Distributed Continuous Aperture Arrays for Multiuser SWIPT

Distrib uted Continuous Aper ture Arrays f or Multiuser SWIPT Muhammad Zeesha n Mumtaz, Mohammadali Mohammadi, Hien Quoc Ngo, a nd Michail Matthaiou Centre for W ireless Innovati on (CWI), Queen’ s Uni versity Belfast, U.K. Email: { mmumtaz01, m.mohammadi, hien.ngo, m.ma tthaiou } @qub .ac .uk Abstract —This paper pro poses a distributed continuous aper- ture array (D-CAP A) to support simultaneous wireless infor - mation and power transfer (SWIP T ) to multip le information users (IUs) and energy users (EUs). Each metasurface sup- ports continuous surface curr ents that radiate electro magnetic (EM) wa ves for in fo rmation an d energy transmission to the users. These wav es propagate t h rough conti n uous EM chann els characterized by the dyadic Gre en’ s function. W e f ormulate a system power consumption (PC) min imization problem subject to spectral efficiency and energy harvesting quality-of-service ( QoS) requirements, where the QoS requirem ents are deriv ed und er the equal power allocation (EP A) scheme. An efficient t wo-layer optimization algorithm is developed to solve this problem by optimizing the powe r allocation subject to the QoS violation penal- ties using augmented Lagrangian transformation. Our numerical results show th at well-optimized current distributions ov er each metasurface in the proposed D-CAP A achiev e up to 65% and 61% reductions in overa ll system PC compared to the EP A and co-located CAP A (C-CAP A) cases, while maintaining the same total aperture size and transmission power . Index T erms —Contin uous aperture array (CAP), power con- sumption (PC), simu ltaneous wireless inform ation and power transfer (SWIPT). I . I N T RO D U C T I O N Continuou s-aperture array (CAP A) transcei vers synthesize continuo us cur rent distributions b y replacing spatially discrete arrays (SPDAs ) with a field-c e n tric manifestation that emu lates the classical Max w e ll’ s EM propag ation mo del [1], [2]. By manipulatin g these cur rent distributions over the p hysical aper- ture, CAP A architectures un lo ck precise energy focu sing and enhanced spatial d egrees o f freedo m com pared to the e le m ent- spaced a r rays of com p arable size [ 3], [ 4]. T h ese uniq ue func- tionalities translate into lo w e r system PC as the transmitted energy is ac curately concen tr ated o n the network user locations with red uced spillover . The EM field-based in te r pretation, that underpins the energy- focusing capability of CAP As, enables u s to pursue SWIPT This work was support ed by the UK E nginee ring and Physical Sciences Researc h Council (EPSRC) grant EP/X04047X/ 2 for TIT AN T elecoms Hub . The work of H. Q. Ngo was supported by the U.K. Research and Innov ation Future Leaders Fellowshi ps under Grant MR/X010635/1 , and a research grant from the Department for the Economy Northern Ireland under the US-Ireland R&D Part nership Programme. This work was supported by the E uropean Researc h Council (ERC) under the European Union’ s Horizon 2020 research and innov ation programme (grant agree ment No. 101001331). M. Z. Mumtaz is also with the Colle ge of Aeronautic al Engineering , Nationa l Uni versity of Scie nces & T echnology (NUST ), Pakistan, (email: zmumtaz@c ae.nust.edu.pk). in commu nication networks [ 5]. In particular, b y effecti ve utilization of these pr opagation character istics, the c urrent distributions over these metasurfaces can be exploited in a manner to a c h iev e EM signal concen tration for both EUs and IUs, while mitigatin g inter-user-interference (IUI) fo r I Us with aggressive sidelo be sup pression [6] . Recently , CAP A technology has attracted significant resear ch interest, including the hardware imp lementation of shared aper- ture metasurfaces [7] . Howe ver, most of the rela ted works hav e considered the c ase of a single CAP A sur face for p roviding services in commu nication networks. I n [8], W ang et a l. c o n- ceiv ed the optimal beamform ing desig n s for a multiuser single- CAP A system with the aim of p ower minimizatio n. The authors also demo nstrated its super ior perfo r mance again st the SPD A variant. In [9 ] , the au thors av ailed of the c a lculus of variation (CoV) techn ique to handle the continu ous na tu re of the ene rgy distribution over a CAP A surface fo r a multigro up mu lticast network, while substantially improving the energy efficiency (EE). Similar ly , metasur face-based ho lograph ic MI M O (H- MIMO) was considered in [5 ] for suppo rting integrated data and energy transfer by using finite-item Fourier series approx - imation for the continuou s beam forming function s. Howev er , these studies d o no t address the effects of distributed arch itec- tures on the PC reductio n. For the given overall apertu r e size and transmission power of the CAP A metasurfaces, we assess the pe r forman ce g ains of spatially D- CAP A over its C-CAP A c o unterpa rt. The m ain contributions of this paper are as follows: • W e propose a D-CAP A system that delivers SWIPT ser- vices using continuou s transmission-current signals over dyadic Green’ s function - b ased EM channels for both IUs and EUs. Th is architectu re e n ables p recise EM wav e manipulatio n across spatially distributed metasur faces, thereby enhancin g both energy transfer efficiency a n d commun ication performan c e. • A dual-routine optimization framew ork is developed to addr ess the PC minimization pr oblem while ensur- ing complian ce with th e bench mark spectral-efficiency (SE) and harvested-energy (HE) QoS requirem ents. T he propo sed method employs the limited-memory Broy- den–Fletche r–Goldfarb– Shanno algorithm with boun d constraints (L-BFGS-B) to optimize p ower allocatio n, while an augmented-Lagrangian (AL) mech a n ism is used to man age Qo S constrain t violation s. • Our nu merical simulation results verify a substan tial re- duction in the collective PC o f the considered o ptimized D-CAP A system in c omparison to the C-CAP A case. This p erform ance enhanc e ment is attributed to the pre- cise b eamform ing ca p ability ac h iev ed due to the fo cused energy zones over the continuou s metasurfaces by the propo sed optimization techn ique. Notations: W e u se bold lower case and up per case letters to denote vectors an d matrices, respectively . The superscrip ts ( · ) ∗ , ( · ) T , a nd ( · ) − 1 denote the conjugate, transpo se, and in verse of a matrix, re sp ectiv e ly ; I N represents the N × N identity matr ix ; k · k return s the n orm of a vector; E {·} rep resents the statistical expectation. Finally , ∇ f ( · ) pe r forms the gradient o peration on a f unction f ( · ) . I I . S Y S T E M M O D E L W e co nsider a multi-metasurface system in volving S spa- tially distributed co ntinuou s aperture a r rays which are indexed by s ∈ S = { 1 , . . . , S } , each having an ef fectiv e aperture S ( s ) T with occu pied area A ( s ) T = |S ( s ) T | = ¯ A T /S , wh ere ¯ A T is the constant to tal aper ture. It is further assum e d that each metasurface is capable to synthesize any curren t distribution posed on its aperture. The downlink operation of these surfaces simultaneou sly serves K = L + M users, repr esented b y the un ion set U = D ∪ E . Here, D = { 1 , . . . , L } denotes the IUs, while E = { 1 , . . . , M } d enotes the EUs. Now , the network to p ology is ba sed o n uniform ly distributed CAP A surfaces with cen ter location s c s ∈ R 3 × 1 . Moreover, the K users are un iformly distributed in the three-dim e nsional (3-D) space surro unding the distrib u ted CAP A surfaces, which ar e located on th e plane z = 0 . A. CAP A T ransmission S ignal an d P ower Con straints Each CAP A sur face operates with contin uous current signals which is the sup e rposition of cu rrent distributions intended fo r each user stream [5 ] . For the CAP A surface s , this pheno menon can b e mathem atically represen ted as J s ( u , Ω ) = X K j =1 Ω sj θ sj ( r j , u ) x sj , (1) where Ω sj ∈ Ω is the wa ve a mplitude which serves as the cor- respond in g P A coefficient, w h ile Ω denotes the matrix of all the P A coefficients; x s = [ x s, 1 , . . . , x s,L , x s,L +1 , . . . , x s,L + M ] T ∈ C K denotes the transmission symbo l vector f ollowing a unit- variance Gaussian distribution, while satisfying the conditio n E { x s x H s } = I K ; θ sj ( r j , u ) , ∀ u ∈ S ( s ) T , is the beamf ormer at- tributed to the current d ensity componen t fo r the mo dulation of symbol x sj intended for user j located at position r j ∈ R 3 × 1 . Additionally , this composite transmitted signal is bounde d by the available power budget at eac h CAP A surface a s Z S ( s ) T k J s ( u , Ω ) k 2 d u ≤ P s , (2) where P s = P t /S is th e maximu m tr ansmit p ower av ailab le at each sur face out of th e overall CAP A system power P t . B. EM Chan nel Model The superimp osed electric field e ( r ) at a position r = ( r x , r y , r z ) ∈ R 3 × 1 can be o btained as [10 ] e ( r ) = X S s =1 Z S ( s ) T G s ( r , u ) J s ( u , Ω ) d u , (3) where G s ( r , u ) ∈ C 3 × 3 is the dya d ic f ree-space Green’ s function used to char acterize the chan n el matrix fro m th e surface aper ture u ∈ S ( s ) T to the user location r k , given by G s ( r , u ) = j κZ 0 4 π e j κ k p k k p k h  I 3 − ˆ p ˆ p H  + j κ k p k  I 3 − 3 ˆ p ˆ p H  + 1 ( κ k p k ) 2  I 3 − 3 ˆ p ˆ p H  i , (4) where p = r − u , with th e u nit vector defined as ˆ p = p / k p k , κ = 2 π / λ is the characteristic wav e number with λ bein g the carrier wa veleng th, while Z 0 represents the free-space impedanc e. The three difference terms in the d e finition o f the Green’ s function relate to the far -field, the middle-field and the near-field EM radiatio n region s. Con sider this function , the continuo us EM chann el from the sur face s to a cer tain user k can be defined by using the co- polar comb ining comp o nents of th e tr i- polarized CAP A framework as h sk ( u ) , ˆ i T k G ( r k , u ) ˆ i s + ˆ j T k G ( r k , u ) ˆ j s + ˆ k T k G ( r k , u ) ˆ k s , ( 5) where h ˆ i s , ˆ j s , ˆ k s i and h ˆ i k , ˆ j k , ˆ k k i are unit vectors fo r the CAP A transmission surfaces and receptio n antennas, r espectiv ely . C. B eamforming Design The continuo us-apertu re fo rmulation allows us to sy nthesize user-specific current distrib utions associated to the beamfo r mer θ sj ( r j , u ) over the span of the co ntinuous E M channels h sk ( u ) . Math ematically , these beamf o rmers are defin e d as θ sj ( r j , u ) , θ sj ( u ) ˆ p w ith nor malized coefficients given θ sj ( u ) = P K k =1 b j k h sk ( u ) q b H j A ( s ) b j , (6) where A ( s ) , [ A ( s ) kk ′ ] k,k ′ ∈ C K × K denotes the full-rank channel correlation m atrix whose in dividual elements encode the cor- relation of c h annel realizatio ns induced on th e CAP A sur face, defined as [ 11, Eq. (6)] A ( s ) kk ′ , Z S ( s ) T h sk ( u ) h ∗ sk ′ ( u ) d u . (7) Note that, b j in (6) repre sents the pr e coding vectors, which can be designed by utilizing linear tech niques, i.e. , regularized zero-fo rcing ( RZF) an d maxim u m ratio transmission (MR T) f or the IUs and EUs, respe c ti vely . For RZF preco ding, we extract the IU block fr om the composite chan nel cor relation matrix as A L ⊂ P S s =1 A ( s ) ∈ C L × L , which is u sed to con struct the matrix B L = ( A L + α ZF I L ) − 1 , whose column elements are used as IU precodin g coefficients b j . The regularization factor α ZF > 0 trades b etween conventional ZF and MR T func tion- alities. On the oth er hand, the EUs emp loy the MR T techn ique derived from vector b j = e j , where e j is the can onical b asis vector with e j m = δ j m . These MR T co efficients perfectly align the incident energy strea m s to the corr esponding EM channels. D. Received Sig nal Each user can b e assumed as a point in the 3-D radiatio n space of the CAP A transmission system, since the effecti ve aperture o f th e r eceiv er is negligible in comp arison to the transmission f ramework, i.e., A R = λ 2 / 4 π ≪ A ( s ) T . T he user k located at the position r k receives the signal ( from (3)) y k = S X s =1  Ω sk x sk J sk ( G , θ ) + K X j 6 = k Ω sj x sj J sj ( G , θ )  + n k , (8) where the fu nction J sk ( G , θ ) = R S ( s ) T L sk ( G , θ ) d u is de- fined over the continu ously dif ferentiable f unction giv e n as L sk ( G , θ ) = ˆ i T k G ( r k , u ) θ sk ( u ) ˆ i s + ˆ j T k G ( r k , u ) θ sk ( u ) ˆ j s + ˆ k T k G ( r k , u ) θ sk ( u ) ˆ k s . Moreover, n k represents the EM noise with zero -mean Gau ssian distributed entr ies n k ∼ C N (0 , σ 2 ) . E. Downlink SE and HE A nalysis For a giv en IU l , the co rrespond ing signal-to-interf erence- plus-noise ratio (SINR) can be derived from (8 ) as Γ l ( G , θ , Ω ) = P S s =1 Ω 2 sk |J sk ( G , θ ) | 2 P S s =1 P K j =1 ,j 6 = k Ω 2 sj |J sj ( G , θ ) | 2 + σ 2 k . (9) According ly , the achievable SE (in bit/s/Hz) can be ex- pressed as R l = log 2 (1 + Γ l ( G , θ , Ω )) . (10) The p ower density incident o n th e anten na o f a EU receiv er m c a n be ev alu a ted as S ( G , θ , Ω ) = 1 2 Z X S s =1 X K j =1 Ω 2 sj |J sj ( G , θ ) | 2 , (11) where Z is the wave impedance of the receiver . Using the Poynting’ s the o rem, the RF power harvested at EU m can b e derived as Q m ( G , θ , Ω ) = Z S ( s ) T S ( G , θ , Ω ) ˆ η · d u = A R cos ϕ m 2 Z X S s =1 X K j =1 Ω 2 sj |J sj ( G , θ ) | 2 , (12) where ˆ η represen ts the norm al vector to the rec ei ver antenna, while ϕ m ∈ [0 , π / 2] d enotes the including angle b etween ˆ η and th e Poynting vecto r at the m -th u ser lo cation. The practical implemen tation of EH corr esponds to th e n on- linear h arvesting energy (NL-HE) Q NL m , w h ich can be modeled using th e following logistic fun ction [12] Q NL m ( G , θ , Ω ) = Q max ς  1 + e − a ( Q m ( G , θ , Ω ) − b )  − ζ , (13) where ς = e ab / (1 + e ab ) and ζ = Q max /e ab with a and b are the EH rectifier param eters, while Q max denotes the satu r ation power of this rectifier circu it. I I I . P O W E R C O N S U M P T I O N M I N I M I Z A T I O N D E S I G N A. Optimization Pr oblem F ormulation The goal of the optimizatio n p roblem is to minimiz e the overall system PC with respect to the beamf ormers, while ensuring that th e SE and H E QoS re quiremen ts—deriv ed from the EP A bench mark—are satisfied fo r all users associated with each CAP A surface. Mathem a tically , this optimization problem with continuo us integra l functio n s-based Qo S constraints can be fo rmulated as min { Ω } X S s =1 X K j =1 Ω 2 sj (14a) s.t. Γ l ( G , θ , Ω ) ≥ Γ EP A l , ∀ l = 1 , 2 , . . . , L , (14b) Q m ( G , θ , Ω ) ≥ Q EP A m , ∀ m = 1 , 2 , . . . , M , (14c) X K j =1 Ω 2 sj ≤ P s , ∀ s = 1 , 2 , . . . , S. (14d) For (14a ) and (1 4d), it is assumed that the beamform e rs satisfy the c ondition R S ( s ) T k θ sj ( r j , u ) k 2 d u = 1 . Moreover, the threshold SE ( Γ EP A l ) and HE ( Q EP A m ) in (1 4b) and (14c) have been computed fo r th e EP A case by using the P A coefficients Ω EP A sj = P t / ( S K ) . B. A ugmented- Lagrangian Optimization F ramew ork Considering the o ptimization problem in (1 4a), we lev er- age the AL assist ed quasi-Newton method based L-BFGS-B approa c h [13], [14]. This pro blem ca n be transformed into a finite sum minim ization pro blem by employing the AL approa c h , which combin es the continu ous co nstraint functions in (14 b) and (14c ). Mathematically , this AL transform a tion c an be de fin ed with respect to the P A coefficients choices ( Ω ) as f ( Ω ) , S X s =1 K X j =1 Ω 2 s,j + λ T SE v SE ( G , θ , Ω ) + ρ 2   v SE ( G , θ , Ω )   2 | {z } IU SI NR penalties + β  λ T EH v HE ( G , θ , Ω ) + ρ 2   v HE ( G , θ , Ω )   2  | {z } EU HE penalties , (15) where, v SE ( G , θ , Ω ) = Γ EP A − Γ ( G , θ , Ω ) and v HE ( G , θ , Ω ) = Q EP A − Q ( G , θ , Ω ) represent the in equality residual vecto rs for I U SINR and E U HE EP A targets, which are arrang ed as Γ EP A and Q EP A , respectively; ρ is the AL penalty; λ SE and λ EH are the Lagran ge multiplier s, and β > 0 balances the IU an d EU pe nalties conside ring the c o rrespon d ing signal power differences. For the box constraints 0 ≤ Ω s,j ≤ √ P s , we use the L-BFGS-B technique to com pute the pro jected quasi-Newton steps of the fo r m Ω t +1 = Π [0 , √ P s ]  Ω t + α t p t  , (16) with search direction d efined a s p t = − H t ∇ f ( Ω t ) , while Π [0 , √ P s ] ( Φ t ) = a rg min Ω ∈ R + {k Ω t − Φ t k} projects the term Φ t = Ω t + α t p t to the closest p oint within the feasible inter val using E uclidean no rm. Here, ∇ f ( Ω t ) ∈ R S × K represents the gra dient of th e Lagrang ian function defined in ( 15) ca lc u lated for th e step t . As both SE ( Γ ) and HE ( Q ) are co mbination s of continuou s in- tegral function s with linear depe n dence on the P A coefficients, the Lang rangian function in (1 5) can be d ifferentiated using the cha in rule. Each e le m ent of ∇ f ( Ω ) can be represented as partial d eriv ative w ith respect to a particular P A v ariable Ω s,j as ∂ f ( Ω ) ∂ Ω s,j =2 Ω s,j − X L l =1 w l ∂ Γ l ( G , θ , Ω ) ∂ Ω s,j − X M m =1 u m ∂ Q m ( G , θ , Ω ) ∂ Ω s,j , (17) where, w l , λ SE ,l + ρ v SE ,l and u m , λ HE ,m + ρ v HE ,m , while v SE ,l ∈ v SE ( G , θ , Ω ) and v HE ,m ∈ v HE ( G , θ , Ω ) are the ineq uality re sid u als for IU l and E U m , respectively . The partial deriv ative of the SE component can be derived using the der iv ative q uotient rule as ∂ Γ ∂ Ω s,j =          2 Ω s,l |J sl ( G , θ ) | 2 P S s =1 P K j =1 ,j 6 = l Ω 2 sj |J sj ( G , θ ) | 2 + σ 2 l , j = l , 2 Ω s,j |J sj ( G , θ ) | 2 P S s =1 Ω 2 sl |J sl ( G , θ ) | 2  P S s =1 P K j =1 ,j 6 = l Ω 2 sj |J sj ( G , θ ) | 2 + σ 2 l  2 , j 6 = l . (18) Now , the partial deriv ati ve of the HE component can be giv en as ∂ Q m ( G , θ , Ω ) ∂ Ω s,j = 2 A R cos ϕ m Ω sj |J sj ( G , θ ) | 2 2 Z . (19) Substituting the expression s in ( 18) and (19 ) in (17), we ob - tain the c losed-for m expre ssion fo r the gradient o f Lagrangian function in (20 ) at the top of the n ext page. On th e other hand, the m a trix H t in (16) represents a semi- positive definite limited-mem ory approxima tio n fo r in verse- Hessian ∇ 2 f ( Ω t ) − 1 constructed fr om the last q curvature pairs { ( s i , y i ) } t − 1 i = t − q in ord e r to av o id intense c o mputation al requirem ents fo r second order deriv a tives. This appr oximation ev aluates the BDFS updates using the following relatio nship [13, Eq . (6) ] , H t +1 =  I q − s t y T t y T t s t  H t  I q − y t s T t y T t s t  + s t s T t y T t s t , (21 ) where, s t , Ω t +1 − Ω t and y t , ∇ f ( Ω t +1 ) − ∇ f ( Ω t ) are the differentials of the successiv e P A upd ates an d the g radients of their Lagrang ian f unction, which is calculated by using th e above closed-form expression in (20 ). C. Du al-Rou tine Optimization Algo rithm Now , we discu ss the dual-routine op timization structure which compu tes the P A co e fficient up dates ( Ω ) using the L- BFGS-B method ( in ner iter ation) and th e L agrange m ultiplier updates for AL-f u nction (outer iteration ), alternatively . 1) L-B FGS-B Up date Ro utine: For a particular inn e r iter- ation t , we generate the generalized Cauchy point ( Ω c t ) by following the steepest fea sib le direction ∇ f ( Ω t ) until the bound s [0 , √ P s ] a re reac h ed. Then, we perform sub space minimization b y app lying L -BFGS-B two-loop recursion [15, Algorithm 7.4], wh ile using dua l search dir ections ( ± p t ) for the Newton step in (16 ). Additiona lly , the pr o jected line-search function is d e fined as φ ( α t ) = f ( Π [0 , √ P s ] ( Ω c t − α t p t ) , which is used to perfor m b a c ktracking search to obtain line-searc h step size α t > 0 . Based on these updates, we evaluate the next P A estimate Ω t +1 and utilize it to up date the feasible curvature pair ( s t , y t ) an d limited-memor y Hessian H t based on the last q curvature pairs stored tem porarily . The initial choice of th is Hessian m atrix is set as H (0) t , s T t − 1 y t − 1 k y t − 1 k 2 I q . It is impo r tant to note that the upd ate step is require d to satisfy the curvature condition y T t s t > 0 , which is regulated by AL penalty terms. Th e inner routine terminates as either the step size s t < δ or grad ient size y t < δ with toleran ce level δ . 2) AL Upda te Routine: During th e iteration u , the ou ter routine updates the Lagra nge multipliers λ SE , λ HE , subject to the QoS constrain t violations. Mathe matically , this can be represented as: λ u +1 SE = λ u SE + ρ v u SE , ( G , θ , Ω ) , (22a) λ u +1 SE = λ u HE + ρ v u HE ( G , θ , Ω ) , (22b) where the QoS violations fo r a particular iteration u are ca lcu lated as v u SE ( G , θ , Ω ) = ma x( Γ EP A ( G , θ , Ω ) − Γ ( G , θ , Ω ) , 0 ) and v u HE ( G , θ , Ω ) = max( Q EP A ( G , θ , Ω ) − Q ( G , θ , Ω ) , 0) . If the max imum of these vio lation criter ia v max = max {k v SE k , k v EH k} redu ces below th e feasibility tolerance ǫ ≥ 0 , the algo rithm conv e rgen ce is achieved. In the ev ent of non - conv ergence, the AL penalty weig ht ρ a d dresses the constrain t violations adaptively over the successive AL- iterations. By adop ting an aggressive policy , we d ouble this parameter for the n ext AL- iteration u + 1 as ρ u +1 = 2 ρ u . The steps o f this pro cess are presented in Algo rithm 1 . Complexity a nalysis: The e xecutio n of the L-BFGS-B routine inv olves the calculation of finite difference gradient ∇ f ( Ω t ) and limited memory Hessian matrix H t , inc urring per- iteration complexity of O ( S K 2 ) and O ( q S K ) , respecti vely . On the other hand, AL routine evaluates QoS violations with complexity of O ( S K 2 ) . Hence, the overall com putational complexity can be gi ven as O ( U ( S 2 K 3 T ( K + q )) for T inner and U ou ter iterations. I V . N U M E R I C A L R E S U LT S In this sectio n, the performance of the D-CAP A m eta- surface fr amew ork with the p roposed optimization algorithm is evaluated. W e have considered S ∈ { 1 , . . . , 6 } m ultiple metasurfaces which serves K = 2 0 to tal users inclu ding L = 14 IUs and M = 6 EUs. Figure 1 depicts a rando m realization of the pro posed CAP A metasurfaces fram ew o r k with D-CAP As shown as blue squar es, while the C-CAP A as grey squa re. T hese surfaces ar e rand omly positioned on the xy -plane ( x, y ∈ [ − 10 , 10] m , z = 0 ) with the IUs (green dots) scattered in region z ∈ [0 . 5 , 20 ] m . The EUs (red d ots) are located in small xy -areas ( dashed blue squares) arou nd the D-CAP As in region z ∈ [0 . 5 , 2] m. For fair an alysis over multiple surface choices, we have con sid e red constant total aper ture area ¯ A T = { 0 . 25 , 0 . 5 , 1 } m 2 with total system transmission power fixed at P t = { 5 , 10 } mA 2 [8]. This cor r oborates the fact that the sam e ap erture area and transmit power co nsidered fo r the C-CAP A is further divided into mu ltiple distributed surfaces, A ( s ) T = ¯ A T /S , P s = P t /S . The system carrier wa velength is fixed at λ = 0 . 1 m , while the free-space chann e l imp edance Z 0 and the EU receiver impedanc e Z ar e set to 120 π Ω and 25 Ω , respectiv ely . The noise power at each user receiver is σ 2 = 10 − 9 A 2 . The NL-HE para m eters are selected as Q max = 24 mW , while ∂ f ( Ω ) ∂ Ω s,j =2 Ω s,j − 2 w l Ω s,l |J sl ( G , θ ) | 2 P S s =1 P K j =1 ,j 6 = l Ω 2 sj |J sj ( G , θ ) | 2 + σ 2 l + L X l =1 ,j 6 = l Ω s,j |J sj ( G , θ ) | 2 P S s =1 Ω 2 sl |J sl ( G , θ ) | 2  P S s =1 P K j =1 ,j 6 = l Ω 2 sj |J sj ( G , θ ) | 2 + σ 2 l  2 ! − X M m =1 2 u m A R cos ϕ m Ω sj |J sj ( G , θ ) | 2 2 Z , (20) Algorithm 1 AL + L- BFGS-B for P A optim ization 1: Initialization: Set iteration indices t = 0 , u = 0 ; initial P A estimates Ω 0 = P t / ( S K ) ; Lagran ge multiplier s λ SE , λ HE ; AL pen alty ρ > 0 ; I U/EU balancing factor β > 0 ; f easibility thre sholds δ = 0 , ǫ > 0 . 2: AL Routine: 3: repeat 4: Define f ( Ω ) as in (15) with current λ SE , λ HE , ρ . 5: Set inner iterate Ω ← Ω u . Initialize limited- memory pairs { ( s i , y i ) } for L -BFGS-B ap p roach. 6: L-BFGS-B Routine: 7: repeat 8: Comp ute gradien t ∇ f ( Ω t ) u sin g (20 ). 9: Follow the steepest direction ∇ f ( Ω t ) to the bo u nds [0 , √ P s ] , ob taining generalized Cauch y point Ω c t = Π [0 , √ P s ] ( Ω t − α ∇ f ( Ω t )) . 10: App ly th e two–loop L-BFGS re cursion to g e t the quasi-Newton direction p t = − H t  ∇ f ( Ω c t )  . 11: Define the pr ojected line-sear ch function φ ( α t ) = f  Π [0 , √ P s ]  Ω c t + α t p t   and per form a back tracking search to g et α t > 0 . 12: Upd a te P A e stima te : Ω t +1 = Π [0 , √ P s ]  Ω c t + α t p t  . 13: Upd a te curvature pair: s t = Ω t +1 − Ω t , y t = ∇ f ( Ω t +1 ) − ∇ f ( Ω t ) . 14: if y T t s t > 0 : 15: Store the f easible cur vature pair ( s t , y t ) 16: Update the limited- memory inverse-Hessian H t 17: else continue 18: Inc r ement inner iteration index t = t + 1 . 19: until k s t k ≤ δ or k y t k ≤ δ . 20: Ev aluate violation s: v SE ( G , θ , Ω u +1 ) = max  Γ EP A − Γ ( G , θ , Ω u +1 ) , 0  , v HE ( G , θ , Ω u +1 ) = max  Q EP A − Q ( G , θ , Ω u +1 ) , 0  . 21: Lagrange multiplier upd a te : λ SE ← λ SE + ρ v SE , λ HE ← λ HE + ρ v HE . 22: Increase pen alty: ρ ← 2 ρ . 23: Increment ou ter iteration index u = u + 1 . 24: until max {k v SE k , k v HE k} ≤ ǫ . 25: Return: Ω ⋆ = Ω u +1 . the rectifier pa r ameters as a = 150 0 , b = 0 . 0022 [12]. Th e optimization convergence thresholds are fixed as δ = 10 − 9 and ǫ = 10 − 3 . For the L-BFGS-B routine, we store only the q = 10 last curvature pair s for Hessian app roximation , while the AL routine sets the initial pena lty weight and IU/EU b alancing factor as ρ 0 = 20 an d β = 10 , respectively . In ord er to an alyze the comparative advantage of th e D- Fig. 1: C-CAP A and D-CAP A network layouts (single random real- ization). CAP A structure with th e proposed du al-routine optimiza tion algorithm , we consider the PC ratio against the benchmark EP A case. In this regard , this metric has been analyzed in Fig. 2 for multiple number o f metasu r faces ( S ) including b oth C-CAP A ( S = 1 ) and D-CAP A ( S = { 2 , 3 , 4 , 5 , 6 } ) cases, while considering different ch o ices of ¯ A t and P t . I t can be clearly no ticed here th a t the PC ratio red uces p rogressively with an increasing nu mber of CAP A surfaces, while satisfy ing the IU SE and EU H E Qo S benc h marks. This observation can be attributed to the smaller distances between the CAP A surfaces and the spatially distributed users, even thou gh the aperture area and transmission p ower for ea ch D-CAP A decr eases lin- early . For the C-CAP A case, we can observe that our pro posed optimization algor ithm reduces the PC up to 10% against th e non-o ptimized EP A case (d otted purple line). On the oth er hand, this re d uction is up to 6 5% and 61 % for the D- CAP A for th e S = 6 scenario with respect to EP A and op timized C-CAP A counterparts, respectively . W e have also analyzed the ef fect of different total trans- mission p ower P t = { 5 , 10 } mA 2 (dotted an d solid lines), along with the aperture sizes ¯ A T = { 0 . 25 , 0 . 5 , 1 } m 2 (red, blue an d gr een line s). Figure 2 demo nstrate that a larger to tal aperture size with higher transmission power re su lts in the most effecti ve perfo r mance for our proposed optimization algo rithm shown by the solid green line. Most importantly , a decreased transmission p ower ha s more dr astic effect on the PC reduction. For example, we can notice a 40% p erform a nce reduction for the case of P t = 5 mA 2 , ¯ A T = 0 . 2 5 m 2 (dotted red line) in compariso n to P t = 10 mA 2 , ¯ A T = 0 . 25 m 2 (solid red line). Now , we examine the peak transmission power density ( ˆ S T ) as the numb er of D-CAP As in c r eases (color ed bars) alon g with the C-CAP A (grey bars) case in Fig. 3, assuming a constant P t = 10 mW 2 . This analysis also evaluates the impact of Fig. 2: PC ratio v ersus the number of CAP A surfa ces. the overall CAP A aperture size ¯ A T = { 0 . 5 , 1 } m 2 , sho wn as red and blue bars, respectively . T he proposed dual-routine optimization framew o rk focuses the tran smission current distri- bution by ef ficiently manip u lating the wa ve amplitude s with out compro mising on the SE and HE r e q uiremen ts, thus, leading to concentr a ted energy ’ h ot-spots’ over the m etasurfaces. For the C-CAP A case, we observe that ˆ S T increases by 32% and 120 % for ¯ A T = 0 . 5 m 2 and ¯ A T = 1 m 2 , respectiv ely , compared with the EP A b enchmar ks, whose average power densities ¯ S T are sho wn as the purple and green dotted horizontal lines. Moreover , the energy focusin g becomes substantially stronger in the D-CAP A configuratio n , achieving up to a six- fold im p rovement f or S = 6 compared with the EP A case. This pr o noun c e d power co ncentratio n over specific r egions of the CAP A sur face can be attr ibuted to the selective spatial EM visibility acr oss d ifferent areas o f the metasur face, which guides the beam f orming design to conc e n trate user-specific current d istributions onto particular ph ysical zo n es of the transmitter . Anothe r key observation is that ˆ S T decreases as the overall aper tu re size ¯ A T increases from 0 . 5 m 2 to 1 m 2 , shown b y the red and blue bars, re sp ectiv e ly . Th is implies that for larger ap ertures, the same tr ansmission p ower is spr ead across a wider phy sical area of th e m etasurface. V . C O N C L U S I O N This paper in vestigated a distributed CAP A (D-CAP A) archi- tecture for m ultiuser SWIPT and developed an optimizatio n framework to minimize system PC und er SE and HE QoS constraints der iv e d fr om the EP A ben chmark. By explo it- ing con tinuous current-signal synth esis and dy adic Green’ s function –based chan nels, the propo sed framework applies the L-BFGS-B me thod to o ptimize the P A co efficients and le ver- ages an AL scheme to ma n age Qo S constraint violations. Sim- ulation resu lts de m onstrate that the op timized D-CAP A design achieves up to a 6 % reduction in system PC compared with an optimized C-CAP A configu ration, sh owcasing the stro ng EM beamfor ming cap abilities of CAP A transmit surfaces. 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