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Transceiver design of optimum wirelessly powered full-duplexMIMO iot devices

Citation for published version:Xue, J, Biswas, S, Cirik, AC, Du, H, Yang, Y, Ratnarajah, T & Sellathurai, M 2018, 'Transceiver design ofoptimum wirelessly powered full-duplex MIMO iot devices', IEEE Transactions on Communications, vol. 66,no. 5, pp. 1955-1969. https://doi.org/10.1109/TCOMM.2018.2801870

Digital Object Identifier (DOI):10.1109/TCOMM.2018.2801870

Link:Link to publication record in Edinburgh Research Explorer

Document Version:Peer reviewed version

Published In:IEEE Transactions on Communications

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Download date: 07. Jun. 2020

IEEE TRANSACTIONS ON COMMUNICATIONS 1

Transceiver Design of Optimum WirelesslyPowered Full-Duplex MIMO IoT Devices

J. Xue, Member, IEEE, S. Biswas, Member, IEEE,A. C. Cirik, Member, IEEE, H. Du, Member, IEEE, Y. Yang, Member, IEEE,

T. Ratnarajah, Senior Member, IEEE and M. Sellathurai, Senior Member, IEEE

Abstract—In this paper, we investigate the energy harvesting(EH) technique and accordingly design transceivers for a K linkmultiple-input multiple-output (MIMO) interference channel.Each link consists of two full-duplex (FD) internet of things (IoT)nodes exchanging information simultaneously in a bi-directionalcommunication channel. All the nodes suffer from interference,in particular strong self-interference and inter-node interference,due to operating in FD mode and simultaneous transmission ateach link, respectively. Further, we divide the received signal ateach node into two parts. While one part of the signal is used forinformation decoding, the other part is used for EH. We jointlydesign the transmit and receive beamforming vectors and receiverpower splitting ratios by minimizing the total transmission powerof the system, subject to both signal-to-interference-plus-noiseratio (SINR) and EH threshold constraints. Furthermore, thecase of multiple-input single-output (MISO) interference channelis also included for the sake of comparison. We also revisit theabove problems for the case when the available channel stateinformation (CSI) at the transmitters is imperfect, where theerrors of the CSI are assumed to be norm bounded. Simulationresults show that the EH technique can harvest enough energyto support power consumption limited IoT devices by aiding inrecharging their respective batteries.

Index Terms—Beamforming, energy harvesting, full-duplex,MIMO interference channels, power splitting, self-interference.

I. INTRODUCTION

Wireless data traffic has exponentially increased over thelast decade and it is projected to increase even further. As aresult, effective and efficient utilization of the scarce spectrumresources has become an extremely important issue. The

J. Xue is currently with the National Engineering Laboratory for Big DataAnalytics, Xi’an International Academy for Mathematics and MathematicalTechnology, School of Mathematics and Statistics, Xi’an Jiaotong University,Xi’an, Shaanxi, 710049, P. R. China. He was with the School of Engi-neering, Institute for Digital Communications, The University of Edinburgh,Edinburgh. The work of J. Xue is supported by the ”Young Talent SupportPlan” of Xian Jiaotong University and the UK EPSRC under grant numberEP/L025299/1.

S. Biswas, and T. Ratnarajah are with the School of Engineering, Institutefor Digital Communications, the University of Edinburgh, Edinburgh, EH93JL, United Kingdom. The work of S. Biswas and T. Ratnarajah are supportedby the UK EPSRC under grant number EP/L025299/1.

A. C. Cirik is with Ofinno Technologies, Herndon, VA 20171, United States.H. Du is with the College of Information Science and Technology, Jinan

University, Guangzhou, China, and also with National Mobile Communi-cations Research Laboratory, Southeast University. The work of HuiqinDu is supported by the National Natural Science Foundation of China(No.61401178) and the Open Research Fund of National Mobile Commu-nications Research Laboratory Southeast University (No.2017D12).

Y. Yang is with the Interdisciplinary Centre for Security, reliability andtrust, University of Luxembourg, L-1855 Luxembourg. The work of Yang issupported by the ERC project AGNOSTIC.

M. Sellathurai is with School of Engineering & Physical sciences, sensors,signals & systems, Heriot-Watt University, United Kingdom.

currently deployed half-duplex (HD) wireless communicationsystems do not utilize the spectrum efficiently as transmissionand reception happen orthogonally, either in time, denotedas time division duplexing (TDD), or in frequency, denotedas frequency division duplexing (FDD). Among the emerg-ing technologies for next-generation wireless networks, full-duplex (FD) communication is considered as a way to poten-tially double the capacity of wireless communications. Simul-taneous transmission and reception of overlapping signals inthe same frequency had generally been assumed impossiblein wireless communications due to the challenges involved inhandling the self-interference [1], which is caused due to thesignal received at the receive antennas of a FD node fromits own transmitter antennas. However, thanks to the recentprogress of cancellation made on self-interference suppression,FD communication systems have triggered enormous researchinterests [2], [3], Consequently, FD is being considered as akey enabling technique for 5G and beyond systems [4], sinceit enables available spectral resources to be fully utilized inboth time and frequency.

A single-antenna FD system was investigated in [5], whilein [6], a multiple-antenna FD system was studied. Manyfeasible solutions including antenna, analog and digital can-cellation have been demonstrated experimentally to mitigatethe overwhelming self-interference, which is the fundamentalchallenge in implementing a full-duplex radio [7]–[10]. How-ever, due to imperfect self-interference channel knowledgeand hardware impairments in the transmitter chain, the self-interference cannot be completely eradicated in practice. Inthis case, the performance is limited by the residual self-interference, which is induced by the imperfection of thetransmit and receive front-end chain [11]–[19]. In addition toself-interference, co-channel interference (CCI) from uplink(UL) to downlink (DL) nodes is another challenge in FDnetworks that needs to be overcome to fully exploit the multi-access nature of the wireless medium in conjunction with FDsystems. To optimize the system performance, self-interferenceand CCI in FD systems should be addressed jointly throughbeamforming [15], [20], [21].

It is a common practice in wireless communication devicesthat energy is supplied to an energy constrained wirelessdevice with the help of rechargeable or replaceable batteries.However, these batteries have limited operation time and theyneed to be replaced or recharged frequently. Not only is thisinconvenient, but also usually incurs high costs. Accordingto the analyses in literature and the fact that radio signalscan potentially carry wireless information and energy simul-taneously, a new promising solution, energy harvesting (EH)

IEEE TRANSACTIONS ON COMMUNICATIONS 2

through simultaneous wireless information and power transfer(SWIPT), has been proposed [22]–[24]. This is seen as a keyenabling technique for next generation wireless devices. Inparticular, the recent advent of low powered internet of things(IoT) devices and device to device (D2D) communications hasmade EH an interesting proposition. Such IoT/D2D deviceswill mainly make use of the current state of the art technolo-gies such as bluetooth 5.0 low energy (LE), IEEE 802.11n,IEEE 802.11ac, etc., along with other future communicationtechnologies such as mmWave transmission. The emergence ofthese low power consumption technologies makes it possibleto realize wireless power transfer, which can act as a cor-nerstone for future green communications. This EH techniquecan be realized by two practical means, namely time-switching(TS) and power splitting (PS) which were introduced in [25].The single-input single-output (SISO) system with additivewhite Gaussian noise (AWGN) was considered in [26], wherea capacity-energy function to characterize the fundamentaltrade-off between wireless information and energy transmis-sion was proposed. Moreover, SWIPT for multiple-antennasystems was studied in [27].

Further from a market perspective (Global Market Forecast(GMF) upto 2023), the EH system market by technology(light, vibration, RF, thermal), component, application (build-ing and home automation, consumer electronics, IoT, indus-trial, transportation, security) is expected to witness a growthfrom USD 311.2M in 2016 to USD 645.8M by 2023, at aCAGR of 10.62% between 2017 and 2023. The key playersin this market are EnOcean GmbH (Germany), Mide Tech-nology Corporation (US), Lord Microstrain (US); secondarybattery and capacitor providers such as Cymbet Corporation(US), Linear Technologies (US), Murata Manufacturing Co.Ltd., (Japan), and Infinite Power Solution Inc. (US); powermanagement IC manufacturers such as Linear Technologies(US), Cypress Semiconductor Corp. (US), STMicroelectronics(Switzerland), Texas Instruments (US), and Fujitsu (Japan).

Motivated by the potential of FD and EH in future wire-less communication systems, such as IoT/D2D, in this paperwe investigate the EH potential of multiple-input multiple-output (MIMO) interference channels consisting of K pairsof IoT/D2D FD nodes with SWIPT. The consideration ofinterference channel is suitable for IoT/D2D devices as mul-tiple low powered devices will interact among each other inan IoT/D2D communication scenario. Each IoT/D2D node isassumed to have Na

i transmit and Mai receive antennas, where

i ∈ {1, 2, ...,K} and a ∈ {1, 2} and operates in FD mode. Theconsideration of FD and EH techniques at each node allows allthe participating nodes to transmit/receive signal and harvestenergy simultaneously. The harvested energy can then be usedby the nodes to recharge their batteries without the need forexternal power supply1.

Indeed, the studied system in our paper shares many sim-ilarities to the traditional interference channel networks, bothin terms of the system concept, service requirements, as wellas the design guidelines. Nevertheless, the application of FD

1This paper does not focus on the modeling of charging of the battery.Analysis of the battery dynamics is beyond the scope of this paper. Interestedreaders can refer to [28], [29] for details on battery modelling.

technology for EH in a IoT/D2D network introduces newfundamental challenges to the traditional interference channelmodels, which is the main focus of this paper. The maindistinctions are summarized as follows:

• The EH requirements of an IoT network need to berevised when operating in a FD mode. This is becausein a FD interference channel network, all the nodesshare the same channel resource for transmission andreception, which results in the imposition of a higherinterference intensity on the network. This issue becomesmore critical considering the fact that the acquisition ofan accurate CSI regarding the interference paths from themultiple users is relatively unrealistic, and calls for theconsideration of a joint robust transmission strategy.

• In an FD MIMO interference channel network, the self-interference at each node is a critical challenge, andstrongly relates the performance/design of the UL/DLreception to the DL/UL transmission. In this respect, theconsideration of an accurate transceiver model, includingthe impacts of transmission and reception distortions arecritical, as it is well-established in the context of FDsystem design and analysis.

• In an FD MIMO interference channel network, apartfrom the self-interference, the interference paths amongall nodes should be additionally taken into account. Thisimpacts both the system performance, as well as thedesign strategy.

Note that the aforementioned considerations regarding thedesign of a robust FD multi-node MIMO IoT network, result ina relatively complicated problem structure. For perfect channelstate information (CSI) at the nodes, the optimization problemis defined according to the SINR and EH constraints, whichresults in a non-convex optimization problem. When the CSIavailability at the transmitters is imperfect, the errors of theCSI are assumed to be norm bounded, resulting in a semi-infinite problem. Moreover, due to the transmit and receivedistortions at the FD nodes in addition to imperfect CSI, EHand SINR become complicated functions, which make thetransformation of the constraints in the optimization problemscomplicated. This, in turn, calls for a rigorous optimization andanalysis, together with a dedicated computational complexitystudy. Numerical results demonstrate the feasibility of EH forFD. It is shown that the harvested energy can support smallIoT devices with minimal power requirements to recharge theirbatteries.

Notations: The following notations are used in this paper.Matrices and vectors are denoted as bold capital and lowercaseletters, respectively. (·)T is the transpose, (·)H represents theconjugate transpose, and (·)∗ the conjugate. E {·} denotesthe statistical expectation and diag (A) is the diagonal matrixwith the same diagonal elements as A. IN is the N by Nidentity matrix, and tr {·} is the trace operator. CN

(µ, σ2

)denotes a complex Gaussian distribution with mean µ andvariance σ2. CN×M denotes the set of complex matrices witha dimension of N by M . |·| and ‖·‖ denote the absolutevalue and the Euclidean norm, respectively. The operator formultidimensional array is denoted by vec(·). A � 0 indicates

IEEE TRANSACTIONS ON COMMUNICATIONS 3

Fig. 1. An illustration of a bi-directional full-duplex MIMO interferencechannel involving K pairs of IoT nodes.

that matrix A is positive semidefinite, and rank (A) is therank of matrix A.

The rest of the paper is organized as follows. The systemmodel is introduced in Section II. Section III and IV presentthe beamforming design problem for the MISO and MIMOinterference channels with perfect CSI, respectively. The im-perfect CSI case is presented in Section V followed by thenumerical results in Section VI. Finally, Section VII providesthe conclusion of the paper.

II. SYSTEM MODEL

As shown in Fig. 1 on the top of next page, the system underconsideration consists of K pairs of FD IoT nodes, where eachIoT node pair exchanges information simultaneously in a twoway communication. For simplification, we assume that theMIMO FD IoT nodes in the ith link have Ni transmit antennasand Mi receive antennas.

The node i(a), where i ∈ {1, . . . ,K} and a ∈ {1, 2} re-ceives signals from all the transmitters in the system. H

(ab)ii ∈

CMi×Ni is the desired channel between node a and b of theith transmitter-receiver pair, where b ∈ {1, 2} and b 6= a.The self-interference channel of the node i(a) is denoted asH

(aa)ii ∈ CMi×Ni , a ∈ {1, 2} and the inter-user interference

channel from the transmitter antennas of the node c in thejth pair to the receiver antenna of the node a in the ithpair, (i, j) ∈ {1, . . . ,K} and j 6= i is denoted as H

(ac)ij ∈

CMi×Nj , (a, c) ∈ {1, 2}.Hereinafter, different mathematical methods will be applied

to obtain optimal EH solutions for both MISO and MIMO FDsystems. We begin our analysis by considering the perfect CSIMISO and MIMO cases, which are then extended to MISOand MIMO cases with imperfect CSI later in the paper.

III. FULL DUPLEX SYSTEMS WITH PERFECT CSI

In this section, we assume that perfect CSI is available atall the IoT nodes. The MISO scenario will be investigated firstfollowed by the MIMO case2.

A. Transceiver design for MISO interference channel

The transmitted data stream of node i(a) is denoted asd

(a)i , i ∈ {1, . . . ,K}, a ∈ {1, 2}, and is assumed to be

2The consideration of both the MISO and MIMO cases is due to the factthat the MIMO case requires certain approximations to derive the covariancematrix of the signal. However, the MISO case doesn’t require any suchapproximations and hence the MISO case may not be a straigtforwardextension of the MIMO case in our analysis.

complex, zero mean, independent and identically distributed(i.i.d.) with unit variance. The Ni×1 signal vector transmittedby node i(a) is given by

x(a)i = v

(a)i d

(a)i , i = 1, . . . ,K, a ∈ {1, 2}, (1)

where v(a)i ∈ CNi×1 represents the precoding vector.

The received signal at node i(a) is a combination of thesignals transmitted by all the IoT nodes plus the additive noise,which is written as

y(a)i =

(h

(ab)ii

)H (x

(b)i + c

(b)i

)+(h

(aa)ii

)H (x

(a)i + c

(a)i

)+

K∑j 6=i

2∑c=1

(h

(ac)ij

)H (x

(c)j + c

(c)j

)+ e

(a)i

+ n(a)i , i ∈ {1, . . . ,K}, (a, b) ∈ {1, 2}, a 6= b. (2)

Here, n(a)i is the additive white Gaussian noise (AWGN) at

node i(a) with zero mean and variance σ2n.

In (2), c(a)i ∈ CNi , i ∈ {1, . . . ,K}, a ∈ {1, 2} is the noise

at the transmitter antennas of node i(a), which models theeffect of limited transmitter dynamic range (DR) and closelyapproximates the effects of additive power-amplifier noise,non-linearities in the DAC and phase noise [12]. The meanof c

(a)i is 0, and the variance of c

(a)i is proportional to the

energy of the intended signal at each transmit antenna, i.e.,

c(a)i ∼ CN

(0, κ diag

(v

(a)i

(v

(a)i

)H)), c

(a)i ⊥ x

(a)i , (3)

where ⊥ denotes the statistical independence3.In (2), e(a)

i , i ∈ {1, . . . ,K}, a ∈ {1, 2} is the additivedistortion at the receiver antenna of node i(a), which modelsthe effect of limited receiver DR and closely approximates thecombined effects of additive gain-control noise, non-linearitiesin the ADC and phase noise. The mean of e(a)

i is 0 andthe variance is proportional to the energy of the undistortedreceived signal at the receive antenna. In particular, e(a)

i ismodeled as

e(a)i ∼ CN

(0, βΦ

(a)i

), e

(a)i ⊥ u

(a)i , (4)

where Φ(a)i = Var{u(a)

i } is the variance of u(a)i , and u

(a)i is

the undistorted received signal at the node i(a), i.e., u(a)i =

y(a)i − e

(a)i .

Node i(a) has the knowledge of the interfering codewordsx

(a)i and the channel h

(aa)ii . So the self-interference term(

h(aa)ii

)Tx

(a)i is known, and thus can be canceled [12]. The

received signal after self-interference cancellation can then bewritten as

y(a)i = y

(a)i −

(h

(aa)ii

)Tx

(a)i

=(h

(ab)ii

)Tx

(b)i + v

(a)i , (5)

3Considering the measurements of various hardware setups which wereshown in [30], [31], the received signal modeled as (2) closely approximatesthe combined effects of additive power-amp noise, non-linearities in the DACand power-amp, and oscillator phase noise. Meanwhile, by the definition oflimited dynamic range, the transmitter-noise variance is dependent on intendedsignal power [16].

IEEE TRANSACTIONS ON COMMUNICATIONS 4

Σ(a)i =

(h

(ab)ii

)H (κ (1 + β) diag

(v

(b)i

(v

(b)i

)H)+ βv

(b)i

(v

(b)i

)H)(h

(ab)ii

)+(h

(aa)ii

)H (κ (1 + β) diag

(v

(a)i

(v

(a)i

)H)+ βv

(a)i

(v

(a)i

)H)(h

(aa)ii

)+∑K

j 6=i

∑2

c=1(1 + β)

[(h

(ac)ij

)H (v

(c)j

(v

(c)j

)H+ κdiag

(v

(c)j

(v

(c)j

)H))(h

(ac)ij

)]+ (1 + β)σ2

n. (7)

where v(a)i is the unknown interference-plus noise component

after self-interference cancellation, and is given by

v(a)i =

(h

(ab)ii

)Tc

(b)i +

(h

(aa)ii

)Tc

(a)i + e

(a)i + n

(a)i

+∑K

j 6=i

∑2

c=1

(h

(ac)ij

)T (x

(c)j + c

(c)j

). (6)

Now, using (3)-(4), Σ(a)i , the variance of v(a)

i is given as (7).By means of a power splitter, the received signal is now

divided into two parts, one for the information decoder andanother for EH. Let ρ(a)

i denote the PS ratio for receiver i(a),which means that a portion ρ(a)

i of the signal power is used forsignal detection while the remaining portion 1−ρ(a)

i is divertedto an energy harvester. Accordingly, the available signal forinformation decoding at receiver i(a) can be expressed as

r(a)i =

√ρ

(a)i y

(a)i +m

(a)i , (8)

where m(a)i is the additional AWGN circuit noise with zero

mean and variance σ2i(a)

due to phase offset and non-linearitiesduring baseband conversion [32]. Using (8), the SINR atreceiver i(a) is given by

SINR(a)i =

ρ(a)i

∣∣∣∣(h(ab)ii

)Tv

(b)i

∣∣∣∣2ρ

(a)i Σ

(a)i + σ2

i(a)

. (9)

Besides, the total harvested energy that can be stored byreceiver i(a) is given as

EH(a)i = ξ

(a)i

(1− ρ(a)

i

)E{∣∣∣y(a)

i

∣∣∣2}= ξ

(a)i

(1− ρ(a)

i

)(∣∣∣∣(h(ab)ii

)Tv

(b)i

∣∣∣∣2+

∣∣∣∣(h(aa)ii

)Tv

(a)i

∣∣∣∣2 + Σ(a)i

), (10)

where ξ(a)i ∈ (0, 1) denotes the energy conversion efficiency

of the i(a)th EH unit.We focus on transmit filter and PS ratio design, in order to

minimize the total transmitted power, subject to SINR and EHconstraints. The optimization scheme is formulated as follows.

minv(b)i ,ρ

(a)i

∑K

i=1

∑2

b=1‖v(b)

i ‖2 (11a)

s.t. SINR(a)i ≥ γ(a)

i , ∀(i, a), (11b)

EH(a)i ≥ δ(a)

i , 0 ≤ ρ(a)i ≤ 1, ∀(i, a), (11c)

where γ(a)i and δ

(a)i are the SINR and EH thresholds at the

i(a)th IoT receiver, respectively. The EH constraint representsthe minimum amount of energy required in order to ensure thatsufficient amount of energy is harvested in each transmissiontime. To obtain the optimal solution, the sufficient conditionsto obtain a feasible solution and the procedure and techniqueswill be discussed in the following sections.

B. Extension to MIMO Interference Channel

In this section, we extend the model proposed in theprevious section to multiple-antenna IoT receivers, where thenodes at the ith link now has Mi receive antennas. Note thatthe channel vectors h

(ac)ij , ∀ (i, j, a, c) in the MISO case is

now replaced with the channel matrices H(ac)ij , ∀ (i, j, a, c)

for the MIMO case. Accordingly, for this case, the SINR(a)i

and EH(a)i are defined as

SINR(a)i =

ρ(a)i

∣∣∣∣(u(a)i

)HH

(ab)ii v

(b)i

∣∣∣∣2ρ

(a)i

(u

(a)i

)HΣ

(a)i u

(a)i + σ2

i(a)‖u(a)

i ‖2, (12)

EH(a)i = ξ

(a)i

(1− ρ(a)

i

)tr

{H

(ab)ii v

(b)i

(v

(b)i

)H (H

(ab)ii

)H+ H

(aa)ii v

(a)i

(v

(a)i

)H (H

(aa)ii

)H+ Σ

(a)i

}. (13)

In (12) and (13), Σ(a)i is the covariance matrix of the total

interference-plus noise components, which is approximated byignoring the terms that include κβ � 1, and is expressed as(14)4

Now, the optimization problem (11) can be reformulated forthe MIMO case, as

minv(b)i ,u

(a)i ,ρ

(a)i

∑K

i=1

∑2

b=1‖v(b)

i ‖2 (15a)

s.t. SINR(a)i ≥ γ(a)

i , ∀(i, a), (15b)

EH(a)i ≥ δ(a)

i , ∀(i, a), (15c)

0 ≤ ρ(a)i ≤ 1, (15d)

‖u(a)i ‖

2 = 1,∀(i, a), (15e)

where u(a)i ∈ CMi×1, i ∈ {1, . . . ,K}, a ∈ {1, 2} is the

linear receiver applied at the receiver i(a).Problem (15) is non-convex and difficult to solve due

to the quadratic terms involving all transmit beamforming

4Note that approximation of Σ(a)i is a practical assumption [12], as the

terms κ and β are much smaller than 1.

IEEE TRANSACTIONS ON COMMUNICATIONS 5

Σ(a)i ≈ κH

(ab)ii diag

(v

(b)i

(v

(b)i

)H)(H

(ab)ii

)H+ κH

(aa)ii diag

(v

(a)i

(v

(a)i

)H)(H

(aa)ii

)H+∑K

j 6=i

∑2

c=1

[H

(ac)ij

(v

(c)j

(v

(c)j

)H+ κdiag

(v

(c)j

(v

(c)j

)H))(H

(ac)ij

)H]+∑K

j=1

∑2

c=1βdiag

(H

(ac)ij v

(c)j

(v

(c)j

)H (H

(ac)ij

)H)+ σ2

nIMi. (14)

Algorithm 1 : SWIPT for MIMO Full-Duplex Systems.

1: Initialize u(a),[n]i , ∀ (i, a) and set n = 0.

2: Repeat3: Compute v

(a),[n+1]i and ρ(a),[n+1]

i , ∀ (i, a) by solving the opti-mal problem with fixed u

(a),[n]i .

4: Compute u(a),[n+1]i , ∀ (i, a) using (19) with fixed v

(a),[n+1]i and

ρ(a),[n+1]i .

5: n = n+ 1.6: Until convergence

vectors. In order to solve this problem, we introduce a new

variable W(b)i , v

(b)i

(v

(b)i

)H, ∀(i, b) with rank(W

(b)i ) = 1,

and relax the corresponding problem by dropping the rankconstraint. The reformulated problem can be expressed as

minW

(b)i ,u

(a)i ,ρ

(a)i

∑K

i=1

∑2

b=1tr{

W(b)i

}(16a)

s.t. A−σ2i(a)‖u(a)

i ‖2

ρ(a)i

≥ 0, (16b)

B −

(δ

(a)i

)2

1− ρ(a)i

≥ 0, (16c)

0 ≤ ρ(a)i ≤ 1, (16d)

W(a)i � 0, ∀(i, a), (16e)

where the variables A and B are obtained after performingsome simple algebraic manipulations, and are defined as

A =1

γ(a)i

((u

(a)i

)HH

(ab)ii W

(b)i

(H

(ab)ii

)Hu

(a)i

)−(u

(a)i

)HΣ

(a)i u

(a)i , (17)

B =ξ(a)i tr

{H

(ab)ii v

(b)i

(v

(b)i

)H (H

(ab)ii

)H+H

(aa)ii v

(a)i

(v

(a)i

)H (H

(aa)ii

)H+ Σ

(a)i

}. (18)

To obtain the optimal v(b)i , we propose an iterative alter-

nating algorithm given on top of this page to solve problem(16). Some insightful discussion for the original problem andits solution is given in the following.

Finding the optimal V: First, under fixed u(a)i , we drop

the constraint rank(W

(b)i

)= 1 for numerical tractability to

solve for the optimal W(b)i and ρ(a)

i . Note that, the rank of thesolution W

(b)i can be guaranteed as rank one in general, and

the randomization procedure is proposed to generate a feasible

but suboptimal solution. Suppose W∗ is the optimal solutionto the relaxed problem. Due to the relaxation, rank(W∗)may not satisfy (15) in general. As a result, we adopt therandomization technique [1], [2], in which the solution W∗ iseigen-decomposed as W∗ = UW∗ΛW∗UH

W∗ and the solutionto (15) is chosen as W = UW∗Λ

1/2W∗vvHΛ

1/2W∗UH

W∗ withNi×1 uniform or Gaussian distributed random vector v [33],[34].

Secondly, under fixed v(b)i and ρ(a)

i , the problem to computethe optimal u

(a)i boils down to a feasibility problem, which

has a closed form solution, and it is given by [16], [35]

u(a)i =

(ρ

(a)i Σ

(a)i + σ2

i(a)IMi

)−1

H(ab)ii v

(b)i∥∥∥∥(ρ(a)

i Σ(a)i + σ2

i(a)IMi

)−1

H(ab)ii v

(b)i

∥∥∥∥ . (19)

The steps of the proposed algorithm are shown in Algorithm1.

Remark: Since the sum transmission power decreases (orstays at the same value) at each iteration, and it is lowerbounded by zero, the sum transmission power will converge.But since the primal problem (15) is non-convex, globaloptimality is not guaranteed.

C. Complexity AnalysisIn this subsection, we discuss the computational complexity

of the proposed algorithm. The number of arithmetic opera-tions required to solve a standard real-valued SDP problem

minx∈Rn

cT x (20)

subject to A0 +∑n

i=1xiAi � 0, and ‖x‖2 ≤ X, (21)

where Ai denotes the symmetric block-diagonal matrices withP diagonal blocks of size el × el, l = 1, . . . , P , is upper-bounded by [36]O (1)

(1 +

∑P

l=1el

)1/2

n

(n2 + n

∑P

l=1e2l +

∑P

l=1e3l

). (22)

The main computational complexity of Algorithm 1 arisesfrom computing the optimal v

(a)i and u

(a)i . For simplicity,

let us assume same number of transmit and receive antennasat each node, i.e., Mi = M and Ni = N , i = 1, . . . ,K.Accordingly the complexity analysis is given as below.

1) Computations required to calculate v(a)i : Since the

proposed algorithm solves a SDP problem in Step 2, thenumber of arithmetic operations required to compute optimalvi is calculated from (22) as follows. In computing vi,the number of diagonal blocks P is equal to 8K. For theconstraint (16b) for each node, the dimension of blocks aree

(a)i = M + 1, i = 1, . . . ,K, a ∈ {1, 2}. For the constraint

IEEE TRANSACTIONS ON COMMUNICATIONS 6

(16c), e(a)i = 3M2 + 1, i = 1, . . . ,K, a ∈ {1, 2}. For the

constraint (16d), and (16e), e(a)i = 1, and e

(a)i = N2, i =

1, . . . ,K, a ∈ {1, 2}, respectively. The unknown variables tobe determined are of size n = 2N2 + 2M + 1, where the firstand second terms correspond to the real and imaginary partsof W

(a)i and u

(a)i , respectively, while the third term represents

the slack variable.2) Computations required to calculate u

(a)i : The compu-

tation of the receive beamformer is calculated from (19) asfollows [37].• Term in numerator inside the inverse:

2∑Kj=1N

(N +

(1− N+1

2

))+ 2M2

• Term in numerator outside the inverse: 2MN −M• Inverse term in numerator: M3 +M2 +M• Product of the terms in numerator (outside the inverse

and the inverse): 2M2 −MAccordingly, the total computational complexity to calculatethe receiver matrix is in the order of O(φ(K2X+KM2(M+5))), where X = 2N

(N +

(1− N+1

2

)).

IV. TRANSCEIVER DESIGN WITH IMPERFECT CSIConsidering a more realistic scenario, it may not be pos-

sible to obtain perfect CSI at all the IoT nodes due to,for example error of channel estimation, quantization errors,feedback delay, etc. Hence, it is necessary and important tooptimize the system under imperfect CSI and design robusttransceivers. In this section, we will investigate the FD MISOand MIMO interference channels again, but with the importantdiscrepancy that the channels are now imperfectly known atthe IoT nodes.

A. FD MISO system with imperfect CSIConsidering the well-known norm-bounded error (NBE)

model [38], the actual CSI from the transmitter antennas ofnode b in the jth tier to the receiver antenna of node a in theith tier is given by

h(ab)ij = h

(ab)ij + eabij , (23)

(i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2},

where h(ab)ij is the estimated channel vector and eabij denotes

the CSI error vector, which is bounded by its known positiveconstant Euclidean norm as∥∥eabij ∥∥ ≤ ηabij , (i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2}. (24)

In this case, the uncertainty set of h(ab)ij can be defined as

S(ab)ij =

{h|h = h

(ab)ij + eabij ,

∥∥eabij ∥∥ ≤ ηabij } , (25)

(i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2}.Accordingly, we can rewrite the optimization problem

in (11) as

minv(b)i ,ρ

(a)i

∑K

i=1

∑2

b=1‖v(b)

i ‖2 (26a)

s.t. SINR(a)i ≥ γ(a)

i , ∀(i, a), (26b)

EH(a)i ≥ δ(a)

i , 0 ≤ ρ(a)i ≤ 1, ∀(i, a), (26c)

h(ab)ij = h

(ab)ij + eabij , (26d)

∥∥eabij ∥∥2 ≤ ηabij2, (27)

(i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2},

where SINR(a)i and EH

(a)i are given by (9) and (10), re-

spectively. Now, introducing several auxiliary variables andapplying the S-procedure [39] lemma, the problem for theMISO case can be expressed as (28), detailed steps of whichare included in Appendix A. In (28), λabij and µabij are slackvariables for (i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2}, andvariables aabii , b

aaii , c

acij , a

abii , b

aaii , c

acij and d

(aa)ii are defined in

(32), (33), (34), (40), (41), (42) and (43) in Appendix Arespectively. The above problem can be solved iteratively byusing the standard CVX toolbox, a package in Matlab forspecifying and solving convex programs [40], [41].

B. FD MIMO system with imperfect CSI

Similarly, for the MIMO scenario the actual CSI from thetransmitter antennas of the node b in the jth tier to the receiverantennas of the node a in the ith tier can be expressed as

H(ab)ij = H

(ab)ij + Eab

ij , (29)

(i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2},

where H(ab)ij is the estimated channel matrix and Eab

ij denotesthe CSI error matrix, which is bounded by its known positiveconstant Frobenius norm as∥∥Eab

ij

∥∥F≤ ηabij ,(i, j) ∈ {1, . . . ,K} and (a, b)∈{1, 2}. (30)

Now reformulating the SINR and EH constraintsand introducing slack variables, the problem for theMIMO case can be expressed as (31), the lengthy proofof which is relegated to Appendix B. The variablesaabii , b

aaii , c

acij , d

(aa)ii , aaaii , a

abii , b

aaii , c

acij and d

(aa)ii are defined

in (69), (70), (71), (72), (80), (81), (82), (83) and (84) inAppendix B, respectively. The optimal solutions can now beobtained via an alternating minimization approach.

Remark: By solving the problem (31), the matrices W(b)i

are not guaranteed to be of rank one mathematically. Generallyspeaking, the solution provides a lower bound to the originalproblem. As we mentioned before, it is worth noting thatour solutions are of rank one in most cases, which meansthe principal eigenvector v

(b)i of W

(b)i is the optimal solution

to the original problem. Otherwise, as explained before ran-domization procedure is proposed to generate a feasible butsuboptimal solution [33], [34], [42].

C. CSI Acquisition

In this paper, we consider the IoT nodes to be low powereddevices. These devices can range from smart wearables tosmart home appliances, which may use bluetooth 4.0 LE/5.0 LE, IEEE 802.11n, IEEE 802.11ac, etc., along withother future communication technologies such as mmWavetransmission.

We assume that the IoT devices are connected to a parentdevice (PD), such as a smartphone with high end signal pro-cessing capabilities. The PD has the knowledge of the nominalchannels and the radius of uncertainty regions. We undertake

IEEE TRANSACTIONS ON COMMUNICATIONS 7

minW

(b)i , ρ

(a)i , λabij , µ

abij , a

abii ,

baaii , cacij , a

abii , b

aaii , c

acij , d

(aa)ii

K∑i=1

2∑b=1

tr{

W(b)i

}(28a)

s.t. (50), (51), (52), (53), (57), (28b)(58), (59), (60), (61), (28c)

W(b)i � 0, 0 ≤ ρ(a)

i ≤ 1, (28d)λabij ≥ 0, µabij ≥ 0, (28e)

aabii ≥ 0, baaii ≥ 0, (28f)cacij ≥ 0, aabii ≥ 0, (28g)

baaii ≥ 0, cacij ≥ 0, d(aa)ii ≥ 0, (28h)

(i, j) ∈ {1, . . . ,K} and (a, b, c) ∈ {1, 2},

minW

(b)i , ρ

(a)i , λabij , µ

abij , a

abii , b

aaii ,

cacij , d(aa)ii aaaii , a

abii , b

aaii , c

acij , d

(aa)ii

K∑i=1

2∑b=1

tr{

W(b)i

}(31a)

s.t. (63), (64), (65), (66), (67), (68), (31b)(73), (74), (75), (76), (77), (78), (79), (31c)

W(b)i � 0, 0 ≤ ρ(a)

i ≤ 1, (31d)λabij ≥ 0, µabij ≥ 0, (31e)

aabii ≥ 0, baaii ≥ 0, (31f)

cacij ≥ 0, d(aa)ii ≥ 0, (31g)

aaaii ≥ 0, aabii ≥ 0, (31h)

baaii ≥ 0, cacij ≥ 0, daaii ≥ 0, (31i)(i, j) ∈ {1, . . . ,K} and (a, b, c) ∈ {1, 2}.

a centralized approach where the PD collects all channelmatrices, computes the beamforming matrices based on theimperfect CSI, and then distributes them to the IoT nodes.The estimation of CSI matrices follows a similar strategy tothat of traditional systems, as the IoT nodes cooperate withthe PD. This is performed via the exchange of the trainingsequences and feedback, and the application of usual CSIestimation methods [43].

Accordingly, from a practical implementation perspective,it is important to note the following points:

• The devices considered for EH in this paper are mainlyindoor devices, where the variation in channel is min-imum. So once acquired, the channel can be estimatedlater with minimum error and less complicated signalprocessing.

• Devices such as smart watches, heart rate monitors,fitness trackers, etc., are usually required to be connectedto a parent smart phone through bluetooth LE. Theselow powered devices are not required to do any complexprocessing to acquire the CSI, which will be the taskof the more capable smart phones. Once acquired, thesmart phones can transmit the information to the wearabledevices.

• Devices such as Nest thermostat, Google home, Microsoft

Hololens, Amazon Echo, Echo Show, etc., are usuallyconnected to a central WiFi device. Centralized algo-rithms can be processed at the central device, whichwill aggregate all CSI and perform the optimization. Thiswould however incur heavy signaling overhead and limitthe network scalability, which is not an issue here asscalability is not a factor in an indoor home network.

V. NUMERICAL RESULTS

In this section, we numerically investigate the SWIPToptimization problem for FD MISO and MIMO interferencechannels involving IoT nodes as a function of SINR constraintsand inter-user interference power. Accordingly, we set thenumber of transmit and receive antennas at each node asNi = 3 and Mi = 2, i = 1, . . . ,K. For simplicity, theEH and SINR thresholds are assumed to be equal5 for allreceivers, i.e., δ = δ

(a)i , γ = γ

(a)i , ∀ (i, a), respectively.

Unless otherwise stated, the parameters used for the simulationare: κ = β = −40dB, ξ(a)

i = 0.5, σ2i(a)

= −70dBm, ∀ (i, a),

5In practice however, thresholds for devices will depend on their transmis-sion power and battery capacity. For example, small devices such as wearables,might only be interested in EH, while larger devices such as smart speakersmay perform EH while also transmit substantial energy for others to harvestwithout its quality of service (QoS) being affected.

IEEE TRANSACTIONS ON COMMUNICATIONS 8

0 5 10 15 20 25

Iteration number

30.0

30.5

31.0

31.5

32.0

32.5

33.0

33.5

34.0

34.5

Cost fu

nction

Fig. 2. Convergence of the proposed algorithm

TABLE ITOTAL PORTION OF POWER DEDICATED TO EH CONSIDERING DIFFERENT

EH THRESHOLDS WHEN K = 2.

τ = 0.1 τ = 0.4 τ = 0.7 τ = 1.0

δ = −5dBm 0.6874 0.8687 0.8831 0.8872δ = 0dBm 0.7568 0.8953 0.9348 0.941δ = 5dBm 0.7986 0.9646 0.9394 0.966

τ = 1.3 τ = 1.6 τ = 1.9

δ = −5dBm 0.8761 0.8949 0.885δ = 0dBm 0.9243 0.9355 0.9306δ = 5dBm 0.966 0.9518 0.9625

and σ2n = −50dBm. Iteration method’s performance may

rely on the initialization state. As a result, it is importantto select good initialization points to achieve a suboptimalsolution with a good performance. While various initializa-tion techniques, such as random initialization, right singularmatrix initialization, etc., are used in literature [44], due tothe problem complexity, in this paper we use the randomintialization method, which is also very common in litera-ture [45]. The tolerance (the difference between cost functionof two iterations) of the proposed iterative algorithm is setto 10−5, the maximum number of iterations is set to 50,and the results are averaged over 1000 independent channelrealizations. The entries of the channel H

(ac)ij , ∀ (i, j, a, c) are

i.i.d. zero-mean with variance σ2ij,ac. For the direct channels,

i.e., H(ab)ii , i = 1, . . . ,K, a 6= b, σabii

2= 10−4. For the inter-

user interference channels, i.e., H(ac)ij , i 6= j, σacij

2 = 10−4

τ ,with τ being the inter-user interference suppression factor andfor the self-interference channel H

(aa)ii , ∀ (i, a), σaaii

2 = 10−3.We begin by showing the evolution of the proposed algo-

rithm, i.e., its convergence in Fig. 2. Here, the SINR threshold= 20 dB. The monotonic decrease of the cost function (sumpower in dBm) can be verified from the figure.

After establishing the convergence of the proposed algo-rithm, we now show the amount of power dedicated to EHwith different EH thresholds in Table I when K = 2 and theSINR threshold γ(a)

i = 10dB. It is worth noting that the valuesin this table are the portions of total power dedicated to EHin the system. The portions increase when the EH threshold

TABLE IIPORTION OF POWER AT EACH NODE DEDICATED TO EH WHEN

γ(a)i = 10dB AND δ

(a)i = 0dBm.

τ = 0.1 a = 1 a = 2 τ = 0.7 a = 1 a = 2K = 1 0.0569 0.0560 K = 1 0.0141 0.0155K = 2 0.0758 0.0545 K = 2 0.0155 0.0201τ = 1.3 a = 1 a = 2 τ = 1.9 a = 1 a = 2K = 1 0.0222 0.0204 K = 1 0.0166 0.0206K = 2 0.0183 0.0148 K = 2 0.0164 0.0158

-20 -15 -10 -5 0

Energy harvesting threshold (dBm)

10

15

20

25

30

35

Pow

er

consum

ed (

dB

m)

SINR Threshold = 5 dB

SINR Threshold = 10 dB

SINR Threshold = 15 dB

SINR Threshold = 20 dB

Fig. 3. Power consumption of FD MIMO interference channel with differentEH and SINR thresholds.

increases which is quite intuitive. However, the variance ofinter-user interference channel decreases with the increase inτ . However, no linear relation can be explicitly seen betweenthe values of τ and the portions of power. Nonetheless, itcan be seen that achieved power portion values are minimumwhen τ = 0.1. In other words, when the inter-user interferenceis more, the amount of dedicated power required for EH isless. This result clearly shows the usefulness of inter-userinterference in EH arising due to operating in FD mode.Furthermore, the optimal portion of power for each node hasalso been derived. For example, the optimal portion of EH foreach node when γ

(a)i = 10dB and δ

(a)i = 0dBm is shown in

Table II.Next, the minimum power consumption of FD MIMO

interference channel with different EH and SINR thresholdsis shown in Fig. 3 when K = 2. It can be observedthat the system power consumption increases for higher EHthreshold requirements. Thus, more power is required as SINRthreshold increases. For a specific EH threshold, the powerconsumption increases slowly with the SINR threshold. Thiscan be attributed to the fact that the system requires morepower for signal detection.

In Fig. 4, the importance of smart channel assignment,at a stage prior to the precoder/decoder design is depictedfor the proposed algorithm. The value of τ represents theprovided isolation among the nodes responsible for inter userinterference. In particular, the power consumption of FDMIMO interference channel with respect to the different values

IEEE TRANSACTIONS ON COMMUNICATIONS 9

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.820

22

24

26

28

30

32

34P

ow

er

consum

ed (

dB

m) EH threshold = -10 dBm

EH threshold = -5 dBm

EH threshold = 0 dBm

Fig. 4. Power consumption of FD MIMO interference channel for differentvalues of τ and EH thresholds.

-40 -35 -30 -25 -20 -15 -10

Energy harvesting threshold (dBm)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ave

rag

e n

orm

aliz

ed

tra

nsm

it p

ow

er

2 Nodes

4 Nodes

6 Nodes

Fig. 5. Average normalized total transmitted power versus EH threshold ofFD MISO interference channel.

of τ and EH threshold when K = 2 is presented. The SINRthreshold is fixed in this figure at 10dB. It can be seen thatthe power consumption increases with τ , but becomes flatlater. It is worth noting that the inter-user interferences arestronger when τ is small and the system can harvest moreenergy from the interferences. However, the system doesn’tharvest enough energy from the interference channels when τincreases. Like before, this figure also illustrates the benefitsof inter-user interference in EH.

In Fig. 5, we give the simulations for the MISO scenario.Assuming SINR threshold to be 20dB, we compare theperformance of average normalized total transmitted powerversus EH threshold for the case of 2, 4 and 6 nodes whenτ = 1. It can be seen that the average value of the totaltransmitted power increases monotonically with the increaseof the EH threshold. Furthermore, with the increase in thenumber of nodes, the average normalized transmitted poweralso increases. It is worthwhile to note that the average

-10 -5 0 5 10

Energy harvesting threshold (dBm)

35

40

45

50

55

60

Pow

er

consum

ed (

dB

m)

SINR threshold = 15 dB, 2=0.1

SINR threshold = 5 dB, 2=0.1

SINR threshold = -15 dB, 2=0.1

SINR threshold = -15 dB, 2=0.2

SINR threshold = -15 dB, 2=0.5

Fig. 6. Power consumption of imperfect FD MIMO interference channel fordifferent SINR thresholds and error bounds.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.636

38

40

42

44

46

48

50

52

54

Po

we

r co

nsu

me

d (

dB

m)

EH threshold = 5 dBm

EH threshold = -5 dBm

EH threshold = -15 dBm

Fig. 7. Power consumption of imperfect FD MIMO interference channel fordifferent values of τ and EH thresholds.

normalized power increases slightly when EH threshold is lessthan −15dB. It means that the EH technique can be appliedwhen the required energy is small, which was also applicablefor the MIMO case.

Finally, Fig. 6 and 7 present the total system power con-sumption of imperfect FD MIMO IoT nodes. The numbersof transmit and receive antennas at each node are set asNi = 2 and Mi = 2, i = 1, . . . ,K and K = 2. In Fig. 6,the power consumption is much higher than the perfect CSIscenario and the difference in power consumption for differentSINR thresholds are more obvious. Meanwhile, it shows thatmore power is needed when the EH threshold increases.Further, the bound of CSI error is verified in Fig. 6 fromη =√

0.1 to η =√

0.5. As expected, the power consumptionincreases with the increase in channel uncertainty. Finally,the total power consumption is plotted with respect to τ anddifferent EH thresholds in Fig. 7, where the SINR thresholdis 10dB and the CSI error is bounded by η =

√0.1. Similar

IEEE TRANSACTIONS ON COMMUNICATIONS 10

to the perfect CSI scenario, the power consumption increaseswith τ , but becomes flat later. More power is needed owingto the uncertainty of the channel, but it also shows that theEH process is the primary beneficiary from the inter-userinterference.

At this point it is worthwhile to mention that the EHthreshold value chosen for the numerical analysis, satisfiesthe power consumption requirement of small devices, suchas small low power IoT devices. Accordingly, the harvestedpower can definitely be considered to be utile if not sufficientto recharge their individual batteries.

VI. CONCLUDING REMARKS

The joint transceiver design problem for the SWIPT FDMISO/MIMO interference channels involving IoT nodes wasstudied under both perfect and imperfect CSI cases. An itera-tive alternating algorithm was proposed to design the transmitand receive filters. Meanwhile, the optimal power splittingratio minimizing the total transmit power was derived.

The simulation results show that the EH technique worksefficiently on FD MIMO IoT systems and the harvestedpower from EH can support charging of batteries of powerconsumption limited IoT devices with a specific guaranteedtransmission quality.

Furthermore, in the current work for the sake of simplicity,we do not consider the dynamics of the battery to be charged.Several discretization techniques can be used to approximatethe battery level after storing the harvested energy. In future,we will use steady-state probability analysis to derive theprobability of having sufficient energy to charge the battery.

Also, we consider linear EH circuits in this work foranalytical simplicity. While in linear EH model, the RF-to-direct current (DC) power conversion efficiency is independentof the input impedence of the EH circuit, in a non-linear end-to-end wireless power transfer circuit, this is not the case. Thenon-linear EH model will ensure better resource allocation forSWIPT, which will be considered in future as an extension tothis work.

APPENDIX APROBLEM REFORMULATION (IMPERFECT CSI-MISO

CASE)

Here, we show the steps for reformulating the problem (26).To begin, for the SINR constraint, we introduce three auxiliaryvariables as

aabii = max∀eab

iiHeab

ii ≤ηabii

2

(h

(ab)ii

)H(κ (1 + β)

× diag(

v(b)i

(v

(b)i

)H)+βv

(b)i

(v

(b)i

)H)(h

(ab)ii

),

(32)

baaii = max∀eaa

iiHeaa

ii ≤ηaaii

2

(h

(aa)ii

)H(κ (1 + β)

× diag(

v(a)i

(v

(a)i

)H)+βv

(a)i

(v

(a)i

)H)(h

(aa)ii

),

(33)

cacij = max∀eac

ijHeac

ij ≤ηacij

2

(h

(ac)ij

)H (v

(c)j

(v

(c)j

)H+κdiag

(v

(c)j

(v

(c)j

)H))(h

(ac)ij

), (34)

where aabii , baaii and cacij are the maximum (worst cases)cross tier and self-interference. By using these three auxiliaryvariables, Σ

(a)i in eq. (7) can be rewritten as

Σ(a)

i = aabii + baaii + (1 + β)

K∑j 6=i

2∑c=1

cacij + σ2n

. (35)

Then, the SINR constraint can be rewritten as

ρ(a)i

(h

(ab)ii + eabii

)HW

(b)i

(h

(ab)ii + eabii

)− γ(a)

i ρ(a)i Σ

(a)

i − γ(a)i σ2

i(a)≥ 0,

∥∥eabii ∥∥2 ≤ ηabii2, (36)

aabii ≥(h

(ab)ii + eabii

)H (κ (1 + β) diag

(W

(b)i

)+ βW

(b)i

)×(h

(ab)ii + eabii

),∥∥eabii ∥∥2 ≤ ηabii

2, (37)

baaii ≥(h

(aa)ii + eaaii

)H (κ (1 + β) diag

(W

(a)i

)+ βW

(a)i

)×(h

(aa)ii + eaaii

), ‖eaaii ‖

2 ≤ ηaaii2, (38)

cacij ≥(h

(ac)ij + eacij

)H (W

(c)j + κdiag

(W

(c)j

))×(h

(ac)ij + eacij

),∥∥eacij ∥∥2 ≤ ηacij

2, (39)

for (i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2}.

Similarly, for EH constraint, we introduce four auxiliaryvariables as (40), (41), (42) and (43), where aabii , baaii , cacij anddaaii denote the minimum (worst cases) available power forEH from all the nodes. The covariance matrix in EH can berewritten as

Σ(a)

i = aabii + baaii + (1 + β)

K∑j 6=i

2∑c=1

cacij + σ2n

. (44)

Then, the EH constraint can be rewritten as

ξ(a)i

(1− ρ(a)

i

)((h

(ab)ii + eabii

)HW

(b)i

(h

(ab)ii + eabii

)+daaii + Σ

(a)i

)− δ(a)

i ≥ 0,∥∥eabii ∥∥2 ≤ ηabii

2, (45)

aabii ≤(h

(ab)ii + eabii

)H (κ (1 + β) diag

(W

(b)i

)+βW

(b)i

)(h

(ab)ii + eabii

),∥∥eabii ∥∥2 ≤ ηabii

2, (46)

baaii ≤(h

(aa)ii + eaaii

)H (κ (1 + β) diag

(W

(a)i

)+βW

(a)i

)(h

(aa)ii + eaaii

), ‖eaaii ‖

2 ≤ ηaaii2, (47)

IEEE TRANSACTIONS ON COMMUNICATIONS 11

aabii = min∀eab

iiHeab

ii ≤ηabii

2

(h

(ab)ii

)H (κ (1 + β) diag

(v

(b)i

(v

(b)i

)H)+ βv

(b)i

(v

(b)i

)H)(h

(ab)ii

), (40)

baaii = min∀eaa

iiHeaa

ii ≤ηaaii

2

(h

(aa)ii

)H (κ (1 + β) diag

(v

(a)i

(v

(a)i

)H)+ βv

(a)i

(v

(a)i

)H)(h

(aa)ii

), (41)

cacij = min∀eac

ijHeac

ij ≤ηacij

2

(h

(ac)ij

)H (v

(c)j

(v

(c)j

)H+ κdiag

(v

(c)j

(v

(c)j

)H))(h

(ac)ij

), (42)

daaii = min∀eaa

iiHeaa

ii ≤ηaaii

2

(h

(aa)ii

)H (v

(a)i

(v

(a)i

)H)(h

(aa)ii

), (43)

W(b)i + λabii I W

(b)i

(h

(ab)ii

)(h

(ab)ii

)HW

(b)i

(h

(ab)ii

)HW

(b)i

(h

(ab)ii

)− γ(a)

i ρ(a)i Σ

(a)

i − γ(a)i σ2

i(a)− λabii ηabii

2

� 0, (50)

−F1 + λabii I −F1

(h

(ab)ii

)−(h

(ab)ii

)HF1 aabii −

(h

(ab)ii

)HF1

(h

(ab)ii

)− λabii ηabii

2

� 0, (51)

−F2 + λaaii I −F2

(h

(aa)ii

)−(h

(aa)ii

)HF2 baaii −

(h

(aa)ii

)HF2

(h

(aa)ii

)− λaaii ηaaii 2

� 0, (52)

−F3 + λacij I −F3

(h

(ac)ij

),

−(h

(ac)ij

)HF3 cacij −

(h

(ac)ij

)HF3

(h

(ac)ij

)− λacij ηacij 2

� 0, (53)

cacij ≤(h

(ac)ij + eacij

)H (W

(c)j + κdiag

(W

(c)j

))×(h

(ac)ij + eacij

),∥∥eacij ∥∥2 ≤ ηacij

2, (48)

daaii ≤(h

(aa)ii + eaaii

)H (W

(a)i

)×(h

(aa)ii + eaaii

), ‖eaaii ‖

2 ≤ ηaaii2, (49)

for (i, j) ∈ {1, . . . ,K} and (a, b) ∈ {1, 2}.

However, this problem is still computationally intractable,because it involves an infinite number of constraints, whichhave to be reformulated into finite convex constraints. Byapplying the S-procedure [39], the SINR constraints in (36),(37), (38) and (39) can be reformulated to finite convexconstraints which are equivalent to (50), (51), (52) and (53),where

F1 = κ (1 + β) diag(W

(b)i

)+ βW

(b)i , (54)

F2 = κ (1 + β) diag(W

(a)i

)+ βW

(a)i , (55)

F3 = W(c)j + κdiag

(W

(c)j

). (56)

Meanwhile, the EH constraints in (45), (46), (47), (48) and(49) can be reformulated to finite convex constraints whichare equivalent to (57), (58), (59), (60) and (61).

Accordingly, after some simple mathematical manipula-tions, we obtain (28).

APPENDIX BPROBLEM REFORMULATION (IMPERFECT CSI-MIMO

CASE)

Here, we show the steps for reformulating the problem (31).To begin, the SINR constraint can be rewritten as (62).

Then, the SINR constraint of this MIMO scenario can bereformulated as (63), (64), (65), (66), (67) and (68), where

Σ(a)

i ,κ(aabii + baaii

)+

K∑j 6=i

2∑c=1

cacij +

K∑j=1

2∑c=1

βdacij

+ (uai )Hσ2nIuai ,

V S1 , min‖E(ab)

ij ‖F≤η(ab)ij

1

γ(a)i

((u

(a)i

)H (H

(ab)ii + E

(ab)ii

)×W

(b)i

(H

(ab)ii + E

(ab)ii

)Hu

(a)i

),

F4 =(W

(c)j + κdiag

(W

(c)j

)),

aabii , max∀‖Eab

ii ‖2F≤ηabii

2(uai )

HH

(ab)ii diag

(W

(b)i

)(H

(ab)ii

)Huai ,

(69)

baaii , max∀‖Eaa

ii ‖2F≤ηaaii

2

(uai )H

H(aa)ii diag

(W

(a)i

)(H

(aa)ii

)Huai ,

(70)

IEEE TRANSACTIONS ON COMMUNICATIONS 12

W(b)i + µabii I W

(b)i

(h

(ab)ii

)(h

(ab)ii

)HW

(b)i

(h

(ab)ii

)HW

(b)i

(h

(ab)ii

)+ d

(aa)ii + Σ

(a)i −

δ(a)i

ξai (1−ρai ) − µabii η

abii

2

� 0, (57)

F1 + µabii I F1

(h

(ab)ii

)(h

(ab)ii

)HF1

(h

(ab)ii

)HF1

(h

(ab)ii

)− aabii − µabii ηabii

2

� 0, , (58)

F2 + µaaii I F2

(h

(aa)ii

)(h

(aa)ii

)HF2

(h

(aa)ii

)HF2

(h

(aa)ii

)− baaii − µaaii ηaaii 2

� 0, , (59)

F3 + µacij I F3

(h

(ac)ij

),(

h(ac)ij

)HF3

(h

(ac)ij

)HF3

(h

(ac)ij

)− cacij − µacij ηacij 2

� 0, (60)

W(a)i + µaaii I W

(a)i

(h

(aa)ii

)(h

(aa)ii

)HW

(a)i

(h

(aa)ii

)HW

(a)i

(h

(aa)ii

)− d(aa)

ii − µaaii ηaaii 2

� 0, (61)

ρ(a)i ≥ σ2

i(a)

((

u(a)i

)H (H

(ab)ii + E

(ab)ii

)W

(b)i

(H

(ab)ii + E

(ab)ii

)Hu

(a)i

)−(u

(a)i

)HΣ

(a)i u

(a)i

γ(a)i

−1

,

⇒

ρ(a)i σi(a)

σi(a)

((u

(a)i

)H(H

(ab)ii +E

(ab)ii

)W

(b)i

(H

(ab)ii +E

(ab)ii

)Hu

(a)i

)−(u

(a)i

)HΣ

(a)i u

(a)i

γ(a)i

� 0. (62)

ρ(a)i σi(a)

σi(a) V S1−(u

(a)i

)HΣ

(a)i u

(a)i

� 0 , (63)

1

γ(a)i

(u

(a)i

)HH

(ab)ii W

(b)i

(H

(ab)ii

)Hu

(a)i − V S1− λabii

(u

(a)i

)HH

(ab)ii W

(b)i((

u(a)i

)HH

(ab)ii W

(b)i

)HW

(b)i +

λabij

η(ab)ii

� 0 , (64)

− (uai )H

diag(W

(b)i

)uai + λabii − (uai )

Hdiag

(W

(b)i

)(H

(ab)ii

)Huai

− (uai )H

H(ab)ii diag

(W

(b)i

)uai aabii − (uai )

HH

(ab)ii diag

(W

(b)i

)(H

(ab)ii

)Huai − λabii ηabii

2

� 0, (65)

− (uai )H

diag(W

(a)i

)uai + λaaii − (uai )

Hdiag

(W

(a)i

)(H

(aa)ii

)Huai

− (uai )H

H(aa)ii diag

(W

(a)i

)uai baaii − (uai )

HH

(aa)ii diag

(W

(a)i

)(H

(aa)ii

)Huai − λaaii ηaaii 2

� 0, (66)

− (uai )H

F4uai + λacij − (uai )

HF4

(H

(ac)ij

)Huai

− (uai )H

H(ac)ij F4u

ai cacij − (uai )

HH

(ac)ij F4

(H

(ac)ij

)Huai − λacij ηacij 2

� 0, (67)

− (uai )H

diag(W

(c)j

)uai + λacij − (uai )

Hdiag

(W

(c)j

(H

(ac)ij

)H)uai

− (uai )H

diag(H

(ac)ij W

(c)j

)uai dacij − (uai )

Hdiag

(H

(ac)ij W

(c)j

(H

(ac)ij

)H)uai − λacij ηacij 2

� 0, (68)

IEEE TRANSACTIONS ON COMMUNICATIONS 13

ξ(a)i (1− ρ(a)

i )

√δ

(a)i√

δ(a)i V S2 + aaaii + Σ

(a)i

� 0 , (73)

vec(H

(ab)ii

)H (Wb

i ⊗ I)

vec(H

(ab)ii

)− V S2− µabii vec

(H

(ab)ii

)H (Wb

i ⊗ I)(

vec(H

(ab)ii

)H (Wb

i ⊗ I))H (

Wbi ⊗ I

)+

µabii

ηabii

2 I

� 0 , (74)

vec(H

(aa)ii

)H(Wa

i ⊗ I) vec(H

(aa)ii

)− aaaii − µaaii vec

(H

(aa)ii

)H(Wa

i ⊗ I)(vec(H

(aa)ii

)H(Wa

i ⊗ I)

)H(Wa

i ⊗ I) +µaaii

ηaaii

2 I

� 0 , (75)

vec(H

(ab)ii

)H (diag(Wb

i )⊗ I)

vec(H

(ab)ii

)− aabii − µabii vec

(H

(ab)ii

)H (diag(Wb

i )⊗ I)(

vec(H

(ab)ii

)H (diag(Wb

i )⊗ I))H (

diag(Wbi )⊗ I

)+

µabii

ηabii

2 I

� 0 , (76) vec

(H

(aa)ii

)H(diag(Wa

i )⊗ I) vec(H

(aa)ii

)− aaaii − µaaii vec

(H

(aa)ii

)H(diag(Wa

i )⊗ I)(vec(H

(aa)ii

)H(diag(Wa

i )⊗ I)

)H(diag(Wa

i )⊗ I) +µaaii

ηaaii

2 I

� 0 , (77) vec

(H

(ac)ij

)H(F4 ⊗ I) vec

(H

(ac)ij

)− cacij − µacij vec

(H

(ac)ij

)H(F4 ⊗ I)(

vec(H

(ac)ij

)H(F4 ⊗ I)

)H(F4)⊗ I) +

µacij

ηacij

2 I

� 0 , (78)

vec(H

(ac)ij

)H (Wc

j ⊗ I)

vec(H

(ac)ij

)− cacij − µacij vec

(H

(ac)ij

)H (Wc

j ⊗ I)(

vec(H

(ac)ij

)H (Wc

j ⊗ I))H (

Wcj ⊗ I

)+

µacij

ηacij

2 I

� 0 , (79)

cacij , max∀‖Eac

ij ‖2F≤ηacij

2

(uai )H

H(ac)ij F4

(H

(ac)ij

)Huai , (71)

dacij , max∀‖Eac

ij ‖2F≤ηacij

2

(uai )H

diag

(H

(ac)ij W

(c)j

(H

(ac)ij

)H)uai .

(72)

and λabij are the slack variables. Note that, the slack variablesin (64) and (65) should be marked differently for simulations,which use the same λabii . Similarly, the EH constraint can bereformulated as (73), (74), (75), (76), (77), (78) and (79),where

Σ(a)i ,κ

(aabii + baaii

)+

K∑j 6=i

2∑c=1

cacij +

K∑j=1

2∑c=1

βdacij +tr(σ2nI),

V S2 , min‖E(ab)

ij ‖F≤η(ab)ij

tr{(

H(ab)ii + E

(ab)ii

)×W

(b)i

(H

(ab)ii + E

(ab)ii

)H},

aaaii , min∀‖Eaa

ii ‖2F≤ηaaii

2

tr

(H

(aa)ii W

(a)i

(H

(aa)ii

)H), (80)

aabii , min∀‖Eab

ii ‖2F≤ηabii

2tr

(H

(ab)ii diag

(W

(b)i

)(H

(ab)ii

)H),

(81)

baaii , min∀‖Eaa

ii ‖2F≤ηaaii

2

tr

(H

(aa)ii diag

(W

(a)i

)(H

(aa)ii

)H),

(82)

cacij , min∀‖Eac

ij ‖2F≤ηacij

2

tr

(H

(ac)ij F4

(H

(ac)ij

)H), (83)

dacij , min∀‖Eac

ij ‖2F≤ηacij

2

tr

(diag

(H

(ac)ij W

(c)j

(H

(ac)ij

)H)).

(84)

and µabij are the slack variables. Accordingly, with some simplemathematical manipulations, we obtain (31).

IEEE TRANSACTIONS ON COMMUNICATIONS 14

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IEEE TRANSACTIONS ON COMMUNICATIONS 15

PLACEPHOTOHERE

Jiang Xue (S’09-M’13) received the B.S. degreein Information and Computing Science from theXi’an Jiaotong University, Xi’an, China, in 2005, theM.S. degrees in Applied Mathematics from LanzhouUniversity, China and Uppsala University, Sweden,in 2008 and 2009, respectively. Dr. J. Xue receivedthe Ph.D. degree in Electrical and Electronic En-gineering from ECIT, the Queen’s University ofBelfast, U.K., in 2012. From 2013 to 2017, He wasa Research Fellow with the University of Edinburgh,UK. Since 2017, Dr. J. Xue has joined the National

Engineering Laboratory for Big Data Analytics, Xi’an International Academyfor Mathematics and Mathematical Technology, School of Mathematics andStatistics in Xi’an Jiaotong University, China, as a professor. His main inter-ests include the machine learning and wireless communication, performanceanalysis of general multiple antenna systems, stochastic geometry, cooperativecommunications, and cognitive radio.

PLACEPHOTOHERE

Sudip Biswas (S’16-M’17) received the B.Tech.degree in electronics and communication engineer-ing from the Sikkim Manipal Institute of Tech-nology, Sikkim, India, in 2010, the M.Sc. degreein signal processing and communications from theUniversity of Edinburgh, Edinburgh, U.K., in 2013and the Ph.D. degree in digital communications atthe University of Edinburgh?s Institute for DigitalCommunications in 2017. Currently, he is workingas a research scientist at theInstitute for Digital Com-munications, University of Edinburgh.His research

interests include various topics in wireless communications and networkinformation theory with particular focus possible 5G technologies such asmassive MIMO, mmWave, full-duplex, NOMA and wireless caching.

PLACEPHOTOHERE

Ali Cagatay Cirik (S’13-M’14) received the B.Sand M.S. degrees in telecommunications and elec-tronics engineering from Sabanci University, Istan-bul, Turkey, in 2007 and 2009, respectively, andPh.D. degree in electrical engineering from Uni-versity of California, Riverside in 2014. He heldresearch fellow positions at Centre for WirelessCommunications, Oulu, Finland, University of Ed-inburgh, U.K and University of British Columbia,Vancouver, Canada between June 2014 and October2017. His industry experience includes internships at

Mitsubishi Electric Research Labs (MERL), Cambridge, MA, in 2012, Broad-com Corporation, Irvine, CA, in 2013 and includes industrial postdoctoralresearcher position at Sierra Wireless, Richmond, Canada between November2015 and October 2017. He is currently working at Ofinno Technologies,Herndon, VA as Senior Technical Staff. His primary research interests are full-duplex communication, 5G non-orthogonal multiple-access (NOMA), MIMOsignal processing, and convex optimization.

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Huiqin Du received the B.Sc. in Electronic Infor-mation Science and Technical from Beijing Univer-sity of Chemical Technology, China in 2004, MSc.with distinction in Radio Frequency CommunicationSystem in 2006 and PhD degree in Signal Process-ing in 2010, from University of Southampton andUniversity of Edinburgh, respectively. From 2010-2013, She worked as a research fellow at QueensUniversity of Belfast and University of Edinburgh.Since 2013, she has been with the College of Infor-mation Science and Technology, Jinan University,

Guangzhou, China as an Associate Professor. Her current research interestsinclude statistical signal processing, wireless MIMO communications, cogni-tive radio network, massive MIMO and hybrid beamforming.

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Yang Yang received the B.S. degree in Schoolof Information Science and Engineering, SoutheastUniversity, Nanjing, China, in 2009, and the Ph.D.degree in Department of Electronic and ComputerEngineering, The Hong Kong University of Scienceand Technology. From Nov. 2013 to Nov. 2015 hehad been a postdoctoral research associate at theCommunication Systems Group, Darmstadt Univer-sity of Technology, Darmstadt, Germany. From Dec.2015 to Oct. 2017 he had been a senior standardsand research scientist in Intel, Germany. He joined

the University of Luxembourg as a research associate in Nov. 2017. Hisresearch interests are in parallel and distributed solution methods in convexoptimization and nonlinear programming, with applications in communicationnetworks and signal processing.

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Tharmalingam Ratnarajah (A’96-M’05-SM’05) iscurrently with the Institute for Digital Communica-tions, University of Edinburgh, Edinburgh, UK, asa Professor in Digital Communications and SignalProcessing and the Head of Institute for DigitalCommunications. His research interests include sig-nal processing and information theoretic aspects of5G and beyond wireless networks, full-duplex radio,mmWave communications, random matrices theory,interference alignment, statistical and array signalprocessing and quantum information theory. He has

published over 330 publications in these areas and holds four U.S. patents.He was the coordinator of the FP7 projects ADEL (3.7Me) in the area oflicensed shared access for 5G wireless networks and HARP (3.2Me) in thearea of highly distributed MIMO and FP7 Future and Emerging Technologiesprojects HIATUS (2.7Me) in the area of interference alignment and CROWN(2.3Me) in the area of cognitive radio networks. Dr Ratnarajah is a Fellowof Higher Education Academy (FHEA), U.K..

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Mathini Sellathurai is presently a Reader with theHeriot-Watt University, Edinburgh, U.K and leadingresearch in signal processing for intelligent systemsand wireless communications. Her research includesadaptive, cognitive and statistical signal processingtechniques in a range of applications including Radarand RF networks, Network Coding, Cognitive Radio,MIMO signal processing, satellite communicationsand ESPAR antenna communications. She has beenactive in the area of signal processing research forthe past 15 years and has a strong international

track record in multiple-input, multiple-output (MIMO) signal processing withapplications in radar and wireless communications research. Dr. Sellathuraihas 5 years of industrial research experience. She held positions with Bell-Laboratories, New Jersey, USA, as a visiting researcher (2000); and with theCanadian (Government) Communications Research Centre, Ottawa Canada asa Senior Research Scientist (2001-2004). Since 2004 August, she has beenwith academia. She also holds an honorary Adjunct/Associate Professorshipat McMaster University, Ontario, Canada, and an Associate Editorship for theIEEE Transactions on Signal Processing between 2009 -2013 and presentlyserving as an IEEE SPCOM Technical Committee member. She has publishedover 150 peer reviewed papers in leading international journals and IEEEconferences; given invited talks and written several book chapters as well as aresearch monograph titled “Space-Time Layered Processing” as a lead author.The significance of her accomplishments is recognized through internationalawards, including an IEEE Communication Society Fred W. Ellersick Best Pa-per Award in 2005, Industry Canada Public Service Awards for contributionsin science and technology in 2005 and awards for contributions to technologyTransfer to industries in 2004. Dr. Sellathurai was the recipient of the NaturalSciences and Engineering Research Council of Canadas doctoral award forher Ph.D. dissertation.