Receiver Antenna Partitioning for Simultaneous Wireless Information and Power Transfer
RReceiver Antenna Partitioning for SimultaneousWireless Information and Power Transfer
Rahul Vaze
School of Tech. & Computer ScienceTata Institute of Fundamental ResearchMumbai, IndiaEmail: [email protected]
Jainam Doshi
Dept. of Electrical Engr.Indian Institute of Technology, MadrasChennai, IndiaEmail: [email protected]
Kaibin Huang
Dept. of Electrical & Electronic Engr.The University of Hong KongPok Fu Lam, Hong KongEmail: [email protected]
Abstract —Powering mobiles using microwave power transfer (PT) avoids the inconvenience of battery recharging by cablesand ensures uninterrupted mobile operation. The integration ofPT and information transfer (IT) allows wireless PT to be realizedby building on the existing infrastructure for IT and also leadsto compact mobile designs. As a result, simultaneous wirelessinformation and power transfer (SWIPT) has emerged to be anactive research topic that is also the theme of this paper. Inthis paper, a practical SWIPT system is considered where twomulti-antenna stations perform separate PT and IT to a multi-antenna mobile to accommodate their difference in ranges. Themobile dynamically assigns each antenna for either PT or IT. Theantenna partitioning results in a tradeoff between the MIMO ITchannel capacity and the PT efficiency. The optimal partitioningfor maximizing the IT rate under a PT constraint is a NP-hardinteger program, and the paper proposes solving it via efficientgreedy algorithms with guaranteed performance. To this end, theantenna-partitioning problem is proved to be one that optimizesa sub-modular function over a matroid constraint. This structureallows the application of two well-known greedy algorithms thatyield solutions no smaller than the optimal one scaled by factors (1 − /e ) and / , respectively. Index Terms —MIMO communications, energy harvesting,power transfer, integer programming, greedy algorithms.
I. I
NTRODUCTION
The arguably most desirable new feature for mobile devicesis wireless power transfer (PT), which eliminates the needof recharging using cables and avoids interruptions of mobileservices due to dead batteries. With rapid advancements inmicrowave technologies, microwave PT has emerged to be apromising solution for wirelessly powering mobiles due to itslong transfer ranges (up to hundreds of meters) and supportof mobility [1]. In contrast, non-radiative technologies forwireless PT e.g., inductive coupling and resonant coupling,suffer from extremely short ranges (less than a meter). Usingmicrowaves as carriers, wireless PT and information transfer(IT) can be seamlessly integrated, which has resulted in theemergence of an active research area called simultaneouswireless information and power transfer (SWIPT) [1]–[3]. Theresearch on SWIPT, however, requires thorough revamping ofclassical theories for wireless communications and networkingto achieve not only high IT rates but also high PT efficiencies.PT and IT concern two different aspects of data bearingmicrowaves, namely their information content and absolute power, respectively. As a result, PT can tolerate much lesspropagation loss and support much shorter transmission dis-tances than IT. Furthermore, depending on the channel andenergy states, a mobile may choose to operate in either theIT, PT, or SWIPT modes. Consideration of such factors in re-alizing SWIPT calls for the design of new algorithms/protocolsfor MIMO transmissions [3], multiple access [4], resourceallocation [5], [6], mobile transceivers [10] and network ar-chitectures [8].A simple design of a SWIPT enabled mobile receiver is tocombine a conventional information receiver and an RF energyharvester. The form factor of this design can be reduced bysharing antennas between the receiver and harvester wherethe output of each antenna is split for data processing andenergy harvesting [7]. However, the addition of a power splitterwith an adjustable splitting ratio for each antenna increasesthe receiver complexity. A simpler SWIPT-receiver design thatallows antenna sharing but requires no splitting, is to partitionthe set of antennas into two sets, one dedicated for IT and otherfor PT. This design builds on the classic antenna selectiontechnique for MIMO communications [9] and requires a smallnumber of RF chains, leading to a high-efficient mobile design.The problem of optimal antenna assignment/partitioning forthe special case of SWIPT with a single-input-multiple-outputIT channel has been explicitly solved in [10] for a simplifiedobjective function. However, the problem for the general casewith a MIMO IT channel is much more challenging to solvethat depends on the eigenmodes of the channel matrix. To bespecific, the problem is a NP-hard integer program.The main contribution of this paper is connect the gen-eral SWIPT antenna-partitioning problem to the rich fieldof efficient sub-optimal integer-programming algorithms withguaranteed performance. This important connection is estab-lished by analyzing the structure of the antenna-partitioningproblem. Specifically, the problem is shown to be equivalent tomaximizing a sub-modular function with a matroid constraint.For a sub-modular function, the incremental gain of adding anew element diminishes with increasing set size. The provenstructure allows two well-known greedy algorithms to be ap-plied for solving the antenna-partitioning problem. Moreover,the resultant solutions are shown to be equal to the optimalones with the scaling factors no smaller than (1 − /e ) and a r X i v : . [ c s . I T ] O c t ase Station Power BeaconMobileData Power Fig. 1. A system supporting simultaneous wireless information and powertransfer / . Simulation results reveal that the performance of the saidantenna-partitioning algorithms substantially outperform thederived worst-case bounds.II. S YSTEM M ODEL
Consider the SWIPT system in Fig. 1 where a mobile isreceiving information/data from the base station and powerfrom a power beacon. Let N t , N r , and N p , denote the numberof antennas at the base station, at the mobile, and at the powerbeacon, respectively. The MIMO channel from the base stationto the mobile is represented by the complex N r × N t matrix H .Power is beamed from the power beacon to the mobile and thebeamforming vector is denoted as f . Let H (cid:48) denote the MIMOchannel from the power beacon to the mobile. The effectiveMISO channel after beamforming is defined as g = H (cid:48) f .Let s n ∈ { , } indicate whether the n -th receiver antennaof the mobile is assigned for IT ( s n = 1 ) or PT ( s n = 0 ).For ease of notation, the indicator variables are grouped intoa vector s = [ s , s , · · · , s N r ] † . Let S be a N r × N r diagonalmatrix with diagonal entries being the elements of s . It isassumed that the transmit channel state information (CSIT)is unavailable at the base station. Thus transmission power isequally allocated over all transmit antennas. The assumptionis relaxed in Section VI where the effect of power control isanalyzed. With equal power allocation, the IT channel capacitycan be written as C ( s ) = log det (cid:18) I + PN t SHH † S † (cid:19) . (1)Note that the effect MIMO IT channel matrix SH in (1) con-sists of the rows from H corresponding to receiver antennasassigned for IT.Given the antenna partition specified by s , maximum-ratiocombining is applied at the energy harvester to maximize theharvested power, denoted P r and given as P r = N r (cid:88) n =1 (1 − s n ) | g n | (2)where g n is the n -th element of the mentioned effective MISOPT channel g . The power P r is required to exceed the thresh-old p c > representing fixed circuit-power consumption,called the circuit-power constraint . III. P ROBLEM F ORMULATION
The optimal antenna partitioning is formulated as the prob-lem of maximizing the IT channel capacity in (1) under thecircuit-power constraint as follows: ( P ) max { s } log det (cid:18) I + PN t SHH † S † (cid:19) s.t. s n ∈ { , } , n = 1 , , · · · , N rN r (cid:88) n =1 (1 − s n ) | g n | ≥ p c . This is an integer programming problem which is typically NP -hard to solve. However, the investigation of the structureof P in the sequel leads to efficient methods for findingapproximate solutions. Remark:
One sub-optimal method for solving P is torelax the binary constraints in P to allow s n ∈ [0 , ∀ n ,that corresponds to allowing dynamic power splitting of thereceived signal at all antennas for the purposes of informationtransfer and power transfer as done in [10] for SISO channel.Since the objective function is a concave function of s , theapproximate problem can be efficiently solved using standardconvex optimization algorithms. However, this method doesnot provide any guarantee on the performance and its degra-dation compared with the optimal one can be unacceptable forcertain channel realizations.IV. (1 − /e ) -A PPROXIMATE S OLUTION
A sub-optimal algorithm for solving P can be designedto have guaranteed worst-case performance specified by anapproximation ratio a ∈ (0 , defined as follows. Definition 1.
An algorithm having an approximation ratio of a ∈ (0 , ensures that the ratio of the capacity (1) evaluatedat its output ˆ s and the optimal solution s (cid:63) satisfies C (ˆ s ) C ( s (cid:63) ) (cid:62) a ,regardless of H . Remark:
The approximation ratio a specifies the worst-caseperformance but the actual capacity C (ˆ s ) can be substantiallylarger than the lower bound a C ( s (cid:63) ) .To proceed further, we need some preliminaries. Let f bea set function defined over all subsets of U : f : 2 U → R + where U denotes the power set of U . Definition 2. (Monotonicity) The function f is monotone if f ( S ∪ { a } ) − f ( S ) ≥ , for all a ∈ U, S ⊆ U, a / ∈ S . Definition 3. (Sub-modularity) The function f is sub-modularif f ( S ∪ { a } ) − f ( S ) ≥ f ( T ∪ { a } ) − f ( T ) , for all elements a ∈ U, a / ∈ T and all pairs of subsets S ⊆ T ⊆ U . Essentially, for a sub-modular function, the incremental gainfrom adding an extra element in the argument set decreasesith the size of the set. The definition of sub-modular functionis introduced for the reason that the capacity function C ( · ) in(1) is sub-modular as shown in Lemma 1 that directly followsfrom [11, Theorem ]. Lemma 1.
The function C ( · ) of the set of receive antennasassigned for IT is sub-modular. It is easy to verify that C ( · ) is a monotone function sinceadding more receiver antennas cannot decrease the capacity.Next, multi-linear extension and matroid are defined asfollows. Let n denote the cardinality of S . Consider a function f that assigns a nonnegative value to each subset of S : f : 2 S → R + . For each subset A ⊆ S , let x A = [ x . . . x n ] be the n -length vector, where x i = 1 if the i th element of S is contained in A and x i = 0 otherwise. Thus, f is a functionassigning a value to each vertex of a { , } n hypercube. Definition 4. (Multi-linear extension) The multi-linear exten-sion F extends f to whole of [0 , n such that for x ∈ [0 , n F ( x ) = E { f (ˆ x ) } = (cid:88) A ⊆ S f ( A ) (cid:89) i ∈ A x i (cid:89) j / ∈ A (1 − x j ) , where ˆ x denotes the random vector comprising n i.i.d.Bernoulli random variables where the j th variable is withprobability x j and otherwise. Note that S in the current case is the set of receiver antennas [1 , · · · , N r ] . Definition 5. (Matroid) Consider a set S of n elements. Afamily I ⊆ S of subsets of S is called a matroid, denoted as M ( S, I ) , and I is called a family of independent sets underthe following two conditions: if X ⊆ Y and Y ∈ I , X ∈ I ; if X ∈ I and Y ∈ I with | X | ≤ | Y | , there exits { e } ∈ Y \ X such that X ∪ { e } ∈ I . Lemma 2.
The circuit power constraint of P is a matroid.Proof: The circuit-power constraint of P is linear andequivalent to − N r (cid:88) n =1 (1 − s n ) | g n | ≤ − p c . Thus, if this inequality holds for a set of receive antennas S , clearly it also holds for a subset A ⊆ S , thus satisfyingcondition of matroid definition. Condition can also beverified immediately, completing the proof.Consider a set S of n elements and a matroid M = ( S, I ) of subsets of S . Let w j for j = 1 , , . . . , n be given weightsof the elements of S . Let T ∈ I and let x T = [ x . . . x n ] bethe n -length vector, where x i = 1 if the i th element of S iscontained in T and x i = 0 otherwise. Definition 6. (Weight of an independent set) The weight ofthe subset T is given by w ( T ) = n (cid:88) i =1 w i x i . (3) With the above definitions and results, P can be readilytransformed into the following general problem of optimizinga sub-modular objective function over the convex hull of amatroid, which taps into the rich literature on algorithms forsolving such problems: ( P ) max x ∈ conv ( M ) f ( x ) (4)where f is a sub-modular function and conv ( M ) repre-sents the convex hull of M . Specifically, conv ( M ) = conv ( { I : I ∈ I} ) where I is the indicator vector of length | S | , where [ I ] j = 1 if j ∈ I . It follows from Lemma 1 and2 that P can be written in the form of P .In the remainder of this section, an approach is presentedfor computing an approximate solution for P (or equivalently P ). This approach comprises three algorithms described inthe sequel as follows. Consider a set S , a matroid M = ( S, I ) ,and the weights of the elements { w j } . The first algorithm,Algorithm , was proposed in [17] for maximizing the weightof an independent set in I , namely solving the maximumthe problem max I ∈I w ( I ) . The procedure for Algorithm is presented as follows. Algorithm : Weight Maximization Algorithm for an Inde-pendent Set.1) Rearrange the elements of S and obtain S = { e , e . . . , e n } such that w e ≥ w e ≥ · · · ≥ w e n .2) Initialize X ← ∅ .3) For i = 1 to n , doif ( X + e i ∈ M ) and w ( X + e i ) ≥ w ( X ) then X ← X + e i .4) Output X . Lemma 3 ([17]) . Algorithm solves the maximum weightindependent set problem max I ∈I w ( I ) . Using Algorithm as a sub-algorithm, the following contin-uous greedy algorithm as proposed [12] yields an approximatesolution for P . Algorithm : Continuous Greedy Algorithm1) Let δ = n , where | S | = n . Start with t = 0 and x (0) = ( x t (1) . . . x t ( n )) = .2) Let R t be a vector of size n , where R t ( j ) = 1 independently with probability x t ( j ) . For each j ∈ S ,estimate ω j ( t ) = E { f R ( t ) ( j ) } say by taking the averageof n samples, where f R ( t ) ( j ) = f ( R ( t ) ∪ { j } ) − f ( R ( t )) .
3) Let I ( t ) be a maximum weight independent set in M computed by Algorithm , according to weights ω j ( t ) .Let x ( t + δ ) = x ( t ) + δ. I ( t ) .4) Increment t := t + δ if t < , go to Step 2. Otherwise,return x (1) .As shown in [12], Algorithm has the useful property thatit has the guaranteed worst-case performance as specified inthe following lemma. emma 4. [12] The fractional solution x of the optimizationproblem P found by Algorithm satisfies F ( x ) = E { f (ˆ x ) } ≥ (1 − /e ) f ( x (cid:63) ) with high probability , where x (cid:63) gives the optimal value. The solution x output by Alogrithm may be fractional defined as follows. Given a y ∈ [0 , n , we say that i isfractional in y if < y i < , and for y ∈ C ( M ) , define y ( A ) = (cid:80) i ∈ A y i . Then, a set A ⊆ S is defined to be tight if y ( A ) = r M ( A ) , where r M ( A ) = max {| I | : I ⊆ A and I ∈ I} is the rank function of the matroid. For the caseof fractional solution x generated by Alogrithm , roundingeach element of x to be integers is necessary. The existenceof an efficient algorithm for this purpose is shown in thefollowing lemma from the result in [14]. Lemma 5.
Given a matroid M = ( S, I ) , and a monotone submodular function f : 2 S → R + , and a fractional solution x ∈C ( M ) , there exists a polynomial time randomized algorithm,which returns an independent set X ∈ I of value f ( X ) ≥ (1 − o (1)) F ( x ) , where F is the multi-linear extension, andthe o (1) term can be made polynomially small in n = | S | . A particular rounding algorithm, which was developed in[13], [14] and called the pipage rounding , is used in thispaper for rounding x to yield an integer solution. The detailedprocedure is presented as Algorithm Algorithm : Pipage Rounding Algorithm1) Let y denote the fractional solution output by Algo-rithm ;2) Find A , the minimal tight set containing at least fractional variables i, j ;3) Let y ij ( (cid:15) ) be the vector obtained byadding (cid:15) to y i , subtracting (cid:15) from y j andleaving the other values unchanged. Define (cid:15) + ij ( y ) = max { (cid:15) ≥ | y ij ( (cid:15) ) ∈ C ( M ) } ; and (cid:15) − ij ( y ) = min { (cid:15) ≤ | y ij ( (cid:15) ) ∈ C ( M ) } ;4) If F ( y ij ( (cid:15) + ij )) > F ( y ij ( (cid:15) − ij )) , then y ← y ij ( (cid:15) + ij ) else y ← y ij ( (cid:15) − ij ) ;5) If y is fractional, go to step 1. Otherwise, return y .It is shown in the following lemma that rounding usingAlgorithm does not compromise the worse-case performanceof Algorithm . Lemma 6. [13] The integer solution x int for problem P obtained by applying pipage rounding to the fractional output x of the continuous greedy algorithm satisfies f ( x int ) ≥ (1 − /e ) f ( x (cid:63) ) with high probability, where x (cid:63) is the optimalinteger solution. Finally, combining the above algorithms and their proper-ties, the main result of this section is presented as follows. A high probability in this paper means one whose difference with diminishes exponentially in n . Theorem 1.
The pipage rounded solution s int of the fractionalsolution s found by the continuous greedy algorithm for theantenna partitioning problem P satisfies C ( s int ) ≥ (1 − /e ) C ( s (cid:63) ) with high probability, where s (cid:63) solves P .Proof: From Lemma 1 and 2, problem P is a specialcase of problem P , and the result follows from Theorem 1. Remark:
Exploiting the sub-modularity of the IT channelcapacity function and matroidal circuit power constraint, Al-gorithms - give a guaranteed worst case approximation ratioof (1 − /e ) that is sufficiently close to . However, thecomplexity for these algorithms is high. The continuous greedyalgorithm starts with t = 0 and increments in the direction ofthe maximum weight independent set with a size of n ( n = N r ). In each increment, the weights of elements are foundby taking average of n independent samples. Thereafter,finding the maximum weight independent set has a complexityof O ( n log n ) . This results in an overall time complexity of O ( n ( n ( n ) + n log n )) = O ( n ) . Subsequently, the pipagerounding algorithm takes n iterations to convert the fractionalsolution given by the continuous greedy algorithm into anintegral solution. Thus, the overall time complexity of the − /e algorithm is O ( n ) . A more efficient approach isdescribed in the next section that trades performance for lowcomplexity. V. / -A PPROXIMATE S OLUTION
A simple greedy algorithm for approximately solving P isas follows. Algorithm : Greedy Algorithm1) Start with set s = ;2) At step i , s i = s i − + [0 . . . (cid:124)(cid:123)(cid:122)(cid:125) i (cid:63) . . . , where i (cid:63) = argmax i ∈{ , ,...,N r } log det (cid:18) I + PN t S i HH † S † i (cid:19) , where s i = s i − + [0 . . . (cid:124)(cid:123)(cid:122)(cid:125) . . . and S i is the di-agonal matrix corresponding to s i as mentioned before;3) If s i (cid:63) satisfies n (cid:80) j =1 (1 − s i (cid:63) ) | g n | ≥ p c , repeat for i = i + 1 , else, output s i − .The worse-case performance of the greedy algorithm isquantified as follows. Let S (cid:63) be a set that maximizes the valueof f over all subsets of the matroid. The following result isknown for using greedy algorithms for maximizing monotonesub-modular functions under a matroid constraint. Theorem 2. [15] For a non-negative, monotone sub-modularfunction f and a matroid constraint, let subset S be obtainedby selecting elements one at a time, each time choosing anelement that provides the largest marginal increase in thefunction value that is feasible with respect to the matroidconstraint. f ( S ) ≥ f ( S (cid:63) ) . orollary 1. The objective function of problem P evaluatedat the greedy algorithm GA’s output ≥ OPT / , where OPT is the value of the optimal solution.Proof:
From Lemma 1, we know that the objectivefunction of P is monotone and sub-modular. Moreover, thelinear circuit power constraint is a matroid from Lemma 2.Thus, using Theorem 2, we have that the greedy algorithmgives a / approximation to P . Remark: If S has n elements, the / approximate algorithmwill terminate in a maximum of n iterations, as a new elementis included in every iteration. In each iteration, the elementwith the highest marginal gain can be obtained by findingthe maximum of n entries. This results in a running timecomplexity of O ( n ) . Thus, even though, the continuousgreedy algorithm gives a better bound on performance than thegreedy algorithm, it has a running time complexity of O ( n ) ascompared to O ( n ) for the greedy algorithm. Hence, both thealgorithms have their own significance and which algorithmto be used depends on the problem at hand.VI. E XTENSION TO THE C ASE OF
CSITIn this section, the results in the preceding section areextended to the case of CSIT on the IT channel H that enablesthe base station to allocate power over transmit antennas forincreasing the IT channel capacity. Again, let s n ∈ { , } indicate whether the n -th receiver antennas at the mobile isassigned for information transfer ( s n = 1 ) or power transfer( s n = 0 ). With CSIT, the power allocation (input covariancematrix) at different antennas of the base station depends onreceiver antenna assignment vector s , and the capacity ismodified from (1) as C ( s ) = max Q , tr( Q ) ≤ P log det (cid:0) I + SHQH † S † (cid:1) , = max p i , (cid:80) Nri =1 p i ≤ P N r (cid:88) i =1 log(1 + s i λ i ( s ) p i ) , (5)where λ i ( s ) are the eigen-values of the submatrix H ( s ) of H which is obtained by keeping all rows of H for which s i = 1 ,and without loss of generality we have assumed that N t ≥ N r .The power allocation p i at the basestation depends on thereceiver antennas allotted for data transfer, i.e., p i depend on s i , and the optimal power allocation is given by waterfilling.Compared to (1), the expression inside the max in (5) is simple(sum of N r parallel channels), however, together with the max,the overall capacity expression is more complicated. Moreover,note that the choice of s i and p i depend on each other. Thus,the antenna partitioning problem in the CSIT case is morecomplex than the CSIR case. It follows that the correspondingproblem of optimal antenna partitioning, denoted as P , is given as ( P3 ) max { s } max p i , (cid:80) Nri =1 p i ≤ P (cid:88) N r i =1 log(1 + s i λ i ( s ) p i ) s.t. s n ∈ { , } , n = 1 , , · · · , N rN r (cid:88) n =1 (1 − s n ) | g n | ≥ p c . Even though the capacity expression (5) involves a maxi-mization, it is still a sub-modular function as shown in [16].We summarize the result of [16] as follows.
Theorem 3.
For a set S , and fixed s i , i ∈ S , the rate R ( S ) = max p i , (cid:80) i ∈ S p i ≤ P (cid:88) i ∈ S log(1 + s i λ i ( s ) p i ) , obtained with a set S of parallel Gaussian channels using theoptimal waterfilling algorithm is a sub-modular function overthe set of channels S . It is easy to show that (5) is a monotone function, sinceadding more receiver antennas cannot decrease the rate. Thus,we have that similar to the CSIR case, P is a special case ofproblem P , and we get results on solving P as describedbelow. Theorem 4.
The pipage rounded solution x int of the fractionalsolution x found by the continuous greedy algorithm on P satisfies f ( x int ) ≥ (1 − /e ) OPT with high probability, where
OPT is the optimal value of themutual information in P . Theorem 5.
The objective function of problem P evaluatedat the greedy algorithm GA’s output ≥ OPT / for solving P ,where OPT is the value of the optimal solution of P . VII. S
IMULATION R ESULTS
In this section, we illustrate the numerical performance ofthe two approximation algorithms presented in this paper tomaximize the mutual information while satisfying the circuitpower constraint. For the CSIR case, in Fig. 2, we plot thethroughput (mutual information) as a function of receiverantennas N r for fixed N t = 5 and circuit power constraintof . N r with total transmit power P = 5 W = 6 . dB under a Rayleigh fading assumption on the channel matrix H . We scale the circuit power constraint with N r since largerthe number of receiver antennas more is the power requiredfor their operation. As can be seen from Fig. 2, both thecontinuous greedy and the greedy algorithm perform betterthan their worst case bound of − /e and / , respectively.In Fig. 3, we plot the throughput as a function of receiverantennas N r for the CSIT case, with identical parameters usedfor Fig. 2. Once again we see that the continuous greedyand the greedy algorithm perform better than their worst casebound of − /e and / .An important observation one can make from Fig. 2 andFig. 3, is that the greedy algorithm outperforms the continuousreedy + pipage rounding algorithm, even though, the worst-case performance guarantees of the continuous greedy +pipage rounding algorithm is better than the greedy algorithm. T h r oughpu t Throughput Comparison as a function of receive antennas N r Bruteforce optimalGreedy 1/2 approximate algoritm(1/2)−optimal throughputContinuous greedy (1−1/e) approximate algorithm(1−1/e)−optimal throughput
Fig. 2. Mutual information comparison with different algorithms for CSIR T h r oughpu t Throughput Comparison as a function of receive antennas N r with CSIT Optimal throughputGreedy 1/2 approximate algoritm(1/2) − optimal throughputContinuous greedy (1 − − − optimal throughput Fig. 3. Mutual information comparison with different algorithms for CSIT
VIII. C
ONCLUSION
In this paper, we have considered a SWIPT system andfound theoretical guarantees on a combinatorial problem ofpartitioning mobile antennas for information and power trans-fers to maximize the IT channel capacity under a circuitpower constraint. We exploited the sub-modular property ofthe mutual information expression for both the CSIR andCSIT cases, as well as the matroidal property of the circuit power constraint. To the best of our knowledge, this is anovel approach in the area of SWIPT, and earlier approachesrelied on relaxed problems, where each antenna is required toperform the dual role of information and power transfer mode,which is hard to realize in practice.R
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