Machine Learning Inspired Energy-Efficient Hybrid Precoding for MmWave Massive MIMO Systems
Xinyu Gao, Linglong Dai, Ying Sun, Shuangfeng Han, I Chih-Lin
aa r X i v : . [ c s . I T ] A ug Machine Learning Inspired Energy-Efficient HybridPrecoding for MmWave Massive MIMO Systems
Xinyu Gao ∗ , Linglong Dai ∗ , Ying Sun ∗ , Shuangfeng Han † , and Chih-Lin I † ∗ Tsinghua National Laboratory for Information Science and Technology (TNList),Department of Electronic Engineering, Tsinghua University, Beijing, China † Green Communication Research Center, China Mobile Research Institute, Beijing 100053, China
Abstract —Hybrid precoding is a promising technique formmWave massive MIMO systems, as it can considerably reducethe number of required radio-frequency (RF) chains withoutobvious performance loss. However, most of the existing hybridprecoding schemes require a complicated phase shifter network,which still involves high energy consumption. In this paper,we propose an energy-efficient hybrid precoding architecture,where the analog part is realized by a small number of switchesand inverters instead of a large number of high-resolutionphase shifters. Our analysis proves that the performance gapbetween the proposed hybrid precoding architecture and thetraditional one is small and keeps constant when the numberof antennas goes to infinity. Then, inspired by the cross-entropy(CE) optimization developed in machine learning, we proposean adaptive CE (ACE)-based hybrid precoding scheme for thisnew architecture. It aims to adaptively update the probabilitydistributions of the elements in hybrid precoder by minimizingthe CE, which can generate a solution close to the optimal onewith a sufficiently high probability. Simulation results verify thatour scheme can achieve the near-optimal sum-rate performanceand much higher energy efficiency than traditional schemes.
I. I
NTRODUCTION
Millimeter-wave (mmWave) massive multiple-inputmultiple-output (MIMO) has been considered as a promisingtechnology for future 5G wireless communications [1], sinceit can provide wider bandwidth [2] and achieve higherspectral efficiency [3]. However, in MIMO systems, eachantenna usually requires a dedicated radio-frequency (RF)chain (including high-resolution digital-to-analog converter,mixer, etc.) to realize the fully digital signal processing (e.g.,precoding) [4]. For mmWave massive MIMO, this will resultin unaffordable hardware complexity and energy consumption,as the number of antennas becomes huge and the energyconsumption of RF chain is high [5]. To reduce the numberof required RF chains, hybrid precoding has been recentlyproposed [6]. Its key idea is to decompose the fully digitalprecoder into a large-size analog beamformer (realized by theanalog circuit) and a small-size digital precoder (requiringa small number of RF chains). Thanks to the low-rankcharacteristics of mmWave channels [2], a small-size digitalprecoder can achieve the spatial multiplexing gains, makinghybrid precoding enjoy the near-optimal performance [5].Nevertheless, most of the existing hybrid precoding schemesrequire a complicated phase shifter network, where each RFchain is connected to all antennas with high-resolution phaseshifters [6], [7]. Although this architecture can provide high design freedom to achieve the near-optimal performance, itrequires hundreds or even thousands of high-resolution phaseshifters with high hardware cost and energy consumption [5].To solve this problem, two categories of schemes have beenproposed very recently. The first category is to directly employfinite-resolution phase shifters instead of high-resolution phaseshifters [8], [9]. It can reduce the energy consumption of phaseshifter network without obvious performance loss, but it stillrequires a large number of phase shifters with considerableenergy consumption. The second category is to utilize theswitch network to replace the phase shifter network [10]–[12]. It can significantly reduce the hardware cost and energyconsumption, but it suffers from an obvious performance loss.In this paper, we propose a switch and inverter (SI)-based hybrid precoding architecture with considerably reducedhardware cost and energy consumption. Instead of using phaseshifters, the analog part of the proposed architecture is realizedby a small number of energy-efficient switches and inverters.Then, we provide the performance analysis to quantify the per-formance gap between the proposed hybrid precoding architec-ture and the traditional ones. After that, inspired by the cross-entropy (CE) optimization developed in machine learning [13],we propose an adaptive CE (ACE)-based hybrid precodingscheme for this new architecture. Specifically, according to theprobability distributions of the elements in hybrid precoder,this scheme first randomly generates several candidate hybridprecoders. Then, it adaptively weights these candidate hybridprecoders according to their achievable sum-rates, and refinesthe probability distributions of elements in hybrid precoder byminimizing the CE. Repeating such procedure, we can finallygenerate a hybrid precoder close to the optimal one with asufficiently high probability. Simulation results verify that ourscheme can achieve the near-optimal sum-rate performanceand much higher energy efficiency than traditional schemes.
Notation : Lower- and upper-case boldface letters denotevectors and matrices, respectively; ( · ) T , ( · ) H , ( · ) − , tr ( · ) ,and k·k F denote the transpose, conjugate transpose, inversion,trace, and Frobenius norm of a matrix, respectively; |·| denotesthe absolute operator; E ( · ) denotes the expectation; ⊗ denotesthe kronecker product; I N is the N × N identity matrix.II. S YSTEM M ODEL
In this paper, we consider a typical mmWave massiveMIMO system, where the base station (BS) employs N an-ennas and N RF RF chains to simultaneously serve K single-antenna users (the extension to users with multiple-antennasis also possible as will be explained later). To fully achievethe multiplexing gains, we assume N RF = K [9]. For hybridprecoding systems as shown in Fig. 1, the K × receivedsignal vector y for all K users can be presented by y = HF RF F BB s + n , (1)where H = [ h , h , · · · , h K ] H is the channel matrix with h k presenting the N × channel vector between the BS andthe k th user, s is the K × transmitted signal vector for all K users satisfying E (cid:0) ss H (cid:1) = I K , F RF of size N × N RF isthe analog beamformer realized by analog circuit (differentarchitectures incur different hardware constraints as will bediscussed later), F BB is the baseband digital precoder of size N RF × K satisfying the total transmit power constraint as k F RF F BB k F = ρ , where ρ is total transmit power. Finally, n ∼ CN (cid:0) , σ I K (cid:1) of size K × is the additive white Gaus-sian noise (AWGN) vector, where σ presents the noise power.For the channel vector h k of the k th user, we adopt thegeometric channel model to capture the characteristics ofmmWave massive MIMO channels as [5] h k = r NL k L k X l =1 α ( l ) k a (cid:16) ϕ ( l ) k , θ ( l ) k (cid:17) , (2)where L k denotes the number of paths for user k , α ( l ) k and ϕ ( l ) k ( θ ( l ) k ) for ≤ l ≤ L k are the complex gain and azimuth(elevation) angle of departure (AoD) of the path l for user k , a ( ϕ, θ ) presents the N × array steering vector. For thetypical uniform planar array (UPA) with N elements inhorizon and N elements in vertical ( N = N N ), we have [6] a ( ϕ, θ ) = a az ( ϕ ) ⊗ a el ( θ ) , (3)where a az ( ϕ ) = √ N (cid:2) e j πi ( d /λ ) sin ϕ (cid:3) T for i ∈ I ( N ) , a el ( θ ) = √ N (cid:2) e j πj ( d /λ ) sin θ (cid:3) T for j ∈ I ( N ) , I ( n ) = { , , · · · , n − } , λ is the signal wavelength,and d ( d ) is the horizontal (vertical) antenna spacing. AtmmWave frequencies, we usually have d = d = λ/ [14].III. E NERGY E FFICIENT H YBRID P RECODING
In this section, we first describe the proposed SI-basedhybrid precoding architecture. Then, we propose an ACE-based hybrid precoding scheme for this new architecture.Finally, the complexity analysis is provided.
A. The proposed SI-based hybrid precoding architecture
Fig. 1 (a) and (b) show the traditional precoding architec-tures, i.e., the one with finite-resolution phase shifters (PS-based architecture) [9] and the one with switches (SW-basedarchitecture) [11], respectively.As shown in Fig. 1 (a), the traditional PS-based architecturerequires a complicated phase shifter network, and the corre-sponding energy consumption can be presented as P PS − architecture = ρ + N RF P RF + N N RF P PS + P BB , (4) RF chainRF chainDigitalprecoder
Finite-resolutionphase shifters (a)(b)
RF ChainRF Chain
Switches
AnalogbeamformerDigitalprecoder Analogbeamformer (c)
RF ChainRF Chain
Inverters andswitches
Digitalprecoder Analogbeamformer
Fig. 1. Hybrid precoding: (a) traditional PS-based architecture; (b) traditionalSW-based architecture; (c) proposed SI-based architecture. where P RF , P PS , and P BB are the energy consump-tion of RF chain, finite-resolution phase shifter, and base-band, respectively. Note that although the PS-based archi-tecture enjoys high design freedom to achieve the near-optimal performance [9], it requires a large number (e.g., N × N RF = 64 ×
16 = 1024 [9]) of phase shifters. Moreover,the energy consumption of finite phase shifter is also con-siderable (e.g., P PS = 40mW for 4-bit phase shifter [11]),These make the traditional PS-based architecture still sufferfrom high energy consumption [14].By contrast, as shown in Fig. 1 (b), SW-based architec-ture can efficiently relieve such problem by employing asmall number ( N RF instead of N × N RF ) of energy-efficientswitches. The energy consumption of SW-based architecturecan be presented as P SW − architecture = ρ + N RF P RF + N RF P SW + P BB , (5)where P SW is the energy consumption of switch, which ismuch lower than P PS (e.g., P SW = 5mW [11]). Nevertheless,ince only N RF antennas are active simultaneously, SW-basedarchitecture cannot fully achieve the array gains of mmWavemassive MIMO, leading to an obvious performance loss [15].To overcome the problems faced by traditional architectures,we propose the SI-based architecture as shown in Fig. 1 (c),which can be considered as a better trade-off between thenear-optimal PS-based architecture and the energy-efficientSW-based architecture. Specifically, in the proposed SI-basedarchitecture, each RF chain is only connected to a sub antennaarray with M = N/N RF (assumed to be an integer) antennasinstead of all N antennas [16]. Moreover, each RF chain isconnected to the sub antenna array via only one inverter and M switches instead of N phase shifters. The energy consumptionof SI-based architecture can be presented by P SI − architecture = ρ + N RF P RF + N RF P IN + N P SW + P BB , (6)where P IN is the energy consumption of inverter. It worthpointing out that the inverters can be realized by the digitalchip with the energy consumption similar to switches (i.e., P IN ≈ P SW ) [15]. As a result, by comparing (4)-(6), we canconclude that the energy consumption of the proposed SI-based architecture is much lower than that of PS-based one.Furthermore, as all antennas are used, SI-based architecturecan also achieve the potential array gains of mmWave massiveMIMO, which will be further proved as follows.To do this, we need to first explain the hardware constraintsinduced by the proposed SI-based architecture, which aredifferent from those of the traditional ones. The first constraintis that the analog beamformer F RF should be a block diagonalmatrix instead of a full matrix as F RF = ¯f RF1 . . .
00 ¯f
RF2 ... . . . ... . . . ¯f RF N RF N × N RF , (7)where ¯f RF n is the M × analog beamformer on the n th subantenna array. The second constraint is that since only invertersand switches are used, all the N nonzero elements of F RF should belong to √ N {− , +1 } . (8)Based on these constraints, we have the following Lemma 1 . Lemma 1 . Assume that the channel h k of user k only has sin-gle path, i.e., L k = 1 [12]. When N → ∞ and N/M = N RF ,the ratio ζ between the array gains achieved by SI-basedarchitecture and that achieved by PS-based architecture withsufficiently high-resolution phase shifters can be presented by lim N →∞ , NM = N RF ζ = 4 N RF π . (9) Proof:
For the traditional PS-based architecture with suf-ficiently high-resolution phase shifters, the phases of theelements in the analog beamformer can be arbitrarily adjustedto capture the power of h k . Therefore, the array gains achieved by PS-based architecture is (cid:12)(cid:12)(cid:12) α (1) k (cid:12)(cid:12)(cid:12) . By contrast, the arraygains achieved by SI-based architecture can be presented by (cid:12)(cid:12) h Hk f RF k (cid:12)(cid:12) = N (cid:12)(cid:12)(cid:12) α (1) k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) a H ( ϕ k ) f RF k (cid:12)(cid:12) (10) = 1 N (cid:12)(cid:12)(cid:12) α (1) k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 e j ¯ φ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 N (cid:12)(cid:12)(cid:12) α (1) k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 cos (cid:0) ¯ φ m (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 sin (cid:0) ¯ φ m (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where f RF k is the k th column of F RF including the zeros and ¯ φ m denotes the phase quantization error. Since the nonzeroelements in f RF k belong to √ N {− , +1 } , ¯ φ m can be as-sumed to follow the uniform distribution U ( − π/ , π/ for ≤ m ≤ M [10]. Then, we have lim N →∞ N/M = N RF ζ = 1 N RF M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 cos (cid:0) ¯ φ m (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 sin (cid:0) ¯ φ m (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 N RF (cid:0) E (cid:2) cos (cid:0) ¯ φ m (cid:1)(cid:3)(cid:1) + (cid:0) E (cid:2) sin (cid:0) ¯ φ m (cid:1)(cid:3)(cid:1) = 4 N RF π , (11)which verifies the conclusion in Lemma 1 . Lemma 1 indicates that although the proposed SI-basedarchitecture suffers from some loss of array gains comparedto the near-optimal PS-based architecture, the performanceloss keeps constant and limited, which does not grow as thenumber of BS antennas goes to infinity. Recalling the lowenergy consumption of SI-based architecture, we can concludethat our scheme is a better trade-off between the traditionalarchitectures, which will be also verified by simulation. Next,we will design a near-optimal hybrid precoding scheme for SI-based architecture with quite different hardware constraints.
B. ACE-based hybrid precoding scheme
We aim to design the analog beamformer F RF and thedigital precoder F BB to maximize the achievable sum-rate R ,which can be presented as (cid:0) F optRF , F optBB (cid:1) = arg max F RF , F BB R, s . t . F RF ∈ F , k F RF F BB k F = ρ, (12)where F denotes the set with all possible analog beamformerssatisfying the two constraints (7) and (8) described above, andthe achievable sum-rate R can be presented by R = K X k =1 log (1 + γ k ) , (13)where γ k presents the signal-to-interference-plus-noise ratio(SINR) of the k th user as γ k = (cid:12)(cid:12) h Hk F RF f BB k (cid:12)(cid:12) K P k ′ = k (cid:12)(cid:12) h Hk F RF f BB k ′ (cid:12)(cid:12) + σ , (14)here f BB k is the k th column of F BB .It is worth pointing out that the constraints (7) and (8) onthe analog beamformer F RF are non-convex. This makes (12)very difficult to be solved. Fortunately, as all the N nonzeroelements of F RF belong to the set √ N {− , +1 } , the numberof possible F RF is finite. Therefore, (12) can be regarded as anon-coherent combining problem [13]. To solve it, we can firstselect a candidate F RF , and compute the optimal F BB accord-ing to the effective channel matrix HF RF without non-convexconstraints. After all possible F RF ’s have been searched, wecan obtain the optimal analog beamformer F optRF and digitalprecoder F optBB . However, such exhaustive search scheme re-quires to search N possible F RF ’s and F BB ’s, which involvesunaffordable complexity as N is usually large in mmWavemassive MIMO systems (e.g., N = 64 , ≈ . × ).To solve this problem, we propose an ACE algorithm, whichcan be considered as an improved version of the CE algorithmdeveloped from machine learning [13].At first, we would like to briefly introduce the conventionalCE algorithm, which is a probabilistic model-based algorithmto solve the combining problem by an iterative procedure. Ineach iteration, the CE algorithm first generates S candidates(e.g., possible hybrid precoders in our problem) according to aprobability distribution. Then, it computes the objective value(e.g., achievable sum-rate in our problem) of each candidate,and selects S elite best candidates as “elite” [13]. Finally, basedon the selected elites, the probability distribution is updated byminimizing the CE. Repeating such procedure, the probabilitydistribution will be refined to generate a solution close tothe optimal one with a sufficiently high probability. However,although the CE algorithm has been widely used in machinelearning [13], it still has some disadvantages. One is that thecontributions of all elites are treated as the same. Intuitively,the elite with better objective value should be more importantwhen we update the probability distribution. Therefore, if wecan adaptively weight the elites according to their objectivevalues, better performance can be expected. Following thisidea, we propose an ACE algorithm to solve (12).The pseudo-code of the proposed ACE-based hybrid pre-coding scheme is summarized in Algorithm 1 , whichcan be explained as follows. At the beginning, we for-mulate the nonzero elements in F RF as N × vector f = h(cid:0) ¯f RF1 (cid:1) T , (cid:0) ¯f RF2 (cid:1) T , · · · (cid:0) ¯f RF N RF (cid:1) T i T , and set the probabilityparameter u = [ u , u , · · · , u N ] T as an N × vector, where ≤ u n ≤ presents the probability that f n = 1 / √ N , f n isthe n th element of f . Then, by initializing u (0) = × N × ( is the all-one vector), we assume that all the N nonzeroelements of F RF belong to √ N {− , +1 } with equal prob-ability, since no priori information is available. During the i th iteration, in step 1, we first generate S candidate analogbeamformers { F s RF } Ss =1 based on the probability distribution Ξ (cid:0) F ; u ( i ) (cid:1) (i.e., generate { f s } Ss =1 according to u ( i ) , andreshape them as matrices belong to F ). Then, in step 2, we Note that the convergence of the proposed ACE-based hybrid precodingscheme can be proved by extending the Theorem 1 in [17].
Input:
Channel matrix H ; Number of iterations I ;Number of candidates S ; Number of elites S elite . Initialization : i = 0 ; u (0) = × N × . for ≤ i ≤ I
1. Randomly generate S candidate analog beamformers { F s RF } Ss =1 based on Ξ (cid:0) F ; u ( i ) (cid:1) ;2. Compute S corresponding digital precoders { F s BB } Ss =1 based on the effective channel H s eq = HF s RF ;3. Calculate the achievable sum-rate { R ( F s RF ) } Ss =1 (13);4. Sort { R ( F s RF ) } Ss =1 in a descend order as R (cid:16) F [1] RF (cid:17) ≥ R (cid:16) F [2] RF (cid:17) ≥ · · · ≥ R (cid:16) F [ S ] RF (cid:17) ;5. Select elites as F [1] RF , F [2] RF , · · · , F [ S elite ] RF ;6. Calculate weight w s for each elite F [ s ] RF , ≤ s ≤ S elite ;7. Update u ( i +1) according to { w s } S elite s =1 and n F [ s ] RF o S elite s =1 ;8. i = i + 1 ; end forOutput: Analog beamformer F [1] RF ; Digital precoder F [1] BB . Algorithm 1:
The proposed ACE-based hybrid precodingcalculate the corresponding digital precoder F s BB according tothe effective channel H s eq = HF s RF for ≤ s ≤ S . Note thatthere are lots of advanced digital precoder schemes [3]. In thispaper, we adopt the classical ZF digital precoder with the near-optimal performance and low complexity as the example [3],and F s BB can be computed as G s = (cid:0) H s eq (cid:1) H (cid:16) H s eq (cid:0) H s eq (cid:1) H (cid:17) − , F s BB = β s G s , (15)where β s = √ ρ/ k F s RF G s k F is power normalized factor.After that, in step 3, the achievable sum-rate { R ( F s RF ) } Ss =1 are calculated by substituting F s RF and F s BB (also a functionof F s RF ) into (13). We sort { R ( F s RF ) } Ss =1 in a descend orderin step 4. Then, the elites can be obtained in step 5. In theconventional CE algorithm, the next step is to using elites toupdate u ( i +1) by minimizing CE, which is equivalent to [13] u ( i +1) = arg max u ( i ) S S elite X s =1 ln Ξ (cid:16) F [ s ]RF ; u ( i ) (cid:17) , (16)where Ξ (cid:16) F [ s ]RF ; u ( i ) (cid:17) denotes the probability to generate F [ s ]RF .As mentioned above, in (16), the contributions of all elites aretreated as the same, leading to performance degradation. Tosolve this problem, we propose to weight each elite adaptivelybased on its achievable sum-rate. Specifically, we first definean auxiliary parameter T presenting the average achievablesum-rate of all elites as T = 1 S elite X S elite s =1 R (cid:16) F [ s ] RF (cid:17) . (17)Then, the weight w s of the elite F [ s ] RF can be calculated instep 6 as w s = R (cid:16) F [ s ] RF (cid:17) /T . Based on { w s } S elite s =1 , (16) can bemodified as u ( i +1) = arg max u ( i ) S S elite X s =1 w s ln Ξ (cid:16) F [ s ]RF ; u ( i ) (cid:17) . (18)ote that Ξ (cid:16) F [ s ] RF ; u ( i ) (cid:17) = Ξ (cid:0) f [ s ] ; u ( i ) (cid:1) , and the n th element f [ s ] n of f [ s ] is a Bernoulli random variable, where f [ s ] n = 1 / √ N with probability u ( i ) n and f [ s ] n = − / √ N with probability − u ( i ) n . Therefore, we have Ξ (cid:16) F [ s ]RF ; u ( i ) (cid:17) = N Y n =1 (cid:16) u ( i ) n (cid:17) ( √ Nf [ s ] n ) (cid:16) − u ( i ) n (cid:17) ( −√ Nf [ s ] n ) . (19)After substituting (19) into (18), the first derivative of thetarget in (18) with respect to u ( i ) n can be derived as S S elite X s =1 w s √ N f [ s ] n u ( i ) n − − √ N f [ s ] n (cid:16) − u ( i ) n (cid:17) . (20)Setting (20) to zero, u ( i +1) can be updated in step 7 as u ( i +1) n = P S elite s =1 w s (cid:16) √ N f [ s ] n + 1 (cid:17) P S elite s =1 w s . (21)Such procedure above will be repeated ( i = i + 1 in step 8)until the maximum number of iterations I is reached, and theanalog beamformer and digital precoder will be selected as F [1] RF and F [1] BB , respectively. Finally, it is worth pointing outthat the proposed ACE-based hybrid precoding scheme canbe also extended to the scenario where users employ multipleantennas. In this case, the analog beamformers at the BS andusers should be jointly searched by the ACE algorithm. C. Computational complexity analysis
In this subsection, the computational complexity of theproposed ACE-based hybrid precoding scheme is discussed.From
Algorithm 1 , we can observe that the complexityof the ACE-based hybrid precoding scheme mainly comesfrom steps 2, 3, 6, and 7. In step 2, we need to compute S effective channel matrices (cid:8) H s eq (cid:9) Ss =1 and digital precoders { F s BB } Ss =1 according to (15). Therefore, the complexity ofthis part is O (cid:0) SN K (cid:1) . In step 3, the achievable sum-rate ofeach candidate is computed. Since we employ the digital ZFprecoder, the SINR γ sk of the k th user for the s th candidate issimplified to γ sk = ( β s /σ ) . As a result, this part only involvesthe complexity O ( S ) . In step 6, we calculate S elite weightsbased on (17), which is quite simple with the complexity O ( S elite ) . Finally, in step 7, the probability parameter u ( i +1) is updated according to (21) with the complexity O ( N S elite ) .In summary, after I iterations, the total computationalcomplexity of the proposed ACE hybrid precoding scheme is O (cid:0) ISN K (cid:1) . Since K is usually small, I and S also do nothave to be very large as will be verified in the next section, wecan conclude that the complexity of the proposed ACE-basedhybrid precoding scheme is acceptable, which is comparablewith the simple least squares (LS) algorithm.IV. S IMULATION R ESULTS
In this section, we provide the simulation results in termsof achievable sum-rate and energy-efficiency to evaluate theperformance of the proposed ACE-based hybrid precoding −20 −15 −10 −5 0 5 10051015202530
SNR (dB) A c h i e v ab l e s u m − r a t e ( bp s / H z ) Fully digital ZF precodingConventional two−stage hybrid precodingConventional AS−based hybrid precodingProposed CE−based hybrid precodingProposed ACE−based hybrid precoding
Fig. 2. Achievable sum-rate comparison.
Number of iterations I A c h i e v ab l e s u m − r a t e ( bp s / H z ) S = 50 S = 100 S = 150 S = 200 S = 250 Fig. 3. Achievable sum-rate against S and I . scheme. The simulation parameters are described as follows:We assume that the BS employs an UPA with antennaspacing d = d = λ/ . For the k th user, we generate thechannel h k based on (2), where we assume: 1) L k = 3 ; 2) α ( l ) k ∼ CN (0 , for ≤ l ≤ L k ; 3) ϕ ( l ) k and θ ( l ) k follow theuniform distribution U ( − π, π ) for ≤ l ≤ L k [12]. Finally,the signal-to-noise ratio (SNR) is defined as ρ/σ .Fig. 2 shows the achievable sum-rate comparison ina typical mmWave massive MIMO system with N = 64 , N RF = K = 4 . In Fig. 2, the proposed CE-based (i.e., usingthe conventional CE algorithm to solve (12)) and ACE-based hybrid precoding schemes are designed for SI-basedarchitecture, where we set S = 200 , S elite = 40 , and I = 20 for Algorithm 1 , the conventional two-stage hybrid precodingscheme is designed for PS-based architecture with 4-bit phaseshifters [9], and the conventional antenna selection (AS)-basedhybrid precoding scheme is designed for SW-based architec-ture [11] with switches. Firstly, we can observe from Fig. 2that the proposed ACE algorithm outperforms the traditionalCE algorithm, where the SNR gap is about 1 dB. Note thatthe ACE algorithm only involves one additional step (i.e., step6 in
Algorithm 1 ) with negligible complexity. Therefore, theproposed ACE algorithm is more efficient. Moreover, Fig. 2also shows that the proposed ACE-based hybrid precoding can
Number of users K E ne r g y e ff i c i en c e ( bp s / H z / W ) Fully digital ZF precodingConventional two−stage hybrid precodingConventional AS−based hybrid precodingProposed ACE−based hybrid precoding
Fig. 4. Energy efficiency comparison. achieve the sum-rate much higher than the conventional AS-based hybrid precoding, as it can achieve the potential arraygains in mmWave massive MIMO systems. Finally, we canobserve that the performance gap between ACE-based hybridprecoding and two-stage hybrid precoding is limited and keepsconstant, which further verifies the conclusion in
Lemma 1 .Fig. 3 shows the achievable sum-rate of the proposed ACE-based hybrid precoding against the number of candidates S and the number of iterations I , when S elite /S = 0 . , N = 64 , N RF = K = 4 , and SNR = 10 dB. From Fig. 3, we canobserve that when S is small, increasing S will lead to anobvious improvement in the sum-rate performance. However,when S is sufficiently large, such trend is no longer obvious.This indicates that the number of candidates S does not haveto be very large, e.g., S = 200 is enough. Furthermore, Fig. 3also indicates that the proposed ACE-based hybrid precodingcan converge with a small number of iterations, e.g., I = 20 .These observations verify the rationality of the parameters for Algorithm 1 we used in Fig. 2.Fig. 4 shows the energy efficiency comparison when N = 64 is fixed and N RF = K varies from 1 to 16. Theparameters for Algorithm 1 are the same as Fig. 2. Ac-cording to [15], [16], the energy efficiency can be definedas the ratio between the achievable sum-rate and the energyconsumption, which should be (4), (5), and (6) for two-stage hybrid precoding, AS-based hybrid precoding, and ACE-based hybrid precoding, respectively. In this paper, we adoptthe practical values ρ = 30mW [16], P RF = 300mW [16], P BB = 200mW [15], P PS = 40mW (4-bit phase shifter) [15],and P SW = P IN = 5mW [15]. From Fig. 4, we can observethat the proposed ACE-based hybrid precoding with SI-basedarchitecture can achieve much higher energy efficiency thanthe others, especially when K is not very large (e.g., K ≤ ).Furthermore, it is interesting to observe that when K ≥ , theenergy efficiency of the two-stage hybrid precoding with PS-based architecture is even lower than that of the fully digital ZFprecoding. This is due to the fact that as K grows, the numberof phase shifters in PS-based architecture increases rapidly. Asa result, the energy consumption of the phase shifter networkwill be huge, even higher than that of RF chains. V. C ONCLUSIONS
In this paper, we propose an energy-efficient SI-basedhybrid precoding architecture, where the analog part is realizedby a small number of switches and inverters. The performanceanalysis proves that the performance gap between the proposedSI-based architecture and the traditional near-optimal onekeeps constant and limited. Then, by employing the idea of CEoptimization in machine learning, we further propose an ACE-based hybrid precoding scheme with low complexity for SI-based architecture. Simulation results verify that our schemecan achieve the satisfying sum-rate performance and muchhigher energy efficiency than traditional schemes.R
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