RSMA-Assisted Multi-Functional 6G: Integrated Sensing, Communication, and Powering

RSMA-Assisted Multi-Functional 6G: Inte grated Sensing, Communication, and Po wering Xiaoxuan Jiang and Y ij ie Mao School of Informatio n Science and T echnolo gy , ShanghaiT ech Univ ersity , Shang h ai 201 2 10, China Email: { jiangxx2 0 24,maoyj } @shanghaitech.edu.cn Abstract —Integrated sensing, communication, and powering (ISCAP) has emerged as a promising solution for enabling multi- functionality in 6G networks. Howev er , it poses a significant challenge in the design of multi-fun ctional wa vefo rms that must jointly consider communication, sensing, and powering perfor - mance. In this paper , we propose a novel rate-splitting multiple access (RSMA)- en abled multi-functional ISCAP netwo rk, w h ere RSMA facilitates the use of communication signals to simultane- ously achiev e all three functionalities. Based on the proposed sys- tem model, we in vestigate the beamf orming optimization problem to expl ore the performance trade-offs among communication, sensing, and power transfer . T o efficiently solve this problem, we d ev elop a n ov el ISCAP-extragradient (ISCAP - EG) algorithm, which transforms the original problem into a sequence of con vex subproblems, refor mulates the d u al pro bl em as a variational inequality , and solv es it using the EG meth od. Numerical r esults show that the proposed ISCAP-EG algorithm achieves perfor - mance equiva len t to that of the con ventional successive conv ex approximation (S CA)-based method, w h ile signifi cantly reducing simulation time. Moreov er , the RSMA-enabled multi-fu nctional ISCAP network enhance the performa nce trade-off compared with the con ventional space-division mult i ple access (S DMA)- based scheme, hi ghlighting RS MA as a promising technique for advancing multi-fun ctional IS CAP development in 6G. Index T erms —integrating sensing, communication, and pow- ering (ISCAP), rate-splitting multiple access (RSMA), multi- functional, max-min fairness (MMF ) , effi cient algorithm I . I N T R O D U C T I O N It is envisioned that futu re wir eless networks will undergo a paradigm shift from the “co m munication -centric 5G” to a “multi-fun ctional 6G” e r a, sup p orting not only commun ication services b ut also new fu nctions suc h as sensing, intelligence, computatio n, localization, navigation, and powering [1] . Un- der th is b ackgrou nd, th e integrated sensing, co mmunica tio n, and powering (ISCAP) network has been pr oposed in [2], which u nifies the se three function alities into a sing le system and leverages the communication wav ef o rm to simultaneously enable sensing and wireless power tr ansfer . Ho wever , the integration of these functionalities p oses severe challenges in managing the interf erence b etween different fu nctionalities, including mu lti-user interferenc e, in terference between co m - munication and r a dar sensing , and in te r ference between radar sensing an d power transfer . T he existence o f these interfer- ences sign ifica n tly degrad es th e perf ormance of I SCAP , which is a m ajor obstacle to its pra c tical imp le m entation in 6G. Existing works h av e m ade some e fforts to address these challenges by foc using on wa veform d esign [2 ]–[5]. Ho wever , they all rely on con vention al space- d ivision multip le access (SDMA), which h andles inter-user interference by simply treating it as n oise. Recent studies have shown that rate- splitting multiple access (RSMA) is a promising interferen ce managem ent technique, enc ompassing conventional SDMA, non-o rthogo n al multiple access (NOMA), an d ortho gonal mul- tiple access (OMA) as special cases [6], [7]. It achieves enhanced spectr a l efficiency , en ergy ef ficien cy , and robust- ness against imper fect chan nel state inf o rmation, making it a promising appro a ch for en suring the r esilience of 6G networks [8]. By le verag ing a common stream to simultaneously man- age inter-user in te r ference, mitigate in ter-functionality inter- ference, and serve as rada r seque n ces f or sensing beampattern requirem ents (or as p ower beam s for en e rgy harvesting at energy users), RSMA has dem onstrated enhance d trade-o ffs between dual functionalities. This improvement applies to both integrated sensing an d c ommun ic a tio n (ISA C) [ 1], [9], [10] and simultaneo us wireless info rmation and p ower transfer (SWIPT) [1 1]. Howe ver, the integration of RSMA into multi- function al ISCAP systems remains largely unexplor e d. Motiv ated by the benefits of RSMA in I SA C an d SWIPT , as well as its research g ap in ISCAP , in this work, we initiate the study of RSMA in ISCAP . The prima r y contributions of this work are summar ized as follows: • W e propo se a novel RSMA-enab led m u lti-function al ISCAP ne twork that allows c o mmunica tio n signals to simultaneou sly support commun ication, sensing, and power tran sfer . Leveraging the co mmon stream intro- duced by RSMA, the system elimin ates th e need for dedicated radar sequ ences o r en ergy-carry ing signals for target sensing and energy receivers ( E Rs), ther e b y mitigating inter-functionality in te r ference. T o th e best of our kn owledge, this is the first w or k to investigate RSMA for multi-fun ctional ISCAP . • W e formu late a joint wa veform design pr oblem that aims to maximize the worst-case r ate o f c o mmunica tio n users and th e Cram ´ er-Rao Bound (CRB) for radar sensin g , while satisfying the en ergy harvesting constraints at the ERs and the power b ud get at the base-statio n (BS). T o solve this pro blem, we propose a novel algo rithm, termed ISCAP-extragradien t (ISCAP-EG), which dec om- poses th e original pro blem into a sequence o f convex subprob lems, reform ulates the dual of each sub p roblem as a v aria tio nal inequ ality , and solves it using the extra- gradient method. • Numerical resu lts validate the ef fectiveness of the pro- posed approac h , sh owing th at I SCAP-EG achieves per- forman ce equiv alent to that o f the co n ventiona l suc- cessi ve conv ex approx imation (SCA)-based m ethod, Messages for K IR s  1    2 … Message Splitter   , 1   , 2 …   ,    , 1   , 2 …   ,  Message combiner Encoder      1  2   … Linear Precoder …  1  2    … … IR- K IR- 1 T arget ER- 1 ER- L ISCAP T ransmitter Fig. 1: A n RSMA-assisted I SCAP system illustratio n. while significantly red ucing simulation time. Further- more, th e RSMA-ena bled multi-fun ctional ISCAP net- work ach ie ves a superior p erform a n ce trad e-off com - pared to conventional SDMA-b ased schemes, h ig hlight- ing RSMA as a pro mising techn o logy for advancing multi-fun ctional ISCAP systems in 6G. I I . S Y S T E M M O D E L A N D P RO B L E M F O R M U L A T I O N As illustrated in Fig. 1, this work focuses on a downlink multi-user multiple- input sing le-outpu t (MU - MISO) network comprising a multi-fu n ctional BS, m ultiple informatio n re- ceiv ers (IRs), ERs, and a sensing target. The BS is equip ped with N t transmit antennas and N r receive antennas, where the rec ei ve anten nas are dedicated to target sen sin g . This work co nsiders separ ate IRs and ERs, wh ich are resp ecti vely indexed b y K = { 1 , 2 , . . . , K } and L = { 1 , 2 , . . . , L } . The BS employs th e basic 1 - layer RSMA mod el [6] which enables message splitting b y di vid ing each IR’ s message into: W c,k (a com mon par t) and W p,k (a pr i vate par t). The c ommon parts of IRs, denoted as { W c, 1 , . . . , W c,K } , are combined and then encoded i n to a c o mmon stream s c for all I Rs. In p arallel, each pr i vate part W p,k , k ∈ K is in depend e ntly enco ded into the private stream s k dedicated to IR- k . The data streams for all IRs s = [ s c , s 1 , . . . , s K ] T ∈ C K +1 are linearly precoded by the p r ecoders in P = [ p c , p 1 , · · · , p K ] ∈ C N t × ( K +1) , leading to the transmit sign al: x = p c s c + X k ∈K p k s k . (1) Here, we assume E  ss H  = I and the transmit power is upper boun ded b y P t , i.e., tr ( PP H ) ≤ P t . Although there is no dedicated radar sequence or energy-car rying signal, the transmit signal in (1) is designed to simu ltaneously fu lfill three function s: deli vering inform ation to th e IRs, transferring energy to the ERs, and sensing targets. Let h k and g j denote the channels fro m the BS to IR- k and ER- j , respectiv ely . Assume the tran smitter fully knows the c hannel state inform ation, the sign al re c e i ved at IR- k is: y k = h H k x + n k , ∀ k ∈ K , (2) where n k with zero mean and variance σ 2 k is the add itive white Gaussian noise (A WGN) r eceiv ed at IR- k . At IR- k , employing the successive interfer e nce can cellation (SIC) metho d, the common stream s c and the de d icated priv ate stream s k are sequentially decoded with the signal- to-interfer ence-plu s-noise ra tios (SINR) respectively gi ven as: γ c,k =   h H k p c   2 P j ∈K   h H k p j   2 + σ 2 k , ∀ k ∈ K , (3) γ p,k =   h H k p k   2 P j ∈K ,j 6 = k   h H k p j   2 + σ 2 k , ∀ k ∈ K . (4) The correspon ding transmission rates at IR- k are expressed a s R i,k = log 2 (1 + γ i,k ) , ∀ k ∈ K , i ∈ { c, p } . (5) As all IRs are re quired to decode s c , its achievable common rate is limited by the worst-case rate as R c = min { R c, 1 , . . . , R c,K } . Denoting the co mmon rate allocated to IR- k as c k , we have P k ∈K c k = R c . T h e overall achiev able rate at IR- k is then expre ssed as R k = c k + R p,k . At ea ch ER, the en ergy carried by all information precod ers is harvested. Follo wing the energy h arvesting model in [12 ] , the resulting harvested energy at E R- l is giv en as: Γ l = M l X l (1 + exp ( − υ l ( Q l − ς l ))) − Y l , ∀ l ∈ L , (6) where Q l =    g H l p c   2 + P k ∈K   g H l p k   2  , X l = exp( υ l ς l ) 1+exp( υ l ς l ) , an d Y l = M l exp( υ l ς l ) , M l is the maximum outp ut DC power under satur ation, υ l and ς l are con stants related to the energy harvesting circuit. The tran smit signal is also utilized fo r target sensing. The echo signal reflected from the sensing target to the BS is: y s = H s x + n , where n r e presents the A WGNs at the BS receiver with zer o mea n and variance σ 2 s , H s = α a r ( θ ) a H t ( θ ) , α A ( θ ) is the chan nel matrix from th e BS to the target and then to the BS receive antennas. Here, α and θ respectively denote the complex r eflection coefficient an d the angle of departure/arr iv al (AoD/AoA), a t ( θ ) ∈ C N t × 1 and a r ( θ ) ∈ C N r × 1 are the tr ansmit a nd receive arr ay steer ing vectors, respectively . In this work, we emp loy the Cram ´ er - Rao B ou nd ( CRB) associated with θ as a sensing perfor mance metric. For b r evity , we den ote H s x as x s . By stacking the param e te r s for estimation as ω , [ θ , b α T ] T ∈ R 3 × 1 , where b α = [ ℜ{ α } , ℑ{ α } ] T denotes the co m plex reflection coefficient, where ℜ{ α } and ℑ{ α } re sp ectiv ely d enote the real and the imaginary par ts o f α . The Fisher in f ormation matrix (FIM) related to ω is th en computed as [13 ] F = ℜ  ∂ x H s ∂ ω R − 1 s ∂ x s ∂ ω  = 2 σ 2 s ℜ  ∂ x H s ∂ ω ∂ x s ∂ ω  , (7) where R s = σ 2 s I N r . F can be furth e r rewritten into the blockwise form as F =  F 11 F 12 F ⊤ 12 F 22  , (8) where each block can b e expre ssed as: F 11 = κ | α | 2 tr ( ˙ A θ R x ˙ A H θ ) , F 12 = κ ℜ n [ α ∗ tr ( AR x ˙ A H θ ) , α ∗ tr ( AR x ˙ A H θ ) j ] o , F 22 = κ tr ( AR x A H ) I . Here, κ = 2 σ 2 s , R x = PP H , α ∗ is the con jugate o f α , an d ˙ A θ is the deriv ative of A ( θ ) with respectiv e to θ : ˙ A θ = ˙ a r ( θ ) a H t ( θ ) + a r ( θ ) ˙ a H t ( θ ) . (9) The trace of the CRB matrix, which is defined as trace of the inverse of the FIM, i.e., tr ( F − 1 ) , is used as the sensin g perfor mance metric. In this work, we aim at jo intly maximizin g th e worst-case commun ication ra te among IRs and min imizing the trace of CRB by o ptimizing the pre coding ma tr ix P and th e co mmon rate allocation c , [ c 1 , c 2 , . . . , c K ] T . This op timization is subject to constrain ts on the minimum harvested en ergy of each energy receiver and the p ower b ud get of BS. The problem is formulated as f ollows: max P , c min k ∈K { c k + R p,k } − λ tr ( F − 1 ) (10a) s.t. 1 T c ≤ min k ∈K { R c,k } , (10b) c ≥ 0 , (10c) tr ( PP H ) ≤ P t , (10d) Γ l ≥ E l , ∀ l ∈ L , (10e) where λ ≥ 0 is the tradeo ff between the MMF rate for th e IRs and ra d ar sensing metric for target detection, 1 is the all-one vector , E l is the harvested e nergy lower b ound at E R- l . I I I . P RO P O S E D O P T I M I Z A T I O N F R A M E WO R K Problem (10) is challen ging to solve d ue to its inheren t non-co n vexity and non-smo oth nature. T o addre ss these dif- ficulties, we pro p ose a novel and e fficient optimizatio n algo- rithm that guarantee s conver gen ce to a subo ptimal solution of (10), while sign ifica n tly reducing simulation tim e comp a r ed to con ventiona l SCA-based meth o ds. Specifically , we first employ fractional prog ramming (FP) an d first-order T aylor approx imation to tr ansform the or iginal pr oblem into a ser ies of co n vex subproblems. However , rather than directly solving these subpro blems using conventional interio r-point metho d s which are u sually implemented by CVX, we focus on the Lagrang ian du a l of each subpr oblem, which is equ iv alent to a variational inequality problem. Base on this o bservation, an extragr adient-based alg orithm is p roposed to effectiv ely tackle the Lagrangian dual problem with itera ti ve closed-for m solutions. A. Pr oblem T ransformatio n T o tackle the non-co nvex rate expressions, the FP tech nique can be employed here. By apply in g the Lagra n gian dual transform ation method as described in [ 1 4], we have: f i,k ( P , ϑ i,k ) , lo g(1 + ϑ i,k ) − ϑ i,k + (1 + ϑ i,k ) γ i,k 1 + γ i,k , i ∈ { c , p } (11) where ϑ i,k is auxiliary variable wth respect to γ i,k , i ∈ { c , p } . f i,k is lower bou nd for R i,k , i ∈ { c , p } . The eq ualities hold when: ϑ ◦ i,k = γ i,k , i ∈ { c , p } (12) By fur ther employin g the quadr a tic transform in [1 4], we have: g i,k ( P , ϑ i,k , ϕ i,k ) , J i,k + 2 p 1 + ϑ i,k S i,k − | ϕ c,k | 2 T i,k , (13) where i ∈ { c, p } , ϕ i,k is auxiliary variable, J i,k , log (1 + ϑ i,k ) − ϑ i,k , S c,k , ℜ{ ϕ c,k h H k p c } , S p,k , ℜ{ ϕ p,k h H k p k } , T c,k , | h H k p c | 2 + P K j =1 | h H k p j | 2 + σ 2 k , T p,k , T p,k − | h H k p c | 2 , g i,k is lo wer bound for f i,k . The equa lities are achieved wh en: ϕ ◦ c,k = p 1 + ϑ c,k h H k p c T c,k , ϕ ◦ p,k = p 1 + ϑ p,k h H k p k T p,k . (14) W ith (11)– (14), the rate expressions in ( 5) are equiv alently transform ed into: R i,k = max ϑ i,k ,ϕ i,k g i,k ( P , ϑ i,k , ϕ i,k ) , i ∈ { c, p } (15) Define ϑ i , [ ϑ i, 1 , ϑ i, 2 , . . . , ϑ i,K ] , ϕ i , [ ϕ i, 1 , ϕ i, 2 , . . . , ϕ i,K ] , i ∈ { c, p } and substitute equatio n (15) into problem (1 0), problem (1 0) is re f ormulated as: max P , c , ϑ c , ϑ p , ϕ c , ϕ p min k ∈K { g p,k ( P , ϑ p,k , ϕ p,k ) + c k } − λ tr ( F − 1 ) (1 6a) s.t. 1 T c ≤ min k ∈K { g c,k ( P , ϑ c,k , ϕ c,k ) } , (16b) (10 c ) , (10 d ) , (10 e ) . Although pro blem (1 6) remains non -conve x, it is block -wise conv ex amon g variable b locks: { P , c } , { ϑ c , ϑ p } , { ϕ c , ϕ p } . Howe ver , for the blo c k of { P , c } , the problem remain s non- conv ex due to (10 e) and the CRB metric in the objective function . Next, we app ly the fir st-order T aylor ap p roximatio n to a d dress th ese non-conve x ities. Specifically , for constraint (10e), we first equiv alently tra n sform it into a simple r for m:   g H l p c   2 + X k ∈K   g H l p k   2 ≥ e E l , ∀ l ∈ L (17) where e E l = ς l − ln  M l ( E l + Y l ) X l − 1  υ l . T hen, w e substitute | g H l p i | 2 , i ∈ { c , k } with its first-or der T ay lor expansion: | g H l p i | 2 ≥ 2 ℜ{ ( g H l p ( t ) i ) H g H l p i } −    g H l p ( t ) i    2 , U ( g l , p i , p ( t ) i ) . (18) Here, p ( t ) i is the value of p i at iteration t . W ith the aid of (18), constraint (10e) is ref ormulated as U ( g l , p c , p ( t ) c ) + X k ∈K U ( g l , p k , p ( t ) k ) ≥ e E l , ∀ l ∈ L (19) Inspired by the first-ord er T aylor appro x imation ap proach propo sed in [13], we obtain that tr ( F − 1 ) can be ap proxima ted at the point P ( t ) by: tr ( PP H Λ ) ≥ 2 ℜ{ tr ( P ( t ) P H Λ ) } − tr ( P ( t ) P ( t ) H Λ ) , (2 0) where Λ = ζ I + 1 2 ( B + B H ) , B = κ ( φ 11 | α | 2 ˙ A H ˙ A + 2( φ 12 + j · φ 13 ) α ∗ ˙ A H A + ( φ 22 + φ 33 ) A H A ) , φ ij denotes the ( i, j ) - th entry of Φ , Φ = F − 2 , and ζ is selected to guarantee that Λ is positive sem i- definite. The equality of (20 ) is ac hiev ed when P = P ( t ) . Readers are r eferred to [13] for the d etailed pro of. W ith (1 9 ) and ( 2 0), the optimiz a tion b lock of { P , c } for problem (16) with fixed ϑ c , ϑ p , ϕ c , ϕ p can be equiv alently transform ed in to a sequence of conve x subpro blems. F o r a feasible P ( t ) , th e co n vex subprob lem is given as: max P , c min k ∈K { g p,k ( P ) + c k } + 2 λ ℜ{ tr ( P ( t ) P H Λ ) } (21a) s.t. (10 c ) , (10 d ) , (16 b ) , (19 ) . Problem (21) is con vex, it can be directly solved usin g the interior-point method , typic a lly implemented thr ough standard optimization toolbo xes such as CVX. Therefore, prob lem (10) can be ad dressed b y iteratively optimizing the variable blocks { P , c } , { ϑ c , ϑ p } , and { ϕ c , ϕ p } in (16). Howe ver , this approa c h incurs significant computational overhea d due to the repeated use of CVX in each iteration . B. V ariatio nal In equality and The EG Algorithm In this sub sectio n, we add ress problem (21) by propo sin g a novel extragr adient-based meth od to its Lagran g ian du al. This approa c h circu mvents the need fo r conv entio nal interior -p oint methods to solve p roblem (21 ) directly , thereby elimin ating reliance on stan d ard optim ization toolboxes such as CVX. Using an auxiliary variable r to rep r esent the MMF rate part, problem (21) is reformu late d as: max P , c ,r r + 2 λ ℜ{ tr ( P ( t ) P H Λ ) } (22a) s.t. r ≤ c k + g p,k ( P ) , ∀ k ∈ K , (22b) (10 c ) , (10 d ) , (16 b ) , (19 ) . Rather than solv ing pr oblem (2 2 ) directly , we focu s on its Lagran gian dual pro blem. By in troducin g the Lagran g ian dual variables β = [ β 1 , · · · , β K ] T , ρ = [ ρ 1 , · · · , ρ K ] T , µ = [ µ 1 , · · · , µ K ] T , ω and η = [ η 1 , · · · , η L ] T , the Lag rangian function for problem (2 2) is d e fin ed as: L ( P , c , r, β , ρ , µ , ω , η ) , r + 2 λ ℜ{ tr ( P ( t ) P H Λ ) } − K X k =1 β k ( r − ( c k + g p,k ( P ))) − K X k =1 ρ k K X j =1 c j − g c,k ( P ) ! + K X k =1 µ k c k − ω ( tr ( PP H ) − P t ) + L X l =1 η l ( U ( g l , p c , p ( t ) c ) + X k ∈K U ( g l , p k , p ( t ) k ) − e E l ) . (23) The Lagrangian d ual problem is then formulated as: min β , ρ , µ ,ω, η max P , c ,r L ( P , c , r, β , ρ , µ , ω , η ) (24a) s.t. β ≥ 0 , ρ ≥ 0 , µ ≥ 0 , ω ≥ 0 , η ≥ 0 . (24b) Inspired by [15 ], we next show th at ( 24) is equivalent to a variational ineq uality pr oblem and can be addressed by the extragradient metho d. First, by stacking the real and imaginary parts of variables, w e d efine: y , [ ℜ{ p c T } , ℑ{ p c T } , ℜ{ p 1 T } , · · · , ℜ{ p K T } , ℑ{ p 1 T } , · · · , ℑ{ p K T } , c T , r ] T , z , [ β T , ρ T , µ T , ω , η T ] T . (25) Due to the co ncavity o f L ( P , c , r, β , ρ , µ , ω , η ) in ( P , c , r ) and its conve xity in ( β , ρ , µ , ω , η ) , the optim a l solutions y ◦ and z ◦ of problem (24) ar e achieved when: ∂ L ∂ y     y = y ◦ ! T ( y ′ − y ◦ ) ≤ 0 ,  ∂ L ∂ z     z = z ◦  T ( z ′ − z ◦ ) ≥ 0 . (26) where y ′ and z ′ denote any other points in the feasible set. T o make it closer to the standard for m of th e classical variational inequa lity , we furth er define: x , [ y T , z T ] T , h ( x ) ,  −  ∂ L ∂ y  T ,  ∂ L ∂ z  T  T . (27) Eventually , problem (24) is refo r mulated into th e f o llowing variational inequa lity problem : Find x ∈ S s.t. h ( x ) T ( x ′ − x ) ≥ 0 , ∀ x ′ ∈ S , (28) where S is the feasible set of p roblem (24): S = { x | P ∈ C N t × K , c ∈ R K , r ∈ R , z ≥ 0 } . (29) According to [16 ] , when a map ping h ( · ) is monotone , extragradient ser ves as an ef ficient method to deal with the variational inequa lity problem . The extragrad ient algorithm typically inv olves two iterati ve steps, responsilble for estima tin g a n d adjusting respe cti vely: Step 1 : Prediction ¯ x ( n ) = Π S  x ( n ) − σ ( n ) h  x ( n )  , (30) Step 2: Cor rection x ( n +1) = Π S  x ( n ) − σ ( n ) h  ¯ x ( n )  , (31) where x ( n ) is the value of x at iter ation n , Π S ( · ) denotes the projection o perator on to S . Here , we follow [17] and select σ ( n ) = min n τ k x ( n ) − ¯ x ( n ) k k h ( x ( n ) ) − h ( ¯ x ( n ) ) k , σ o as the step size to guaran tee th e conv ergence of the algo rithm, where τ ∈ (0 , 1) , σ is the in itial step size. The detailed pre diction an d correc tion steps to update the variables in ( 2 8) are detailed as fo llows . First, u tilizing the W irtinger calculus, the update o f P is given by: ¯ p ( n ) i = p ( n ) i + 2 σ ( n ) ∂ L ∂ p ∗ i     x = x ( n ) , (32a) p ( n +1) i = p ( n ) i + 2 σ ( n ) ∂ L ∂ p ∗ i     x = ¯ x ( n ) , (32b) where i ∈ { c , k } , ∂ L ∂ p ∗ c = λ q 1 + P L l =1 η l g l g H l p ( t ) c − ω p c + P K j =1 ρ j  ϕ c,j p 1 + ϑ c,j h j − | ϕ c,j | 2 h j h H j p c  , ∂ L ∂ p ∗ k = λ q k + β k ϕ p,k p 1 + ϑ p,k h k − P K j =1 ( β j | ϕ p,j | 2 + ρ j | ϕ c,j | 2 ) h j h H j p k − ω p k + P L l =1 η l g l g H l p ( t ) k , q i is th e i -th coloum of ΛP ( t ) . Here, ¯ x ( n ) is the v alue of x n by substituting all variables u sing results calculated in ( 32a), ( 33a), and ( 34a). The updates of c k and r a r e re sp ectiv ely gi ven by: ¯ f ( n ) = f ( n ) + σ ( n ) ∂ L ∂ f     x = x ( n ) , (33a) f ( n +1) = f ( n ) + σ ( n ) ∂ L ∂ f     x = ¯ x ( n ) , (33b) where f ∈ { c k , r } , ∂ L ∂ c k = β k + µ k − P K j =1 ρ j , ∂ L ∂ r = 1 − P K j =1 β j . Note that the p rojection operation s fo r P , c and r are omitted in (3 2) and (3 3 ) becau se there are no oth er constraints on them. The update of z is given by : ¯ z ( n ) =  z ( n ) − σ ( n ) ∂ L ∂ z     x = x ( n )  + , (34a) z ( n +1) =  z ( n ) − σ ( n ) ∂ L ∂ z     x = ¯ x ( n )  + , (34b) where ∂ L ∂ z =   ∂ L ∂ β  T ,  ∂ L ∂ ρ  T ,  ∂ L ∂ µ  T , ∂ L ∂ ω ,  ∂ L ∂ η  T  T , ∂ L ∂ β k = − ( r − c k − g p,k ) , ∂ L ∂ ρ k = −  P K j =1 c j − g c,k  , ∂ L ∂ µ k = c k , ∂ L ∂ ω = −  tr ( P H P ) − P t  , ∂ L ∂ η l = U ( g l , p c , p ( t ) c ) + P k ∈K U ( g l , p k , p ( t ) k ) − e E l , ( · ) + represents max(0 , · ) , which is the simplified result of the pr o jection o peration f o r z because there is o nly non-negativity constrain t on it. The whole p rocess of the prop rosed algorithm is sum - marized in Algorithm 1, where obj ( · ) den o tes th e o b jectiv e function in (10a), ǫ 1 , ǫ 2 , ǫ 3 are the c on vergence tolerances for each iteration loop, an d k · k F is the Frobenius n orm. In Algorith m1, we fir st initiate P , c , r and use them to up d ate ϑ c , ϑ p , ϕ c and ϕ p using (1 2) an d (14). Then, we optimize { P , c } by emp loying (19) and (2 0) to tran sform the orig inal problem into a ser ies of con vex subpro blems. Finally , a two- step extragradie n t-based alg orithm is design e d to obtain an iterativ e closed-fo rm solution. C. Complexity Ana lysis The dom inant complexity of Algorithm 1 lies in the gradien t calculation for P , scaling at O ( K 2 N t ) . Considering the three loops, the complexity o f the whole Algorithm 1 is O ( I 1 I 2 I 3 K 2 N t ) , wher e I 1 , I 2 , I 3 are th e iteration n umbers of each loop respectively , an d are constants relev ant to the tolerance of each loop ǫ 1 , ǫ 2 , ǫ 3 . It is evidently mo re efficient than SCA, which req uires a computation a l complexity of O (( K N t ) 3 . 5 ) in each iter ation [18]. I V . N U M E R I C A L R E S U LT S In this section, we provide n u merical results to demon strate the efficiency of o ur proposed ISCAP-EG algor ithm. Th r ee schemes are compared in th e resu lts: • RSMA-assisted ISCAP-EG: This is th e algorithm pro- posed in section III for a RSMA-assisted multi- function a l ISCAP network. • SDMA-assisted ISCAP-EG: This applies our propo sed ISCAP-EG algo rithm to address the problem of a SDMA-assisted multi-fun ctional ISCAP network. This is achieved b y turn ing off p c in our pro posed Algorithm 1 . • RSMA-assisted SCA: This refers to th e classical FP and SCA based algorithm that addresses problem (21) directly using the CVX toolb ox. For the sensing target, we con sid er a unifor m linear arr ay (ULA), with the transm it and receive steer ing vector s as: a t ( θ ) =  e − 1 j N t − 1 2 π sin θ , e − 1 j N t − 3 2 π sin θ . . . , e 1 j N t − 1 2 π sin θ  T and a r ( θ ) has the similar form. The numbe r o f ERs is fixed Algorithm 1 : Th e p r oposed ISCAP-EG algo rithm. 1 Initialize: P , c , r, α, τ , λ, υ , ς , ǫ 1 , ǫ 2 , ǫ 3 2 i = 0 , P ( i ) = P , f ( i ) 1 = obj ( P ( i ) ) . 3 repeat 4 Update ϑ c , ϑ p , ϕ c and ϕ p by (12) an d ( 14). 5 t = 0 , P ( t ) = P , f ( t ) 2 = obj ( P ( t ) ) . 6 repeat 7 Update U and Φ b y ( 19), (20). 8 n = 0 , P ( n ) = P , f ( n ) 3 = obj ( P ( n ) ) . 9 Initialize: c , r, β , ρ , µ , ω , η , σ 10 repeat 11 Update ¯ P ( n ) , ¯ c ( n ) , ¯ r ( n ) , ¯ z ( n ) by (32a), (33a), (34a). 12 Calculate σ ( n ) = min { β k x ( n ) − ¯ x ( n ) k k h ( x ( n ) ) − h ( ¯ x ( n ) ) k , σ } . 13 Update P ( n +1) , c ( n +1) , r ( n +1) , z ( n +1) by (32b) , (33b ), (34b) , f ( n +1) 3 = obj ( P ( n +1) ) , n = n + 1 . 14 until | f ( n ) 3 − f ( n − 1) 3 | < ǫ 3 ; 15 Update P ( t ) = √ P t P ( n ) k P ( n ) k F , f ( t +1) 2 = obj ( P ( t +1) ) , t = t + 1 . 16 until | f ( t ) 2 − f ( t − 1) 2 | < ǫ 2 ; 17 Update P = √ P t P ( t ) k P ( t ) k F , f ( i +1) 1 = obj ( P ) , i = i + 1 . 18 until | f ( i ) 1 − f ( i − 1) 1 | < ǫ 1 ; to L = 2 . The noise variance is u n iformly set to σ 2 k = 1 for all k ∈ K and σ 2 s = 1 . Th e trad e-off pa r ameter λ is set to be 0.1. T he convergence tolerance for all iter a tion loops is fixed to 10 − 3 for all schemes. All resu lts are averaged over 100 channels. 2 3 4 5 6 Number of transmit antennas 0.5 1 1.5 2 Objective function RSMA-assisted ISCAP-EG,SNR=25dB RSMA-assisted ISCAP-EG,SNR=15dB RSMA-assisted ISCAP-EG,SNR=5dB RSMA-assisted SCA,SNR=25dB RSMA-assisted SCA,SNR=15dB RSMA-assisted SCA,SNR=5dB (a) Objec tiv e functio n versus N t . 2 3 4 5 6 Number of IRs 0.7 0.8 0.9 1 1.1 1.2 1.3 Objective function RSMA-assisted ISCAP-EG,SNR=25dB RSMA-assisted ISCAP-EG,SNR=15dB RSMA-assisted ISCAP-EG,SNR=5dB RSMA-assisted SCA,SNR=25dB RSMA-assisted SCA,SNR=15dB RSMA-assisted SCA,SNR=5dB (b) Obj ecti ve function ver sus K . Fig. 2: Th e objectiv e function versus numb e r of transmit antennas and I Rs In Fig. 2, we com pare the objectiv e function of RSMA- assisted ISCAP-EG and RSMA-assisted SCA schemes with respect to the number of transmit antennas N t and IRs K , respectively . In Fig. 2 ( a), we show the c omparison o f the two schemes und er different SNRs while setting K = 4 , E l = 6 mW , l ∈ { 1 , 2 } . In Fig. 2(b ), we show the comparison considerin g N t = 4 , E l = 6 mW , l ∈ { 1 , 2 } . Both subfig ures demonstra te that ou r p roposed alg orithm a c hiev es the sam e perfor mance as the co n vention a l SCA-based alg orithm. Both algorithm s gu arantee convergence to a subo ptimal solution of the original problem. 2 3 4 5 6 Number of transmit antennas 10 1 10 2 CPU Time(s) 91.51% RSMA-assisted ISCAP-EG RSMA-assisted SCA (a) CPU ti me versus N t . 2 3 4 5 6 Number of IRs 10 0 10 1 10 2 CPU Time(s) 90.24% RSMA-assisted ISCAP-EG RSMA-assisted SCA (b) CPU time versus K . Fig. 3: Th e CPU time versus the number of transmit antennas and the number of IRs . SNR= 15 dB, K =4 in (a), N t =4 in (b). In Fig. 3, we c o mpare the CPU time o f RSMA-assisted ISCAP-EG and RSMA- a ssisted SCA schem es with re spect to N t and K , u nder SNR=15 dB, E l = 6 mW , l ∈ { 1 , 2 } . In both Fig . 3 ( a) and Fig. 3(b), we observe that our proposed algorithm takes less CPU time than the conventional SCA al- gorithm. Compared with SCA, ISCAP-EG redu ces the average CPU time b y 91 . 51% in Fig. 3 (a) and 9 0 . 24% in Fig . 3( b). 2 3 4 5 6 Energy harvested threshold (mW) 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 MMF Rate (bit/s/Hz) 0.06 0.08 0.1 0.12 0.14 0.16 0.18 CRB RSMA-assisted ISCAP-EG SDMA-assisted ISCAP-EG RSMA-assisted ISCAP-EG SDMA-assisted ISCAP-EG Fig. 4: The MMF Rate and CRB versus energy harvested threshold E l . N t = 4 , K = 4 , SNR= 25 dB. Fig. 4 shows the MMF rate an d CRB fo r RSMA-assisted ISCAP-EG an d SDMA-assisted ISCAP-EG schemes versus the energy harvested thr eshold E l . W e observe that the MMF rate of RSMA-assisted sch eme is always h igher than that o f the SDMA-assisted scheme, while the CRB of R SMA-a ssisted scheme is always lower than th at of SDMA-assisted scheme. This demonstrates that RSMA possesses a significan tly en - hanced capab ility to simu ltaneously support co mmunicatio n , sensing, an d power transfer . This advantage mainly attr ibutes to the emp loym ent of com m on stream . Th e commo n stream fulfills th r ee key roles: (1) interf erence manag ement among commun ication users, (2) a dedicated radar sequence, and (3) an unified wa vefor m for wireless power transfe r . This integrated appro ach inh erently mitigates in ter-functionality interferen ce. V . C O N C L U S I O N This paper pro poses an advanced RSMA-enabled multi- function al ISCAP network. Within this novel system model, we study the b eamform in g optimizatio n pr oblem to explore the perfo rmance tra de-offs amon g co mmunica tio n, sensing, and p ower tran sfer fun ctionalities. T o solve the beamfo rming problem ef ficiently , we develop a novel ISCAP-EG algorithm, which tr ansforms the o riginal p roblem in to a seque nce of conv ex subp r oblems, reformulates its dual problem as a varia- tional inequality , and solves it using the extrag radient meth od. Numerical results demonstrate that the pr o posed ISCAP-EG algorithm ach iev es perfo rmance equ iv alent to that of the conv entio nal SCA-based m ethod, while significantly red u cing simulation time. Furth ermore, the proposed RSMA-assisted multi-fun ctional I SCAP network outper forms conventional SDMA-based scheme, showing that RSMA is a prom ising technique for advancing multi-functional ISCAP de velop m ent for 6G. R E F E R E N C E S [1] B. 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