , c It = [ c I ,t , . . . , c IK,t ] T . In particular, anyfeasible solution to Problem (9) is a feasible solution to Problem (5) .Also, Problem (9) is a convex quadratic program. In this paper we will focus on Problem (9). Note that this newformulation is restrictive owing to the inequality in (6) and the im-plication in (7). In practice, such restriction is considered mild espe-cially if the SEP requirement ε i is small. Problem (9) is similar tothe formulation in [8] in which a more intuitive idea of “relaxed deci-sion region” was introduced to guarantee certain SNR performance.Our formulation, in comparison, provides a more precise control onSEP performance guarantees.
4. MAIN RESULT
Our main result is described as follows.
Proposition 1
Suppose that H has full row rank. The optimal solu-tion x ⋆t to Problem (9) is given by x ⋆t = H † ( Ds t + u ⋆t ) , where H † = H H ( HH H ) − is the pseudo-inverse of H , and u ⋆t is the solution to min u t ( Ds t + u t ) H R ( Ds t + u t )s . t . − d + a Rt ≤ ℜ ( u t ) ≤ d − c Rt , − d + a It ≤ ℑ ( u t ) ≤ d − c It . (10) with R = ( HH H ) − .Proof: Note that R is nonsingular. Thus, any x t ∈ C N can berepresented by x t = H H R ( Ds t + u t ) + η t , for some u t ∈ C K , η t ∈ R ( H H ) ⊥ . Here R ( H H ) ⊥ denotes theorthogonal complement of the range space of H H . By the abovechange of variable, and using Hη t = , Problem (9) can be equiv-alently transformed to min u t ∈ C K , η t ∈R ( H H ) ⊥ ( Ds t + u t ) H R ( Ds t + u t ) + k η t k s . t . − d + a Rt ≤ ℜ ( u t ) ≤ d − c Rt , − d + a It ≤ ℑ ( u t ) ≤ d − c It . It is seen from the problem above that η t = must be true at theoptimum. The proof is complete. (cid:4) Proposition 1 reveals a hidden connection between SLP and lin-ear beamforming: under the total energy minimization formulationconsidered above, SLP is equivalent to a symbol-perturbed ZF beam-forming scheme —which takes the form x t = H † ( Ds t + u t ) —withthe symbol perturbation u t being optimized according to s t . It isalso interesting to note the following identity: Fact 2
Consider the linear beamforming scheme x t = P Ki =1 w i s i,t .Suppose that H has full row rank, and that w i ∈ R ( H H ) for all i . Then the linear beamforming scheme is equivalent toa symbol-perturbed ZF scheme x t = H † ( Ds t + u t ) where u t = ( HW − D ) s t , and W = [ w , . . . , w K ] . Fact 2 can be easily shown by putting u t = ( HW − D ) s t into thesymbol-perturbed ZF scheme. Fact 2 suggests that a linear beam-forming scheme, under a mild assumption, can be regarded as aninstance of symbol-perturbed ZF. Some further discussions are asfollows.1. While the original SLP problem (9) and its equivalent for-mulation (10) are both convex, the latter is easier to handle.Problem (10) is a quadratic program with bound constraints,which has been extensively studied and has efficient solversreadily available [19].2. We see from Problem (10) that the signal gain factors d i ’s alsocontrol the bounds of the perturbations u t ’s. In particular,if we choose d i = α i for all i , then, for instances where | s i,t | < L i − for all i , we have u t = and the SLPscheme reduces to the ZF.
5. FURTHER ISSUES
Thus far, in our development, we have assumed that the signal gainfactors d i ’s are given. A question arising is how we may choose d i ’s.An optimal way of doing so is to consider the following problem min x ,..., x T , d T P Tt =1 k x t k s . t . SEP i,t ≤ ε i , i = 1 , . . . , K, t = 1 , . . . , T, d ≥ ; (11)where we seek to optimize SLP and the signal gain factors simulta-neously by minimizing the total power over the transmission block.Using Proposition 1, we can recast the problem (in a restrictivesense) as min u ,..., u T , d T P Tt =1 ( Ds t + u t ) H R ( Ds t + u t )s . t . − d + a Rt ≤ ℜ ( u t ) ≤ d − c Rt , t = 1 , . . . , T, − d + a It ≤ ℑ ( u t ) ≤ d − c It , t = 1 , . . . , T, d ≥ , (12)The above problem is convex. We can also consider an alternativeformulation wherein the peak energy, rather than the average, is min-imized: min u ,..., u T , d max t =1 ,...,T ( Ds t + u t ) H R ( Ds t + u t )s . t . − d + a Rt ≤ ℜ ( u t ) ≤ d − c Rt , t = 1 , . . . , T, − d + a It ≤ ℑ ( u t ) ≤ d − c It , t = 1 , . . . , T, d ≥ , (13)The above formulation may be useful when one desires to reduce theenergy spread of the transmitted signals over symbol time. Again,the above problem is convex.Although the two design problems in (12) and (13) are convex,they are by no means easy to deal with. The reason is that the numberof variables of Problems (12) and (13) scales with the block length T . As such, they are large-scale problems when T is large (which isypical in standards), and development of fast algorithms is required.We leave the latter as an open problem for future work. In this paperwe will use general purpose convex optimization software (such as CVX ) to solve Problems (12) and (13), and our emphasis will beplaced on demonstrating performance gains of Problems (12) and(13) by simulations.On the other hand, one can use heuristics to determine d . Sup-pose that L i > for all i . Also, let us make a mild assumptionthat there exist s i,t such that | s i,t | < L i − for all i . From Prob-lem (10), one can verify that it must hold that d i ≥ α i , for all i .Since using smaller d i ’s should be helpful in reducing the total trans-mission energy, we can choose d i = ζ · α i , i = 1 , . . . , K, (14)where ζ ≥ is a manually chosen scaling factor.
6. SIMULATION RESULTS
In this section, we present simulation results to show the perfor-mance of SLP and compare it with other schemes. In the simula-tions, we use the total transmission power T P Tt =1 k x t k and peaktotal energy max t =1 ,...,T k x t k as the performance metrics. Thesimulation settings are as follows. The number of transmit anten-nas N = 16 ; the number of users K = 16 ; the elements h ij of H are randomly generated at each trial and follow CN (0 , in ani.i.d. manner; the power of noise is σ v = 1 ; the transmission blocklength is T = 50 ; the symbols s i,t ’s are uniformly generated fromthe 16-QAM constellation; we set ε = · · · = ε K = ε . For eachsimulation scenario, we generate 100 channel realizations to get anaverage result of the performance metrics.Here we consider two benchmark schemes: the ZF scheme [20]and optimal linear beamforming scheme [14]. The two schemes aredesigned such that the SEP requirements in (5) are satisfied. For theZF scheme, it can be verified that x ZFt = H H ( HH H ) − Ds t with d i = α i for all i achieves the requirements. For the optimal linearbeamforming scheme, we have the following fact: Fact 3
Consider the linear beamforming scheme in (2) - (3) . Supposethat the multiuser interferences are approximated as complex circu-lar Gaussian random variables. The optimal beamforming design in (3) guarantees the SEP requirements in (5) if we choose γ i = ρ i (cid:20) Q − (cid:18) − √ − ε i (cid:19)(cid:21) , i = 1 , . . . , K. We skip the proof of the above fact. In fact, the result is almost afolklore.We first examine total transmission power performance. Weconsider the optimal SLP design in Problem (12). We also try SLPdesigns using the heuristic choice of d i ’s in (14). The results areshown in Fig. 1. As can be seen, the SLP schemes consume muchless power than the two benchmark schemes. Moreover, we see thatthe heuristic SLP schemes can also achieve surprisingly good per-formance. In particular, when ζ = 1 , the performance is almostoptimal. As a future work, it would be interesting to further studywhy this is so.Next, we turn to peak total energy performance. We replacethe average power minimization design (12) with the peak totalenergy minimization design (13). The previously tested heuristic SLP schemes are also tried. The results, shown in Fig. 2, illustratethat the SLP schemes, even the heuristic SLP ones, outperform thebenchmark schemes significantly. However, unlike the previous totalpower result, there is a large performance gap between the optimalSLP and heuristic SLP schemes. We thus conclude that optimal SLPis powerful in the case of peak total energy minimization. -4 -3 -2 -1 T o t a l t r an s m i ss i on po w e r ( d B ) ZFOpt. BFSLP; opt. d SLP; ζ =1SLP; ζ =1.5SLP; ζ =2.5 Fig. 1 . Total transmission power T P Tt =1 k x t k with respect to theSEP requirement ε . -4 -3 -2 -1 P ea k t o t a l ene r g y ( d B ) ZFOpt. BFSLP; opt. d SLP; ζ =1SLP; ζ =1.5SLP; ζ =2.5 Fig. 2 . Peak total energy max t =1 ,...,T k x t k with respect to the SEPrequirement ε .
7. CONCLUSIONS
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