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Optimal power allocation for diffusion‑typesensor networks with wireless information andpower transfer

Yang, Gang; Tay, Wee Peng; Guan, Yong Liang; Liang, Ying‑Chang

2019

Yang, G., Tay, W. P., Guan, Y. L. & Liang, Y.‑C. (2019). Optimal power allocation fordiffusion‑type sensor networks with wireless information and power transfer. IEEE Access,32408–32422. doi:10.1109/ACCESS.2019.2904084

https://hdl.handle.net/10356/82497

https://doi.org/10.1109/ACCESS.2019.2904084

© 2019 IEEE. Translations and content mining are permitted for academic research only.Personal use is also permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for moreinformation.

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Received February 12, 2019, accepted March 4, 2019, date of publication March 11, 2019, date of current version March 26, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2904084

Optimal Power Allocation for Diffusion-TypeSensor Networks With Wireless Informationand Power TransferGANG YANG 1,2, (Member, IEEE), WEE PENG TAY 3, (Senior Member, IEEE),YONG LIANG GUAN 3, (Senior Member, IEEE), AND YING-CHANG LIANG 2, (Fellow, IEEE)1National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China (UESTC), Chengdu611731, China2Center for Intelligent Networking and Communications (CINC), University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China3School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798

Corresponding author: Ying-Chang Liang (liangyc@ieee.org)

This work was supported by the National Natural Science Foundation of China under Grant U1801261, Grant 61631005, andGrant 61601100.

ABSTRACT This paper investigates the problem of power allocation for distributed estimation via diffusionin wireless sensor networks (WSNs) with simultaneous wireless information and power transfer (SWIPT).We consider a WSN consisting of smart sensor nodes (SNs) and common sensor nodes (CNs), and eachSN is capable of performing SWIPT via multi-antenna beamforming to its neighboring (i.e., near-tier) CNs.In each diffusion iteration, all nodes collect measurements and exchange intermediate estimates with theirneighbors. We first analyze the effect of each SN’s beamforming design and each near-tier CN’s harvestedpower allocation on the steady-state network-wide mean square deviation (MSD) of the diffusion least-mean-squares (LMSs) strategy. Then, we formulate a problem to minimize an upper bound MSD by jointlyoptimizing the global power allocation weights for each SN to perform beamforming, and the local powerallocation proportion for each CN to perform measurement collection. We further show that the formulatednon-convex problem is decomposable and propose a gradient-based iterative algorithm to find the optimalsolution. In addition, for practical implementation, we propose adaptive online approaches to estimate someparameters required for system optimization. Finally, extensive simulation results demonstrate that withoptimal power allocation, our proposed scheme improves the MSD performance significantly, compared tothe conventional diffusion LMS strategy without wireless power transfer (WPT).

INDEX TERMS Diffusion, least-mean-squares (LMS), mean-square-deviation (MSD), simultaneous wire-less information and power transfer (SWIPT), wireless power transfer (WPT), problem decomposition.

I. INTRODUCTIONWireless sensor networks (WSNs) are widely deployedfor various Internet-of-Things (IoT) applications such asenvironmental monitoring, industrial sensing, and track oflocalized phenomenon like a sound source [1]–[4]. Typi-cally, sensors have limited energy, storage, computation abil-ity, and bandwidth to communicate with their neighbors.To reduce the bandwidth requirement and energy consump-tion, distributed estimation strategies forWSNs have recentlydrawn growing research interest [5]–[10]. Various distributed

The associate editor coordinating the review of this manuscript andapproving it for publication was Lei Jiao.

estimation strategies have been proposed, including consen-sus strategies [7] and diffusion strategies [10]–[14]. For dis-tributed estimation, each sensor in the network exchangesinformation (e.g., neighborhood data and intermediate esti-mates) with its neighbors in each iteration. By fusing itsown local information with information received from itsneighbors, each sensor aims to estimate a global parameterof interest. Comparatively, the diffusion strategies are moreattractive than the consensus strategies, because they endownetworks with real-time adaptation and learning abilities, andperform better under mild conditions [10].

Since the operation of WSNs is limited by fixed energybudgets, various methods were proposed to reduce the energy

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consumption for distributed estimation strategies, such asselective collaboration [15]–[18], bandwidth allocation [19],and data reservation [20]. In particular, for neighboring sen-sor selection, Zhao and Sayed [15] proposed for each sensorto select only one neighbor for consultation, based on itsall neighbors’ current deviation estimates and a variance-product metric. The joint optimization of neighbor selectionand power allocation was studied in [16]. A hierarchical-averaging consensus scheme was proposed in [17], in whichsensors are divided into layered clusters. Hu and Tay [18] pro-posed a multi-hop diffusion strategy to perform distributedestimation under local and network-wide energy constraints.Each node’s information neighborhood is optimized beforethe diffusion process. A data-reserved periodic diffusionleast-mean-squares (LMS) algorithm was proposed in [20]to reduce power consumption without significant degrada-tion of convergence performance. In the above literature,the network-widemean square deviation (MSD) performance(i.e., the average MSD of the sensors’ estimates for the trueparameter of interest) is fundamentally limited by the fixedenergy budgets of all sensor nodes, which limit the quality ofsensor measurements and information exchanges. To addressthis energy bottleneck, an approach is to introduce additionalenergy replenishment via energy harvesting at sensor nodes.

Among the technologies of harvesting energy from var-ious sources like solar energy and wireless energy, radio-frequency (RF)-enabled wireless power transfer (WPT) hasemerged as a promising approach for IoT applications,1 dueto its flexibility, low cost and long supporting distance up totens of meters [21], [22]. An RF-energy harvester is based ona rectifying circuit, consisting of a diode and a passive low-pass filter, which converts the received RF signal to a direct-current (DC) signal to charge an energy storage component(e.g., battery, (sup-)capacitor). The RF energy can be radi-ated from the transmitter either isotropically or toward somedirection(s) through beamforming. The beamforming trans-mission, referred to as energy beamforming [23], [24], canachieve higher power transmission efficiency than isotropictransmission, by using an antenna array to generate a beamtoward the energy receiver(s) [25]–[28]. In [25], the fea-sibility of WPT by using massive transmit antennas wasjustified under practical system parameters. In [26], the opti-mal channel training and beamforming scheme was designedto maximize the net harvested energy. The optimal energybeamforming was studied for point-to-point backscattercommunication system in [27], and backscatter relay com-munication system in [28]. Furthermore, RF-enabled WPThas recently been integrated with wireless communicationsystems, such as simultaneous wireless information andpower transfer (SWIPT) [23], [29]–[33], and wireless pow-ered communication networks (WPCN) [34]–[39]. In partic-ular, Zhou et al. [29] proposed two architectures, namelypower splitting and time switching, for the receiver of

1Notice that magnetic induction only transfers energy in tens of centime-ters and is thus not applicable to WSN applications [21].

a SWIPT system. The SWIPT system was also studiedfrom other aspects such as rate-energy tradeoff [23], energyefficiency [30], energy harvesting models [32], and wave-form design [33].

RF energy harvesting and energy sharing are recently uti-lized to enhance the performance of centralized estimationin WSNs [40]–[42]. Psomas and Krikidis [40] studied thecentralized estimation of a random event in a wireless pow-ered millimeter-wave sensor network, in which an accesspoint (AP) first performs energy beamforming to multi-ple sensors in the downlink and then estimates a randomsource after receiving measurements from those sensors inthe uplink. In [41], a remote fusion center (FC) first receivesmeasurements of a random source from multiple sensors thatcan harvest energy and also share energy with others, andthen uses them to estimate the random source in a centralizedmanner. A distortion minimization problem was studied andthe optimal energy allocation policy for transmission andsharing is derived. Similarly, in [42], the distortion minimiza-tion problem for centralized estimation using multiple sensorwith only the ability of energy harvesting was formulated asa Markov decision process based stochastic control problem,and the optimal energy allocation policies were obtained byusing dynamic programming techniques.

To the best of our knowledge, exploiting SWIPT toimprove the performance of distributed estimation has notbeen studied in the literature. The harvested RF energy can beused to enhance the accuracy of measurement collection andthe reliability of information exchange between sensors overwireless fading channels, thus leading to better distributedestimation performance.

In this paper, we consider a hybrid diffusion-type WSNwith SWIPT, which consists of common sensor nodes (CNs)each with a single antenna and smart sensor nodes (SNs) eachwith multiple antennas. Each SN has unlimited energy2 andis capable of performing RF-enabled WPT via beamform-ing to its neighbors, while each CN has a limited energystorage and can harvest RF power to replenish its energy.In each iteration of the diffusion LMS strategy, all nodescollect measurements and exchange intermediate estimateswith their neighbors [10]; specifically, each SN performsSWIPT to its neighboring (namely, near-tier) CNs via beam-forming. The energy budgets of all near-tier CNs can thusbe increased. This is in sharp contrast to the aforementionedliterature [41], [42], in which the energy budgets are fixed.The basic idea of this paper is to use WPT to improve thediffusion performance [43]. Our main contributions are asfollows:• We establish the models for information exchange andWPT for the hybrid diffusion-type WSN. Specifically,we design the beamformer for each SN to performSWIPT to its near-tier CNs over wireless fading chan-nels. We further analyze the effect of the designed SNbeamformer’s (global) power allocation weights and

2For instance, an SN can be plugged into the power grid.

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each near-tier CN’s (locally) harvested power allocationproportion on the reliability of information exchangeand the accuracy of measurement collection, both ofwhich determine the steady-state network-wide MSD ofthe diffusion LMS strategy.

• We derive both the exact expression and an upper boundof the steady-state network-wide MSD for such a hybriddiffusion-typeWSN. Furthermore, we formulate a prob-lem to minimize the upper-bound MSD, by jointly opti-mizing the global power allocation weights for SNs toperform SWIPT via beamforming, and the local powerallocation proportions for near-tier CNs to use har-vested energy for measurement collection. The formu-lated problem is non-convex and thus difficult to besolved directly.

• We show that the proposed steady-state network-wideMSD minimization problem is decomposable into opti-mization subproblems. We propose a gradient-basediterative algorithm to find the optimal solutions. More-over, for practical implementation, we propose adap-tive online approaches to estimate various parametersrequired for system optimization, such as the link-noisevariance and the measurement-noise variance.

• Extensive simulation results demonstrate that with opti-mal power allocation, the proposed scheme improvesthe steady-state network-wide MSD performance sig-nificantly. Under typical parameter settings, the MSDof the proposed scheme is reduced by about ten times(i.e., 9.85 dB) than that of the conventional scheme.We obtain useful insights for the optimal power allo-cation, as well as the effect of system parameters likethe inter-node distances and the SN’s antenna numberon the MSD performance. The proposed scheme is alsonumerically shown to be robust to the estimation errorof the measurement-noise power.

This paper is organized as follows: In Section II, we estab-lish the system model and describe the diffusion strategywe adopt in this paper. In Section III, we present the mod-els for information exchange and WPT over wireless fad-ing channels. In Section IV, we formulate an optimizationproblem to minimize the steady-state network-wide MSD.In Section V, we show that the problem is decomposable,and propose a gradient-based iterative algorithm to find theoptimal solutions, followed by adaptive online approachesfor estimating some parameters required for implementingthe proposed algorithm. Simulation results are presented inSection VI. Finally, Section VII concludes this paper.Notations: For readers’ convenience, the main notations

used in this paper and their meanings are listed in Table 1.Moreover, we use the following math operations in this paper.(·)∗, (·)T and (·)H denotes complex conjugation, transposeand complex conjugate transposition, respectively. λmax(A)and ρ(A) denotes the largest eigenvalue and the spectralradius of a matrix A, respectively. The operation A ⊗ Bdenotes the Kronecker product of two matrices A and B.vec(·) denotes a column vector in which the columns of its

FIGURE 1. An example of a hybrid diffusion-type WSN. (Nodes 3 and 7 areSNs, while nodes 1, 5 and 10 are far-tier CNs, with all left near-tier CNs.)

matrix argument are stacked on top of each other. diag{·}means a diagonal matrix constructed from its arguments.The operator E[·] denotes the mathematical expectation. Theabbreviation ‘‘s. t.’’ stands for ‘‘subject to’’. C denotes the setof complex numbers.

II. SYSTEM MODELIn this section, we first present the hybrid diffusion-type WSN with information-plus-power transfer, and thendescribe the diffusion strategy we adopt, followed by theMSD performance analysis for the diffusion strategy.

A. SYSTEM DESCRIPTIONWe consider a hybrid diffusion-type WSN consisting of Nsensors (denoted by the setN ) which includes SNs (denotedbyN s) and CNs (denoted byN c), i.e.,N , N s

∪N c. Oneexample of such hybrid WSNs is illustrated in Fig. 1. Twosensor nodes are defined to be neighbors if they can exchangeinformation (i.e., intermediate updated estimates). The neigh-borhood of node k including node k itself, denoted byNk , canbe predefined according to the network topology and inter-node distances. We assume that the coverage of each pair ofSNs do not overlap, such that the number of SNs is reducedfor lower cost and the interference among SNs is negligible.Each SN has L (L ≥ 1) antennas, and is thus able to performSWIPT to its neighbors via beamforming. Each CN has asingle antenna that can be used to exchange information withneighbors and to harvest RF energy. In particular, each CNin the neighborhood of an SN m can receive energy3 fromSN m. We assume that the CN associated with SN m cannotreceive energy from other SNs, since other SNs are far awayfrom the CN and the efficiency of energy transfer decaysexponentially with the distance. For convenience, we use theset N c

n (and N cf ) to denote all near-tier (and far-tier) CNs

each of which is (not) within the neighborhood of an (any)SN, i.e.,N c , N c

n ∪N cf . Each near-tier CN has both battery

power and additional power from the energy harvester, whileeach far-tier CN has only battery power.

In the considered hybrid diffusion-type WSN, each SNperforms SWIPT to its associated near-tier CNs, each near-

3It is demonstrated in state-of-the-art literature [22], [40] that the har-vested RF energy is able to power wireless sensors at distance of tens ofmeters.

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TABLE 1. Main notations and meanings.

tier CN transfers information to its associated SN and/or itsneighboring CN(s), and each far-tier CN transfers informa-tion to its neighboring CN(s). This paper aims to minimizethe whole network’s diffusion MSD, by jointly optimizingthe global power allocation for SWIPT from the SNs to theirnear-tier CNs, as well as the local allocation of harvestedpower at each near-tier CN.

We consider the diffusion strategy and adopt the samenotation as in [44]. In each iteration (i.e., time instant)i, i ≥ 0, sensor k uses a regression data vector uk,i ∈ CMto collect a scalar measurement dk,i. All vectors in this paperare defined to be column vectors. Note that we use subscriptsto refer to the time-dependence of scalar variables (e.g., as indk,i) and vector variables (e.g., as in uk,i), respectively. Theparameter of interest is denoted by a vector wo

∈ CM .The measurements across all sensors are assumed to followthe following linear regression model:

dk,i = uTk,iwo+ vk,i, (1)

where vk,i denotes the measurement-noise with zero meanand variance or measurement-noise-power σ 2

v,k . For adiffusion-type WSN, the objective is to compute an estimatew of wo in a distributed manner by solving the followingconstrained LMS problem

minw

N∑k=1

E[∣∣∣dk,i − uTk,iw

∣∣∣2] , (2)

where the expectation is taken with respect to dk,i and uk,i.

B. DIFFUSION ALGORITHMS WITH NOISY INFORMATIONEXCHANGEDiffusion strategies enable the solution of the problemin (2) in a distributed and adaptive manner. Specifically, fornoisy information exchange between nodes, the adapt-then-combine (ATC) diffusion strategy [44] is performed using thefollowing update equations at each node k and time instant i

ψk,i = wk,i−1 + µku∗k,i[dk,i − uTk,iwk,i−1

],

wk,i =∑l∈Nk

alkψl,i + v(ψ)k,i ,(3)

where µk is a small positive step-size parameter, alk arecombination weights, and v(ψ)k,i is the aggregate noise over theneighborhood of sensor k and given by

v(ψ)k,i ,∑

l∈Nk\{k}

alkv(ψ)lk,i, (4)

where v(ψ)lk,i is the noise of the link from node l ∈ Nk to node k .

Each element of v(ψ)lk,i is of zeromean, and has variance or link-

noise power σ 2(ψ)lk . Here, the combination weights satisfy

alk ≥ 0, A1N = 1N , and alk = 0 if l /∈ Nk , (5)

where A is the combination weight matrix. In this paper,we assume that A is predefined and fixed.In each iteration, the first operation in (3) is an adaptation

step where each node k uses its data {dk,i, uk,i} to obtain theintermediate estimate ψk,i, and the second operation in (3) isa combination step where node k aggregates the intermediateestimates ψl,i’s from its neighbors l ∈ Nk to obtain its

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updated estimate wk,i. The second operation in (3) involvesthe sharing of information between sensor k and its neigh-bors, i.e., each neighbor l of sensor k sends its intermediateestimator ψl,i to node k over a noisy wireless link. In thispaper, we consider only the ATC form of diffusion since itoutperforms other alternative diffusion strategies under mildtechnical conditions [10].

We make the following assumptions on the measurementdata and noise.

1) The regression data uk,i are temporally white and spa-tially independent random vectors with zero mean andcovariance matrix Ru,k , E

[u∗k,iuk,i

]. The corre-

sponding block covariance matrices is denoted as fol-lows

Ru , diag{Ru,1, · · · , Ru,N }. (6)

2) The link-noise signal v(ψ)lk,i are temporally white andspatially independent random vectors with zero meanand covariance matrix

R(ψ)v,lk = σ

2(ψ)lk IM . (7)

3) The regression data {uk1,i1} and the link-noise {vl2 k2,i2}are mutually-independent random variables for all{k1, i1} and {l2, k2, i2}.

The above assumptions imply that the aggregate noise v(ψ)k,ihas zero mean and the following covariance matrix

R(ψ)v,k =

∑l∈Nk\{k}

a2lkR(ψ)v,lk =

∑l∈Nk\{k}

a2lkσ2(ψ)lk IM . (8)

C. MEAN-SQUARE-DEVIATION PERFORMANCEBefore introducing some useful results on the MSD of thediffusion strategy, we denote the block matrixM of the step-sizes {µn}, the weighted block matrix S of the regressionvectors’ covariance matrices {Ru,n}, and the block matrixR(ψ)v of the aggregate noise’ covariance matrices {R(ψ)

v,n },as follows

M , diag{µ1IM , · · · , µN IM }, (9)

S , diag{σ 2v,1Ru,1, · · · , σ

2v,NRu,N }, (10)

R(ψ)v , diag{R(ψ)

v,1 , · · · , R(ψ)v,N }. (11)

To guarantee the mean and mean-square convergence ofthe ATC strategy, the following matrix B is required to bestable

B , AT (INM −MRu) . (12)

That is, the spectral radius of B, denoted as ρ(B), is less than1.

As shown in [44], the ATC diffusion strategy is both meanand mean-square convergent if the step-sizes {µk} are suffi-ciently small, i.e.,

µk <2

λmax(Ru,k

) , (13)

for k = 1, 2, · · · ,N , where λmax(A) denotes the largesteigenvalue of the matrixA. In the sequel, we assume the step-sizes {µk} are chosen subject to (13).

We define the error vector for sensor node k in the i-thiteration to be

wk,i , wo− wk,i. (14)

The network-wide MSD is then defined as

MSD , limi→∞

1N

N∑k=1

E[∥∥wk,i

∥∥2] . (15)

From [10, eq. (457)] and [44, eq. (119)], we have the fol-lowing lemma on network-wide MSD.Lemma 1: Assuming the step-sizes {µk} satisfy (13),

the network-wide MSD is given by

MSD =1N

[vec

(ATMSMA+R(ψ)

v

)]T· (IN 2M2 − F)−1vec(INM ), (16)

where F , B ⊗ B∗ and ⊗ is the Kronecker product.Moreover, the MSD in (16) is upper bounded as follows

MSD ≤c2

N·

Tr(ATMSMA+R(ψ)

v

)1− [ρ(INM −MRu)]2

, (17)

where c is some positive scalar in [44, eq. (118)], and ρ(·)denotes the spectral radius of its matrix argument.

From (8), (10) and (17), we note that the upper bounddepends on the measurement-noise power {σ 2

v,k} and the

link-noise power{σ2(ψ)lk

}. The overall MSD can thus be

reduced, if the measurement-noise power or link-noise poweris reduced. The measurement-noise power can be reduced bypractical methods like aggregating multiple samples in eachmeasurement. Typically, higher power is required for takingmore samples. Also, the link-noise power can be equiva-lently reduced by increasing the transmission power. Theadditionally required power can be replenished by harvestingRF-energy from the associated SN. This motivates us to useRF-enabled WPT from SNs to CNs to reduce the network-wide MSD, which will be studied in the sequel sections.

III. WIRELESS INFORMATION EXCHANGE ANDPOWER TRANSFERIn this section, we establish our models for informationexchange and WPT over wireless fading channels for thedifferent categories of sensor nodes: SNs, far-tier CNs andnear-tier CNs.

A. WIRELESS INFORMATION AND POWER TRANSFERFROM SNSIn our system framework, each SN m ∈ N s isequipped with L antennas, and performs beamforming forSWIPT to its neighboring CNs. Each neighboring CN per-forms information decoding and RF-energy harvesting usingpower-splitting technique [29]. We make the following twoassumptions.

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• We assume independent block Rayleigh fading,4 i.e., thechannel from node l to node k in iteration i, denotedby hlk,i, follows the L-dimensional Gaussian distribu-tion with zero mean and variance βlk , i.e., hlk,i ∼CN (0L , βlkIL), where βlk is the path loss.

• All nodes have perfect knowledge of the channels fortheir neighbors, and all near-tier CNs know the beam-former used by its associated SN. The resulting overheadis assumed to be negligible and thus omitted in thispaper.

• We assume that all nodes operate in the half-duplexmode (i.e., each node transmits and receives in dif-ferent time slots), and all nodes transmit informationin an orthogonal multiple-access-channel (MAC) (e.g.,in a time-division-multiplexing-access (TDMA) man-ner), to avoid interference among both neighboringnodes and non-neighboring nodes.

SN m first quantizes its intermediate estimate ψm,i, per-forms coding and digital modulation. We denote one mod-ulated symbol by sm,i, and assume that sm,i has unit power.Then, it performs beamforming to all its neighboring CNs.The beamformer5 is given by

bm,i =∑t∈Nm

√ξmt

hmt,i‖hmt,i‖

, (18)

where the power allocation coefficients ξmt ’s are subject to∑t∈Nm

ξmt = 1. The beamformer in (18) is a maximum-ratio-transmit (MRT)-type beamformer, with one beamtowards each neighbor. The coefficients ξmt ’s of the (global)power allocation among neighboring CNs are adjustable andwill be optimized in Section IV. Each SN is assumed to havesufficient power supply, and thus the power consumption forperforming beamforming is not considered in this paper.

Let pm be the fixed transmission power of SN m, whichis set according to RF regulations, for purpose of safety. Thetransmitted signal is thus written as

xm,i =√pmsm,ibm,i. (19)

At the neighboring near-tier CN k , the received signal rk,1,iis given by

rk,1,i = xHm,ihmk,i + nmk,1,i=√pmbHm,ihmk,ism,i + nmk,1,i, (20)

where nmk,1,i is the noise introduced at the receive antenna.CN k splits the received signal rn,1,i into two streams,i.e.,√ρkrk,1,i for RF-energy harvesting and

√1− ρkrk,1,i for

4To focus on the power allocation for diffusion-type WSN, thispaper considers Rayleigh fading channels as in SWIPT/WPCN sys-tems [23], [26], [37], which provides worst-case power-transfer andinformation-transfer performances for other line-of-sight channels likeRician channel. The performances under Rician channel can be easilyobtained by following similar steps in [25], [33], and [36], which will bethe future work.

5This beamformer is asymptotically optimal, as the number L of antennasat SNs tends to infinity [35]. And it enables flexible power allocation fordifferent neighboring CNs.

information decoding. The power splitting ratio ρk ∈ [0, 1]is assumed to be fixed.

1) RF-ENERGY HARVESTINGAs common assumptions in the RF-based WPT litera-ture [23], [24], [33]–[35], we assume that the harvested poweris linearly proportional6 to the received power and the noisepower cannot be harvested. From [35], the harvested power(i.e., average harvested energy normalized to the time dura-tion of one iteration) by near-tier CN k is given by

pk (ξmk ) = αkEbm,hmk,i

[∣∣∣√pmρkbHmhmksm,i∣∣∣2] , (21)

where αk ∈ (0, 1) is the energy conversion efficiency forsensor k .Remark 1: In practice, the RF power can be harvested only

when it exceeds some threshold [21]. For convenience ofanalysis, we ignore such threshold in the energy harvestingmodel (21). In practice, the set of near-tier CNs can be definedsuch that the received power at neighboring near-tier CNsexceed such threshold.

We further have the following lemma on the harvestedpower at each near-tier CN k .Lemma 2: Given the power allocation weights ξm of SN

m, the average harvested power by near-tier CN k is given by

pk (ξmk ) = αkρkpmβmk((L − 1)ξmk + 1

). (22)

In (22), the term Lξmk is from the beam towards CN k , andthe term (1 − ξmk ) is from beams towards other neighboringCNs.

Proof: See proof in Appendix A.Remark 2: Notice that the power in (22) is the harvested

power by averaging over all possible channel realizations, forgiving the power allocation weights ξmk ’s. It can be treatedas the average energy level in an energy storage componentwhich has infinite capacity and is recharged by the harvester.The optimal allocation weights to be obtained in the sequelwill be optimal in the average sense. The focus of this paperis to jointly optimize each SN’s global power allocationamong near-tier CNs, as well as each near-tier CN’s localpower allocation for measurement collection and informationtransmission. To state it more clearly, we ignore the dynamicsof energy storage component in this paper.

2) INFORMATION DECODINGFor information decoding at near-tier CN k , the stream√1− ρkrk,1,i is further corrupted with baseband noise nmk,2,i

with power σ 2k,2. Typically, the noise nmk,1,i is negligible

compared to the noise nmk,2,i, and is thus ignored in theanalysis. That is, the signal for information decoding is givenby

rk,2,i =√pm(1− ρk )bHm,ihmk,ism,i + nmk,2,i. (23)

6We adopt the linear model for analytical tractability and performanceoptimization in this paper, although there are some non-linear energy-harvesting models [33], [45].

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The recovered data can be rewritten as sm,i = sm,i + nmk,i,where the equivalent noise nmk,i has zero mean and powergiven by

σ 2mk (ξmk ) =

σ 2k,2pmβmk (1−ρk )((L−1)ξmk+1)

. (24)

After performing digital demodulation and decoding,the information vector is recovered by near-tier CN k asψmk,i = ψm,i + v(ψ)mk,i, where the overall noise v

(ψ)mk,i includes

the noises from digital decoding and quantization.We assumethat the quantization noise is negligible, and the power of theoverall noise is proportional to the power of decoding noisenmk,i. From (24), the power of noise v(ψ)mk,i is thus obtained as

σ2(ψ)mk (ξmk ) =

εσ 2k,2pmβmk (1−ρk )((L−1)ξmk+1)

, (25)

where the positive constant ε depends on the digital modula-tion and coding schemes and can be measured by each nodein practice.

B. WIRELESS INFORMATION TRANSFER FROMFAR-TIER CNSWith only battery power, we assume that each far-tier CN l ∈N c

f uses the default power pdefl,1 to collect measurements, and

uses the default power pdefl,2 to broadcast its intermediate esti-

mate ψl,i to all its neighbors. With the default transmissionpower pdef

l,2 , the resulting noise power of the link from node l

to its neighboring node k is σ 2(ψ)lk . We assume that the battery

power for each far-tier CN is sufficient for measurementcollection and information broadcasting; otherwise, it can beremoved from the network topology.

C. WIRELESS INFORMATION TRANSFER FROM NEAR-TIERCNSWith the battery power, each near-tier CN has the samedefault measurement power and transmission power as afar-tier CN. However, with the additional harvested powerpk (ξmk ) in (22), each near-tier CN k uses additional powerpk,1 out of the harvested power for measurement collection.The remaining harvested power is used as additional powerfor information transmission, and given by

pk,2(ξmk , pk,1) = αkρkpmβmk((L − 1)ξmk + 1

)− pk,1.

(26)

We assume that the maximal transmission power for node k ispk,2,max. Hence, the constraint pk,2(ξmk , pk,1)+ pk,2(def) ≤pk,2,max, ∀k, should be satisfied in practice.

For near-tier CN k , higher power for collecting measure-ments leads to lower measurement-noise power. This can beachieved, for example, by taking multiple observation sam-ples, and taking the sample mean as the final measurement.The measurement-noise power of CN k is given by

σ 2v,k (pk,1) =

pdefk,1σ

2v,k

pdefk,1 + pk,1

. (27)

The measurement-noise powers for all sensor nodes can thusbe summarized as

σ 2v,k =

σ2v,k (pk,1) =

pdefk,1σ

2v,k

pdefk,1 + pk,1

, if k ∈ N cn

σ 2v,k , if k ∈ N c

f ∪Ns

(28)

Also, higher power for wireless transmission increases thereliability of information transfer over wireless links. In par-ticular, for the link from near-tier CN k to its neighbor q,the link-noise power is equivalently given by

σ2(ψ)kq (ξmk , pk,1)

=pdefk,2σ

2(ψ)kq

pdefk,2 + pk,2(ξmk , pk,1)

=pdefk,2σ

2(ψ)kq

pdefk,2 + αkρkpmβmk

((L − 1)ξmk + 1

)− pk,1

. (29)

From (22), (26) and (29), the link-noise power for anysensor node q can be summarized as in (30), shown at thetop of the next page.

By endowing the SNs with the ability of SWIPT, the addi-tionally harvested power by its associated near-tier CNs isused to increase their accuracy of measurement collection,as indicated in (28). Also, when other communication modessuch as modulation and coding are fixed, the additionallyharvested energy can further increase the reliability of infor-mation exchanges between sensors, as indicated in (30).

IV. PROBLEM FORMULATION FOR MSD MINIMIZATIONIn this section, we formulate a problem to minimize theupper-bound MSD in (17) for the considered diffusion-typeWSN with information-plus-power transfer.

With WPT from SNs to near-tier CNs, the matrix Sin (17) is given by (10) with σ 2

v,k replaced by the reduced

measurement-noise power σ 2v,k in (28), and the matrixR(ψ)

v is

given by (11) with σ 2(ψ)lk replaced by the reduced link-noise

power σ 2(ψ)lk in (30).

With fixed step-size matrix M and regression vectors’covariance matrices {Ru,k}, our objective is to minimize thenumerator of the upper-bound MSD in (17), given by thefollowing expression

Tr(ATMSMA+R(ψ)

v

)=

N∑k=1

u2kTr(Ru,k)σ2v,k

∑j∈Nk

a2kj+M∑

q∈Nk\{k}

a2kqσ2(ψ)kq

,(31)

where σ 2v,k and σ

2(ψ)kq are given in (28) and (30), respectively.

In (31), the first term in the summation is for the deviationresulting from measurement-noise, whereas the second termis for the deviation resulting from link-noise. From (28)and (30), by abandoning the unchanged terms in the sum-mation of (31), it further suffices to minimize the following

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σ2(ψ)kq =

σ2(ψ)kq (ξmk , pk,1) =

pdefk,2σ

2(ψ)kq

pdefk,2 + αkρkpmβmk

((L − 1)ξmk + 1

)− pk,1

, if k ∈ N cn , q ∈ N \ {k}

σ2(ψ)kq (ξkq) =

ασ 2kq

pkβkq(1− ρq)((L − 1)ξkq + 1

) , if k ∈ N s, q ∈ N cn

σ2(ψ)kq , if k ∈ N c

f

(30)

expression∑k∈N c

n

u2kTr(Ru,k)σ2v,k

∑j∈Nk

a2kj

+M∑k∈N c

n

∑q∈Nk\{k}

a2kqσ2(ψ)kq +M

∑k∈N c

n

a2m(k)k σ2(ψ)m(k)k ,

(32)

in whichm(k) is the index of the SN associated with the near-tier CN k . In the sequel of this paper, the argument k in m(k)is ignored for notational simplicity.

In our framework, the SNs can balance the power trans-ferred to their neighboring CNs by adjusting the globalpower allocation weights

{ξm = [ξm1 ξm2 · · · ξm|Nm|]

T}of

the beamformer in (18); the near-tier CNs can locally adjusttheir additional powers {pk,1} used for measurement collec-tion. To minimize the network-wide MSD, we jointly opti-mize both the global power-allocation weight vectors {ξm}and the local power allocation values {pk,1}. In particular,we have the following problem formulation

min{ξm},{pk,1}

∑k∈N c

n

[c1,k

pdefk,1+pk,1

+c2,k

c3,k (ξm,k )−pk,1+

ασ 2mk

c4,k (ξm,k )

](33a)

s. t. 0 ≤ pk,1 ≤ c3,k (ξm,k )− pdefk,2 (33b)

0 ≤ αkpkρmβmk((L−1)ξmk+1

)−pk,1 ≤

pmaxk,2 − p

defk,2, ∀ k ∈ N c

n (33c)∑k∈Nm\{m}

ξmk = 1, ∀ m ∈ N s (33d)

0 ≤ ξmk < 1, ∀ m ∈ N s, ∀ k ∈ N cn , (33e)

where the constants are given by

c1,k = u2kTr(Ru,k)pdefk,1σ

2v,k

∑j∈Nk

a2kj, (34)

c2,k = Mpdefk,2

∑q∈Nk\{k}

a2kqσ2(ψ)kq , (35)

c3,k (ξm,k ) = pdefk,2+αkρkpmβmk

((L−1)ξmk+1

), (36)

c4,k (ξm,k ) = pmβmk (1− ρk ) ((L − 1)ξmk + 1) . (37)

The constraint (33b) means the additionally allocated powerfor measurement collection cannot exceed the totally har-vested power by near-tier CN k . The constraint (33c) meansthe total transmit power of near-tier CN k which is the sumof the default power pdef

k,2 and the remaining harvested power

used for information transmission, cannot exceed the maxi-mal transmit power pmax

k,2 . The constraints in (33d) (33e) arethe normalization conditions for power allocations weights{ξm} of the beamformer.

Notice that the optimization problem (33) is non-convexand thus difficult to be solved directly. The reasons are twofold. First, the objective function in (33a) is not jointly convexwith respect to the variables {ξm} and {pk,1} both of whichappear in the denominators. Second, the variables {ξm} and{pk,1} are coupled in (33b) and (33c). In the next section,we will propose an efficient iterative algorithm to find theoptimal solution.

V. OPTIMAL SOLUTION FOR MSD MINIMIZATIONIn this section, we first show that the MSD minimizationproblem is decomposable in subsection V-A, and then pro-pose a gradient-based method to find the optimal solution insubsection V-B. Also, we propose adaptive online approachesto estimate some required parameters online, to facilitate thealgorithm implementation in subsection V-C.

A. PRIMAL DECOMPOSITIONIn the coverage of SN m, the wireless power delivered fromSN m is allocated among all near-tier CNs by adjusting ξm.The power harvested by near-tier CN k ∈ Nm is furtherallocated between measurement collection and informationtransmission by adjusting pk,1. We observe from (33) thatthe variables ξmk and pk,1 are coupled in (33b). However,suppose that the global power allocation variables {ξm} arefixed. Then, the rest of the optimization problem (33) decou-ples into several subproblems for local power allocation. Theoptimization problem (33) can thus be decomposed into twolevels of optimizations. At the lower level, the subproblems,one for each CN k ∈ N c

n ∩ Nm, optimize pk,1 for given ξm,i.e.,

minpk,1

gk (pk,1) ,c1,k

pdefk,1 + pk,1

+c2,k

c3,k (ξm,k )− pk,1(38a)

s. t. 0 ≤ pk,1 ≤ c3,k (ξm,k )− pdefk,1, (38b)

0 ≤ αkρkpmβmk((L − 1)ξmk + 1

)− pk,1 ≤

pmaxk,2 − p

defk,2. (38c)

At the higher-level, the master problems, one for each m ∈N s, are in charge of updating the coupling variables ξm, i.e.,

minξm

fm(ξm) =∑k∈Nm

[c1,k

pdefk,1 + p

?k,1(ξmk )

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+c2,k

c3,k (ξm,k )− p?k,1(ξmk )+

ασ 2mk

c4,k (ξm,k )

](39a)

s. t.∑

k∈Nm\{m}

ξmk = 1, (39b)

0 ≤ ξmk < 1, ∀ k ∈ N sm \ {m} (39c)

where p?k,1(ξmk ) is the optimal solution of the subprob-lem (38) that depends on ξmk . The above discussion is sum-marized in the following result.Theorem 1: Problem (33) is primal decomposable.The primal decomposition corresponds to a direct power

allocation, since the master problems allocating the wire-less power directly by giving each subproblem the amountof wireless power it can use. Unlike conventional primarydecomposition, we have multiple master problems, as eachSN will allocate wireless power to all its neighbors. Fortu-nately, all master problems can be solved independently, as itis assumed that there is no overlap between the coverage ofany two SNs.

In the next subsection, we propose a gradient-based itera-tive algorithm to find the optimal solutions to the decomposedproblem.

B. GRADIENT-BASED ITERATIVE ALGORITHMBefore giving the gradient-based algorithm, we first derivethe derivative of the objective function of the subproblem (38)as in (40), shown at the bottom of the next page.

As stated before, the global power allocation for each SNis independent of other SNs. Hence, it suffices to solve themaster problem for each SN m and its associated subprob-lems. For each SN m ∈ N s, we propose Algorithm 1 to findthe optimal power allocation solution.

Algorithm 1 Gradient-Based Iterative Algorithm1: Initialization: Let 4 be the feasible set defined by (39b)

and (39c). Choose ξm(0) ∈ 4, a small step-size α, smallpositive constants ε, sufficiently large fm(ξm(0)),∀m, andlet the iteration index t = 0.

2: while |fm(ξm(t))− fm(ξm(t − 1))| > ε do3: For each k ∈ Nm \ {k}, find the solution p?k,1(t) to the

subproblem (38), and compute the gradient g′k (p?k,1(t)).

4: Update ξm by using the following gradient method

ξm(t + 1) =[ξm(t)− αg

′k (p

?k,1)

]4, (41)

where g′k (p?k,1) is the gradient of gk (pk,1) at the point

p?k,1, and [·]4 denotes the projection onto the feasibleset 4.

5: Compute the objective value fm(ξm(t + 1)) in (39a).6: Set t = t + 1.7: end while8: return ξ ?m = ξm(t + 1), p?k,1 = pk,1(t + 1), ∀k ∈ Nm.

Notice that the gradient-based Algorithm 1 will alwaysconverge, since all expressions in the lower-level

problem (38) and the higher-level problem (39) are differ-entiable and smooth. This will be numerically verified inSection VI.

On the other hand, the complexity of Algorithm 1 is low.In each iteration, each near-tier CN k just performs one-dimension optimization in step 3, and the SN m updates ξmsimply using the gradient method in step 4 and updates theobjective value. It will be numerically shown in Section VIthat with ε = 1×10−10, the computation time of Algorithm 1to approach the optimal solution is just about 0.1 second inMatlab software.

C. ONLINE PARAMETER ESTIMATIONPractical implementation of Algorithm 1 requires infor-mation about the link-noise variance σ

2(ψ)lk and the

measurement-noise variance σ 2v,k .

First, we propose a scheme to estimate the link-noisevariance σ 2(ψ)

lk during the phase for estimating the wirelesschannels. Suppose that the channel hlk from each antenna atnode l to each antenna at node k is estimated in T successivemini-slots. Denote the pilot signal by some constant z. Thereceived pilot signal in the j-th, j = 1, 2, · · · , T , slot isgiven by

rlk,j = hlkz+ v(ψ)lk,j . (42)

The channel hlk can be estimated as follows

hlk =1T

T∑j=1

rlk,jz= hlk +

1T

T∑j=1

v(ψ)lk,j

z. (43)

Define the intermediate random variable

Yj ,rlk,jz− hlk =

(T − 1)v(ψ)lk,j

Tz−

T∑i=1, i 6=j

v(ψ)lk,i

Tz. (44)

As a typical assumption, the link-noises v(ψ)lk,j’s are assumed tobe independent and identically distributed random variablesthat followGaussian distributionwith zeromean and varianceσ2(ψ)lk . We thus have

Yj ∼ CN(0,T − 1T |z|2

σ2(ψ)lk

). (45)

Moreover, the link-noise variance σ 2(ψ)lk can be estimated

as follows

σ2(ψ)lk =

T |z|2

T − 1

T∑j=1

∣∣Yj∣∣2 = T |z|2

T − 1

T∑j=1

∣∣∣∣ rlk,jz − hlk∣∣∣∣2 . (46)

In practice, the channel can be estimated accurately dueto relatively short distance between each pair of nodes. Thelink-noise variance can thus be obtained accurately by usingthe estimator in (46).

Second, we provide the online estimation of themeasurement-noise variance σ 2

v,k . The estimate of σ 2v,k ,

denoted by σ 2v,k (i), is estimated by time-averaging as follows

σ 2v,k (i) = θσ

2v,k (i− 1)+ (1− θ )

∣∣dk,i − uk,iwk,i−1∣∣2 , (47)

where θ is some positive constant less than 1.

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FIGURE 2. Network topology and sensor locations.

VI. NUMERICAL RESULTSIn this section, we present numerical results on the MSD per-formance. As shown in Fig. 2, we consider a connected net-work with N = 12 sensor nodes among which sensor 4 and6 are SNs and other sensors are CNs. In Fig. 2, both x-axisand y-axis has the unit of meter (m). The MSD performanceof larger inter-node distance will be given in subsection VI-B.The carrier frequency for wireless information and powertransfer is 900M Hertz (Hz). We assume that the path lossmodel is βlk = 10−2d−2lk , where dlk is the distance betweensensor l and sensor k , and the path loss exponent is assumedto be 2. For each CN, we assume that the default power forcollecting measurements and transmission are pdef

k,1 = 10−4

Watt (W) and pdefk,2 = 10−3 W, respectively. We set the

maximal transmission power of any near-tier CN as pmaxk,2 =

1 W, power-splitting efficiency ρk = 0.95, and the energyconversion efficiencyαk = 0.8, for each near-tier sensor nodek . All the MSD performance is obtained by averaging over1000 experiments.

We follow the simulation parameters in [44]. The unknowncomplex parameter wo of length M = 2 is randomly gener-ated; its value is [−0.6356 + 0.1304i,−0.6780 − 0.3455i].We adopt uniform step-sizes, µk = 0.01, ∀k , and uniformregression data with covariance matrices Ru,k = 1.6862 ×I2, ∀k . We assume that with a reference level of 1 W,the measurement-noise power of CNs are uniform in therange [−25,−10] dB, and the measurement-noise powerof SNs are uniform in the range [−35,−25] dB. We alsoassume that the link-noise power are uniform in the range of

FIGURE 3. Noise profiles.

[σ 2(ψ)−5, σ 2(ψ)

+5] dB, in which σ 2(ψ) is the average link-noise power.

A. RESULTS FOR FIXED L, PM AND INTER-NODEDISTANCESIn this subsection, we assume that the average link-noisepower is σ 2(ψ)

= −65 dB. The power profiles of the defaultmeasurement-noise and link-noise are shown in Fig. 3. EachSN is equipped with L = 4 antennas, and the transmissionpower for SNs are p4 = p6 = 1 W.First, we consider the uniform (or averaging) combination

rule [10], i.e.,

alk =

1nk, if l ∈ Nk

0, if l /∈ Nk

(48)

where nk is the number of neighbors of node k , i.e.,nk , |Nk |.

The optimally global and local power allocation solutionsare given in Table 2. We observe that all near-tier CNs allo-cate a larger fraction of the harvested power for collectingmeasurements, to reduce the measurement-noise power. Theexact power allocation depends on the noise profile and thedistances between each near-tier CN to its associated SN.For instance, for the farther and noisy CN 1, almost all theharvested power is allocated for collecting measurements.

Fig. 4 compares the network-wide MSD of the proposedscheme with optimized solutions, to that of the conventionalscheme without WPT. The theoretical results (16) are alsoplotted. We see that with optimized solutions, the stead-state MSD of the proposed scheme is 9.85 dB less thanthat of the conventional scheme. This verifies the effective-ness of the proposed scheme for adaptive diffusion. Thisdiffusion performance enhancement is because the near-tier

g′k (pk,1) =(c2,k − ck,1)p2k,1 + 2(ck,1c3,k (ξm,k )+ pdef

k,1c2,k )pk,1 + (pdefk,1)

2c2,k − ck,1c23,k (ξm,k )

(pdefk,1 + pk,1)

2(c3,k (ξm,k )− pk,1)2. (40)

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TABLE 2. Optimally global and local power allocation solutions (uniform rule).

FIGURE 4. Comparison of MSD (uniform combination rule).

CNs utilize their harvested RF energy to enhance the accuracyof measurement collection and the reliability of informationexchange. Notice that the simulated results differ from thetheoretical ones in the first 400 iterations, which is becausethe diffusion algorithm needs some iterations to converge andthe theoretical results are for the MSD performance in theconverged state [44].

With uniform combination rule and the optimal globalpower allocation {ξm}, Fig. 5 compares the steady-state MSDfor different local power allocation schemes: 1). optimizedlocal power allocation; 2). allocating all harvested power formeasuring, i.e., pk,1 = pk , ∀k ∈ N c

n ; 3). allocating allharvested power for transmission, i.e., pk,2 = pk , ∀k ∈ N c

n .We first observe that for smaller link-noise power, allocatingmost power for sensing leads to minimum MSD. Then weobserve that as the link-noise power increases, the scheme ofallocating all harvested power for measuring (transmission)suffers larger (smaller) increase in the MSD, compared tothe optimized scheme. This is because for higher link-noisepower, the link-noise has more significant effect on the MSDperformance, and more harvested power should be allocatedfor transmission. This observation shows the importance ofbalancing the local power allocation, especially for the caseof higher link-noise power.

Then, we consider two other combination rules: relative-degree and metropolis combination rule [10]. The relative-degree combination rule is given by

alk =

nl/

∑m∈Nk

nm

, if l ∈ Nk

0, if l /∈ Nk

(49)

FIGURE 5. Comparison of steady-state MSD for different local powerallocation schemes.

FIGURE 6. Comparison of MSD (different combination rules).

while the metropolis combination rule is given by

alk =

1/max{nl, nk}, if l ∈ Nk \ {k}

1−∑

l∈Nk\{k}

alk , if l = k

0, otherwise

(50)

Fig. 6 compares the MSD for different combination rule.In general, with the optimized solutions, for each combina-tion rule, the MSD is significantly reduced compared to theconventional case without WPT. Moreover, we observe thatthe uniform combination rule achieves slightly smaller MSDthan the relative-degree rule, and the metropolis rule has arelatively larger MSD.

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FIGURE 7. Steady-state MSD versus different L and pm.

B. RESULTS FOR VARYING L, PM AND INTER-NODEDISTANCESIn this subsection, we simulate the MSD for varying param-eters. Fig. 7 plots the steady-state MSD versus each SN’snumber of antennas L, and each SN’s transmit power of pm,respectively. We observe that for smaller L and pm, the MSDreduces as L or pm increases. This is because the noisepowers of near-tier CNs are reduced due to more harvestedpower. However, we further observe that for larger L and pm,the MSD reduces very slowly as L or pm increases. This isbecause that the measurement-noise powers of SNs and far-tier CNs are not decreased, and results into the MSD floor forlarger L and pm.In all previous simulations, we keep the same inter-sensor

distances as in Fig. 2. Now we scale the distances by amultiplicative factor κ . Keeping the same transmit power forall sensors, the link-noise σ 2(ψ)

lk is equivalently scaled by κ2.Fig. 8 plots the steady-state MSD versus the distance scalingfactor κ . We observe that the reduction in the MSD reducesas κ increases. This is because the reduced harvested powerleads to weak ability of reducing measurement and link-noisepowers, as distances increase. This is why we consider WPTfrom each SN to only its near-tier CNs. However, we see thatfor κ = 4 that implies the inter-sensor distance is between6 meters and 10 meters, the MSD can still be reduced by 2.5dB, which shows the effectiveness of the proposed scheme.Notice that when line-of-sight channels like Rician channelsare considered, the supporting inter-sensor distance can befurther enhanced.

C. RESULTS FOR ESTIMATED MEASUREMENT-NOISEVARIANCE σ2

V ,KIn this subsection, we consider the practical scenario in whichthe measurement-noise variance σ 2

v,k is estimated on-lineas in (47). All the link-noise variances are assumed to beestimated perfectly by using the estimator in (46) during thechannel estimation phase, since the distance between any pairof nodes is relatively short. The other parameters includingthe L, p and inter-node distances are fixed as in Section VI-A.The uniform combination rule is adopted.

FIGURE 8. Steady state MSD versus distance scaling factor.

FIGURE 9. Comparison of MSD (Nest = 100).

First, we use Nest = 100 iterations to estimate σ 2v,k . With

the perfect or estimated σ 2v,k , Fig. 9 compares the network-

wide MSD of different time-averaging coefficient θ . Thetheoretical results (16) for perfect σ 2

v,k are also plotted. In gen-eral, we observe that with estimated σ 2

v,k , the MSD increasesslightly compared to the case of perfect σ 2

v,k . Moreover,we see that smaller MSD can be obtained for larger θ . Forinstance, for θ = 0.9, the degradation ofMSD performance isnegligible. This verifies that the proposed scheme is practical.

VII. CONCLUSIONIn this paper, we consider the adaptive diffusion LMSestimation over a hybrid WSN in which each SN is capa-ble of performing simultaneous wireless information andpower transfer (SWIPT) to its neighbors via beamformingover fading channels. The steady-state network-wide MSDis minimized, by jointly optimizing the global power allo-cation weights for SNs to perform SWIPT, and the localpower allocation proportion for near-tier CNs to use har-vested energy for measurement collection. The formulated

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non-convex problem is shown to be primal decomposable,and a gradient-based iterative algorithm is proposed to findthe optimal solutions. To facilitate the implementation of thisalgorithm, we propose adaptive approaches to estimate link-noise variance and measurement-noise variance by utilizingchannel-estimation pilot and time-averaging, respectively.Numerical results show that with optimal global and localpower allocation, the proposed diffusion LMS estimationscheme improves the MSD performance significantly, com-pared to the conventional scheme without WPT. We observethat for smaller or moderate link-noise power, allocatingmostpower for sensing leads to minimum MSD. The proposedscheme is also numerically shown to be robust to the estima-tion error of the measurement-noise power. We believe thatthe proposed diffusion LMS estimation scheme has the poten-tial to be used in future low-power IoTs. The future workinclude the joint optimization of power allocation and power-splitting efficiency, as well as the diffusion performanceunder line-of-sight channels like Rician channel, imperfectchannel estate information, etc.

APPENDIX APROOF OF LEMMA 2For notational convenience, we ignore the iteration (or time)index i in this proof, and define H = [hm1 hm2 · · · hm|Nm|].Substituting (18) into (21), the harvested energy is

pk (ξm) = αkρkEH

[pmξmk‖hmk‖2

+ pm√ξmk‖hmk‖2

∑i 6=k

√ξmihHmkhmi‖hmi‖2

+ pm√ξmk‖hmk‖2

∑i 6=k

√ξmi

(hHmkhmi

)H‖hmi‖2

+

∑i 6=k

∑j 6=k

pm√ξmiξmj

(hHmkhmi

)H hHmkhmj‖hmi‖2‖hmj‖2

]. (51)

In the sequel, we investigate the four terms in (51).From the fact that hmk ∼ CN (0L , βmkIL), we have thatEhmk

[hHmkhmk

]= Lβmk . Hence, the first term in (51) is

obtained as

αkρkEH

[pmξmk‖hmk‖2

]= αρkpmξmkLβmk . (52)

Define hmi , hmi‖hmi‖2

, and hmk , ‖hmk‖2hmk . The secondterm in (51) is thus rewritten as

αkρkEH

pm√ξmk‖hmk‖2∑i 6=k

√ξmihHmkhmi‖hmi‖2

= αkρkpm

√ξmk

∑i 6=k

√ξmiEhmk ,hmi

[hHmk hmi

](a)= 0, (53)

where (a) is from the fact that hmi and hmk are independentzero-mean random vectors, for any i 6= k .

The third term in (51), which is the conjugate of the secondterm in (51), is similarly obtained as

αkρkEH

pm√ξmk‖hmk‖2∑i 6=k

√ξmi

(hHmkhmi

)H‖hmi‖2

=0. (54)

The fourth term in (51) is rewritten as

αkρkEH

∑i 6=k

∑j 6=k

pm√ξmiξmj

(hHmkhmi

)H hHmkhmj‖hmi‖2‖hmj‖2

= αkρkpmEH

[∑i 6=k

ξmihHmi(hmkhHmk )hmi

+

∑i 6=k

∑j 6=i,k

√ξmiξmjhHmi(hmkh

Hmk )hmj

](a)= αkρkpmβmk

∑i 6=k

ξmi, (55)

where (a) is from the fact that EH(hmkhHmk

)= βmkIL , and

hmi and hmj are independent zero-mean random vectors, forany i 6= j.Substituting (52), (53), (54) and (55) into (51), we obtain

the harvested energy as

pk (ξm) = αkρkpmξmkLβmk + αkρkpmβmk (1− ξmk )

= αkρkpmβmk((L − 1)ξmk + 1

). (56)

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GANG YANG (S’13–M’15) received the B.Eng.and M.Eng. degrees (Hons.) in communicationengineering, communication and information sys-tems from the University of Electronic Sci-ence and Technology of China, Chengdu, China,in 2008 and 2011, respectively, and the Ph.D.degree from Nanyang Technological University,Singapore, in 2015. In 2015, he joined the Depart-ment of Electrical and Computer Engineering,National University of Singapore, as a Research

Fellow. He is currently an Associate Professor with the National Key Labora-tory of Science and Technology on Communications and the Center for Intel-ligent Networking and Communications (CINC), University of ElectronicScience and Technology of China. His current research interests include theInternet of Things communications, backscatter communications, wirelesspowered communications, far-field and near-field wireless power transfer,and compressive sensing. He was a recipient of the IEEE CommunicationsSociety Transmission, Access, and Optical Systems (TAOS) Technical Com-mittee Best Paper Award, in 2016, and the Chinese Government Awardfor Outstanding Self-Financed Students Abroad, in 2015. He serves as aPublicity Co-Chair for the IEEE Globecom’17.

WEE PENG TAY (S’06–M’08–SM’14) receivedthe B.S. degree in electrical engineering and math-ematics, and the M.S. degree in electrical engi-neering from Stanford University, Stanford, CA,USA, in 2002, and the Ph.D. degree in electricalengineering and computer science from the Mas-sachusetts Institute of Technology, Cambridge,MA, USA, in 2008. He is currently an AssociateProfessor with the School of Electrical and Elec-tronic Engineering, Nanyang Technological Uni-

versity, Singapore. His research interests include information and signalprocessing over networks, distributed inference and estimation, informa-tion privacy, machine learning, information theory, and applied probability.He was a Technical Program Committee Member of various internationalconferences. He was a recipient of the Tan Chin Tuan Exchange Fellowship,in 2015. He is a Co-Author of the Best Student Paper Award at the Asilomarconference on Signals, Systems, and Computers, in 2012, and the IEEESignal Processing Society Young Author Best Paper Award, in 2016. He iscurrently anAssociate Editor of the IEEETRANSACTIONSON SIGNAL PROCESSING

and an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.He serves for the MLSP TC of the IEEE Signal Processing Society.

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G. Yang et al.: Optimal Power Allocation for Diffusion-Type Sensor Networks

YONG LIANG GUAN received the B.Eng. degree(Hons.) from the National University of Singaporeand the Ph.D. degree from the Imperial CollegeLondon, U.K. He is currently a tenured AssociateProfessor with the School of Electrical and Elec-tronic Engineering, Nanyang Technological Uni-versity, Singapore. His research interests broadlyinclude modulation, coding and signal process-ing for communication systems, and data storagesystems. He is an Associate Editor of the IEEE

TRANSACTIONS ON VEHICULAR TECHNOLOGY.

YING-CHANG LIANG (F’11) was a Professorwith The University of Sydney, Australia, a Prin-cipal Scientist and a Technical Advisor with theInstitute for Infocomm Research, Singapore, anda Visiting Scholar with Stanford University, USA.He is currently a Professor with the Universityof Electronic Science and Technology of China,China, where he also leads the Center for Intelli-gent Networking and Communications. He is alsothe Deputy Director of the Artificial Intelligence

Research Institute. His research interests include the general area of wire-less networking and communications, cognitive radio, dynamic spectrumaccess, the Internet of Things, artificial intelligence, and machine learningtechniques.

Dr. Liang received the Prestigious Engineering Achievement Award fromthe Institute of Engineers Singapore, in 2007, the Outstanding ContributionAppreciation Award from the IEEE Standards Association, in 2011, andthe Recognition Award from the IEEE Communications Society TechnicalCommittee on Cognitive Networks, in 2018. He has also received numerouspaper awards, the recent ones including the IEEE ICC Best Paper Award,in 2017, the IEEE ComSoc’s TAOS Best Paper Award, in 2016, and theIEEE Jack Neubauer Memorial Award, in 2014. He was the Chair of theIEEE Communications Society Technical Committee on Cognitive Net-works. He served as the TPC Chair and an Executive Co-Chair for IEEEGLOBECOM 2017. He served as a Guest/Associate Editor for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE JOURNAL OF SELECTED

AREAS IN COMMUNICATIONS, IEEE Signal Processing Magazine, the IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY, and the IEEE TRANSACTIONS ON

SIGNAL AND INFORMATION PROCESSING OVER NETWORK. He was an AssociateEditor-in-Chief of Random Matrices: Theory and Applications (World Sci-entific). He is the Founding Editor-in-Chief of the IEEE JOURNALON SELECTED

AREAS IN COMMUNICATIONS-Cognitive Radio Series, the Key Founder and theEditor-in-Chief of the IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND

NETWORKING, and an Associate Editor-in-Chief of China Communications.He was a Distinguished Lecturer of the IEEE Communications Society andthe IEEE Vehicular Technology Society. Since 2014, he has been a HighlyCited Researcher recognized by Thomson Reuters.

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