Title Wireless power and information transmission

Title

Wireless power and information transmission : nonlinear model and beamforming optimization

Advisor(s)

Wu, YC

Author(s)

Wang, Shuai; 王帅

Citation

Issued Date

URL

Rights

Wang, S. [王帅]. (2018). Wireless power and information transmission : nonlinear model and beamforming optimization. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. 2018

http://hdl.handle.net/10722/265391

The author retains all proprietary rights, (such as patent rights) and the right to use in future works.

Abstract of thesis entitled

“Wireless Power and Information Transmission: Nonlinear Model and Beamforming Optimization” submitted by Shuai Wang for the degree of Doctor of Philosophy at The University of Hong Kong in May 2018 In Internet-of-Things (IoT) paradigm, massive amount of low-power electronic devices need to be connected, but providing energy for these devices is a challenging task. For tackling this issue, wireless power and information transmission (WPIT) is a promising technique as it enables data and energy to be transmitted from a distance via radio frequency signals. Currently, the major obstacle to implement WPIT is the low transmission efficiency, and it is known that beamforming is a key technique to combat against the low efficiency. However, the effectiveness of beamforming design is dependent on the accuracy of energy harvesting model. While existing works mostly adopt a linear energy harvesting model due to its simplicity, a practical energy harvester contains nonlinear elements such as diodes. This leads to a mismatch between model and circuit, and consequently the degradation of the performance of WPIT. In order to address the above problem, a practical nonlinear energy harvesting model, which matches the experimental data of energy harvesting products very well, is proposed in this thesis. The impact of nonlinear model on beamforming design is illustrated in a point-to-point WPIT system. It is shown that the beamforming design based on the proposed model significantly improves

the WPIT transmission efficiency compared to traditional designs based on linear model. With the nonlinear model, the beamforming designs in two other WPIT systems are further investigated. In the first system, the beamforming design of a mobile charger is considered, with the aim of maximizing the data-rate subject to a given energy budget. Due to the additional degree of freedom brought by moving, the beamforming vector needs to be jointly designed with stopping times along the moving path. To this end, a convergence guaranteed iterative algorithm based on difference of convex (DC) programming is proposed, and it is shown that the corresponding beamforming solution achieves a data-rate close to the upper bound. Furthermore, it is found that with an appropriate moving path and the proposed beamforming design, the data-rate gain could be large compared to the fixed charger case. In the second system, the beamforming design under massive antenna arrays at the transmitter is considered. While existing WPIT beamforming algorithms could achieve good performance, they involve the inverse of Hessian matrices. This leads to extremely time consuming computations and cannot be used in practice if the number of antennas is in the range of hundreds or more. To this end, an accelerated first-order algorithm, which only needs to compute the gradients, is proposed. Both theoretical and numerical results show that the proposed method reduces the computation time by orders of magnitude while still guaranteeing the same performance compared to state-of-the-art methods. (Total words: 438)

Wireless Power and Information Transmission: Nonlinear Model and Beamforming Optimization

by

Shuai Wang B. Eng., M. Eng., Beijing University of Posts and Telecommunications

A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy

at

The University of Hong Kong August 2018

This page is intentionally left blank

Declaration

I declare that this thesis and the research work thereof represent my own work, except where due acknowledgement have been made. I further declared that it has not been previously included in a thesis, dissertation, or report submitted to this University or any other institution for a degree, diploma or other qualification.

Signed:

i

To the loving memory of my grandmother

ii

Acknowledgements I am most grateful to my supervisor Dr. Yik-Chung Wu. His insightful comments broaden my horizons and his patient guidance lights up my research career. I would also like to thank Dr. Minghua Xia for his great support, Dr. Zhigang Wen for his continuous encouragement, and Dr. Kaibin Huang for his interesting ideas. Moreover, I would like to thank all the members of my research team and my friends in Hong Kong, Paris, Beijing, and Huhhot: my Ph.D study becomes so wonderful because of you. Finally, I want to express my deepest love to my parents and my wife Shenglan.

iii

Abstract In Internet-of-Things (IoT) paradigm, massive amount of low-power electronic devices need to be connected, but providing energy for these devices is a challenging task. For tackling this issue, wireless power and information transmission (WPIT) is a promising technique as it enables data and energy to be transmitted from a distance via radio frequency signals. Currently, the major obstacle to implement WPIT is the low transmission efficiency, and it is known that beamforming is a key technique to combat against the low efficiency. However, the effectiveness of beamforming design is dependent on the accuracy of energy harvesting model. While existing works mostly adopt a linear energy harvesting model due to its simplicity, a practical energy harvester contains nonlinear elements such as diodes. This leads to a mismatch between model and circuit, and consequently the degradation of the performance of WPIT. In order to address the above problem, a practical nonlinear energy harvesting model, which matches the experimental data of energy harvesting products very well, is proposed in this thesis. The impact of nonlinear model on beamforming design is illustrated in a point-to-point WPIT system. It is shown that the beamforming design based on the proposed model significantly improves the WPIT transmission efficiency compared to traditional designs based on linear model. With the nonlinear model, the beamforming designs in two other WPIT systems are further investigated. In the first system, the beamforming design of a mobile charger is considered, with the aim of maximizing the data-rate subject to a given energy budget. Due to the additional degree of freedom brought by moving, the beamforming vector needs to be jointly designed with stopping times along the moving path. To this end, a convergence guaranteed iterative algorithm based on difference of convex (DC) programming is proposed, and it is shown that the corresponding beamforming solution achieves a data-rate close to the upper iv

bound. Furthermore, it is found that with an appropriate moving path and the proposed beamforming design, the data-rate gain could be large compared to the fixed charger case. In the second system, the beamforming design under massive antenna arrays at the transmitter is considered. While existing WPIT beamforming algorithms could achieve good performance, they involve the inverse of Hessian matrices. This leads to extremely time consuming computations and cannot be used in practice if the number of antennas is in the range of hundreds or more. To this end, an accelerated first-order algorithm, which only needs to compute the gradients, is proposed. Both theoretical and numerical results show that the proposed method reduces the computation time by orders of magnitude while still guaranteeing the same performance compared to state-of-the-art methods. (Total words: 438)

v

Contents Declaration

i

Acknowledgements

iii

Abstract

iv

List of Figures

ix

List of Tables

xi

List of Abbreviations

xii

List of Symbols

xiv

1 Introduction

1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Wireless Power and Information Transmission . . . . . . . . . . .

3

1.3 Beamforming Design . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4 Contributions of This Thesis . . . . . . . . . . . . . . . . . . . . .

6

1.4.1

Energy Harvesting Model . . . . . . . . . . . . . . . . . .

6

1.4.2

Beamforming Algorithms . . . . . . . . . . . . . . . . . . .

7

2 Nonlinear Energy Harvesting Model 2.1 Proposed Nonlinear Model . . . . . . . . . . . . . . . . . . . . . .

vi

10 11

2.2 Impact of Nonlinear Model on WPIT . . . . . . . . . . . . . . . .

15

2.2.1

Problem Formulation . . . . . . . . . . . . . . . . . . . . .

16

2.2.2

Optimal Solution for the Case of M = 1 . . . . . . . . . .

17

2.2.3

Iterative Algorithm for the Case of M ≥ 2 . . . . . . . . .

18

2.2.4

Performance Lower Bound . . . . . . . . . . . . . . . . . .

20

2.2.5

Simulation Results and Discussions . . . . . . . . . . . . .

21

2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.4.1

Proof of Equation (2.5) . . . . . . . . . . . . . . . . . . . .

24

2.4.2

Proof of Property 2.1 . . . . . . . . . . . . . . . . . . . . .

25

3 Beamforming Algorithm For WPIT With Mobile Charger 3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26 26

3.1.1

Charger Movement . . . . . . . . . . . . . . . . . . . . . .

26

3.1.2

Uplink Signal Model . . . . . . . . . . . . . . . . . . . . .

28

3.1.3

Downlink Signal Model . . . . . . . . . . . . . . . . . . . .

28

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.3 Optimal Solution for Fixed Relay Case . . . . . . . . . . . . . . .

31

3.3.1

Recasting F1 as a Feasibility Problem . . . . . . . . . . . .

32

3.3.2

Dimension Reduction . . . . . . . . . . . . . . . . . . . . .

34

3.3.3

Closed-Form Solution of F3 Given γ

. . . . . . . . . . . .

36

3.4 Iterative Algorithm For Mobile Relay Case . . . . . . . . . . . . .

39

3.5 Performance Upper Bound . . . . . . . . . . . . . . . . . . . . . .

46

3.6 Numerical Results and Discussions . . . . . . . . . . . . . . . . .

48

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.8.1

Proof of Property 3.2 . . . . . . . . . . . . . . . . . . . . .

53

3.8.2

Proof of Convexity for H(x, t) . . . . . . . . . . . . . . . .

54

3.8.3

Proof of Property 3.3 . . . . . . . . . . . . . . . . . . . . .

54

vii

3.8.4

Initialization for Algorithm 3.2 . . . . . . . . . . . . . . .

55

3.8.5

Proof of Equation (3.41) . . . . . . . . . . . . . . . . . . .

55

4 Beamforming Algorithm For WPIT With Massive MIMO

57

4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.3 Proposed First-Order Method for P4.1 . . . . . . . . . . . . . . .

62

4.3.1

Solving the Uplink Problem . . . . . . . . . . . . . . . . .

64

4.3.2

Solving the Downlink Problem . . . . . . . . . . . . . . . .

66

4.3.3

Overall Algorithm and Complexity Analysis . . . . . . . .

73

4.4 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . .

75

4.4.1

Hybrid Beamforming . . . . . . . . . . . . . . . . . . . . .

75

4.4.2

Time Allocation . . . . . . . . . . . . . . . . . . . . . . . .

77

4.5 Simulation Results and Discussions . . . . . . . . . . . . . . . . .

77

4.5.1

Convergence Behavior . . . . . . . . . . . . . . . . . . . .

78

4.5.2

Performance and Running Time . . . . . . . . . . . . . . .

81

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.7.1

Proof of Property 4.2 . . . . . . . . . . . . . . . . . . . . .

84

4.7.2

Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . .

85

5 Conclusions and Future Research

89

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Bibliography

92

List of Publications

104

viii

List of Figures 1.1 Forbes IoT report in 2017 [4]. . . . . . . . . . . . . . . . . . . . .

1

1.2 The first RF energy harvester in history. . . . . . . . . . . . . . .

2

1.3 Illustration of two types of WPIT (a) type-I WPIT systems; (b) type-II WPIT systems. . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4 Illustration of beamforming design in WPIT. . . . . . . . . . . . .

5

2.1 Circuit diagram of a typical energy harvester. . . . . . . . . . . .

11

2.2 The prototype experiment at the University of Hong Kong. The transmitter is Powercast TX91501-3W-ID. The energy harvester is implemented by using Powercast P2110. . . . . . . . . . . . . . .

12

2.3 Comparison between the experimental data and the nonlinear energy harvesting model. The parameters in the model are given by τ = 330, ν = 0.62, Pmax = 0.0049 W, and P0 = 0.000064 W. . . . .

14

2.4 Comparison between the Avago HSMS-286x data and the nonlinear energy harvesting model. The parameters in the model are given as follows. (i) Vbr = 3V: τ = 411, ν = 2.2, Pmax = 0.0082 W, and P0 = 0.00008 W; (ii) Vbr = 7V: τ = 116, ν = 2.3, Pmax = 0.0375 W, and P0 = 0.00008 W; (iii) Vbr = 12V: τ = 47, ν = 2.4, Pmax = 0.11 W, and P0 = 0.00008 W. . . . . . . . . . . . . . .

15

2.5 Power consumption versus energy harvesting requirement in the case of (a) M = 1 and K = L = 2; (b) M = 10 and K = L = 2. The vertical lines indicate the 95% confidence intervals. . . . . . . ix

23

3.1 System model of wirelessly powered TWRC with a mobile charger. 28 3.2 Illustration of

ξ1 A1 (ρ)

and

ξ2 . A2 (ρ)

. . . . . . . . . . . . . . . . . . . .

38

3.3 Illustration of Θnl and Λ. . . . . . . . . . . . . . . . . . . . . . . .

46

3.4 Two moving trajectories of relay with M = 4. . . . . . . . . . . .

50

3.5 Sum-rate versus number of iterations for Algorithm 3.2. . . . . . .

50

3.6 Data-rates of wirelessly powered TWRC for fixed relay and mobile relay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.7 Sum-rate versus available relay energy for fixed relay and mobile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.1 System model of WPIT network with large number of antennas. .

58

4.2 Illustration of acceleration. . . . . . . . . . . . . . . . . . . . . . .

72

relay.

4.3 User transmit powers versus the number of outer iterations in Algorithm 4.1 for the case of K = 20 and N = 256. . . . . . . . . .

79

4.4 Function value Ξ[0] of (4.18) versus number of inner loop iterations in Algorithm 4.2 when K = 20 and N = 256. . . . . . . . . . . . .

79

4.5 Transmit power at access point versus number of outer loop iterations in Algorithm 4.2 when K = 20 and N = 256.

. . . . . . . .

80

4.6 Total transmit power versus number of antennas for the case of K = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.7 Average execution time versus number of antennas for the case of K = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.8 Total transmit power versus number of users for the case of N = 256. 83

x

List of Tables 2.1 Output Power versus Input Power . . . . . . . . . . . . . . . . . .

13

4.1 Summary of Complexity . . . . . . . . . . . . . . . . . . . . . . .

74

xi

List of Abbreviations IoT

Internet of Things

RFID

Radio frequency identification

RF

Radio frequency

WPIT

Wireless power and information transmission

MIMO

Multiple-input multiple-output

DC

Difference of convex

CG

Conjugate gradient

QCQP

Quadratically constrained quadratic programming

KKT

Karush-Kuhn-Tucker

EH

Energy harvester

ID

Information decoder

SDP

Semi-definite programming

TWRC Two-way relay channel SNR

Signal-to-noise ratio

SINR

Signal-to-noise-plus-interference ratio

SDMA

Spatial division multiple access

TDMA

Time division multiple access

QoS

Quality-of-service

SOC

Second-order cone

AM

Alternating minimization

xii

SDR

Semi-definite relaxation

SLA

Successive linear approximation

APDG

Accelerated primal-dual gradient

ADMM Alternating direction method of multipliers TDD

Time division duplex

FDD

Frequency division duplex

xiii

List of Symbols x

Scalar

x

Vector

X

Matrix

X

Set

Tr (X)

Trace of a matrix X

XT

Transpose of a matrix X

XH

Hermitian of a matrix X

X−1

Inverse of a matrix X

Rank (X) Rank of a matrix X conj(·)

Conjugate

Re(·)

The real part of the argument

E( · )

Expectation of its argument

∥·∥

The l2 norm of the argument

exp( · )

Exponential function of a scalar

vec(X)

Vectorization of a matrix X

IN

N × N Identity matrix

1N

N × 1 Vector with all the elements being 1

∂

Partial derivative

∇f

Gradient of function f

[x]+

The function max(x, 0)

xiv

X≽0

X is symmetric (Hermitian) and positive semidefinite.

O(·)

Complexity order of arithmetic operations.

xv

Chapter 1 Introduction 1.1

Background

Internet-of-Things (IoT) is a future communication paradigm that connects billions of low-power devices [1]. In particular, capitalizing on massive connectivity, IoT is considered as a promising solution for smart cities, smart grids, smart logistics, and smart homes [2]. With such a wide range of commercial and industrial applications, IoT market is continuously growing [3], as evidenced from the Forbes report in Figure 1.1 that the number of global spending on IoT is

Global Spending ˄10 billions Euro˅

Forbes Report 2017 45 40 35 30 25 20 15 10 5 0

40

40

40

25 15 10

10

7

5

5

2015

12 3

2020

Figure 1.1: Forbes IoT report in 2017 [4].

1

12 2

12 5

expected to increase by a factor of 4 from 2015 to 2020 [4]. However, due to the limited size, vast volume, and sporadic nature of the IoT devices such as sensors [5,6] and radio frequency identification (RFID) tags [7], the powering of IoT devices is challenging [8]. For tackling the energy scarcity in IoT applications, wireless powering of devices is a viable solution. It involves no wire, no contact, fewer batteries, and represents a reliable energy supply (as opposed to the unpredictable nature of conventional energy-harvesting technologies such as solar, thermal, vibration) [9]. In particular, wireless power can be transmitted via both magnetic resonant coupling and radio frequency (RF) signals. While the first approach has become a reality with several commercially available products and standards [10], its transmission range is severely limited. For instance, the maximum distance of the Qi standard (which is the most popular standard of magnetic resonant coupling) from Wireless Power Consortium is only 1.6 inches (4 centimeters) [11], which is too small for IoT applications. Therefore, the second approach, namely the RF power transmission, which transmits energy from a distance via electromagnetic waves [12], is a unique solution for wireless powering of IoT devices. The concept of RF power transmission was first proposed by Nikola Tesla in

Figure 1.2: The first RF energy harvester in history.

2

1899, and was later implemented by William C. Brown in 1963 [12]. As shown in Figure 1.2, the first RF energy harvester in history could generate direct current power of 7 Watts and achieve a maximum conversion efficiency of 40%. Following the experiment of Brown, a variety of RF power transmission experiment was demonstrated: in 2013, the Washington University designed two IoT devices, which could harvest energy from the TV tower (located 10 kilometers away) and communicate with each other (with their distance being 0.45 meters) at the datarate of 1 kbps; in 2014, the University of Catania fabricated a battery-free energy harvesting sensor node based on 90 nm COMS technology, which can be applied to autonomous wireless sensor networks. All these works demonstrate the high potential of RF power transmission in IoT networks.

1.2

Wireless Power and Information Transmission

Wireless signal is however not limited to power transmission: RF signals can also carry information. In fact, the transmission of wireless information has become the foundation of our modern society in the past decades. In the future IoT networks, to make best use of the RF spectrum for the dual purpose of energizing and communicating, the transmission of power and information must be jointly treated, and this leads to wireless power and information transmission (WPIT) [13–33]. WPIT can be categorized into two types: (I) type-I WPIT involves no information transmitted from energy harvesters [13–21]; (II) type-II WPIT involves information transmitted from energy harvesters [22–33]. In particular, in type-I WPIT systems, power and information are transmitted from the same source, and WPIT can be completed in only one transmission phase [13–21]. This is the case shown in Figure 1.3a, in which the Powercast corporation demonstrated 3

(a)

(b)

Figure 1.3: Illustration of two types of WPIT (a) type-I WPIT systems; (b) type-II WPIT systems.

its WPIT system in New York city in 2017. It can be seen that the Powercast transmitter TX91501-3W-ID (the black device at the corner of the desk) can simultaneously charge a lot of low-power devices while broadcasting control signals at the frequency of 915 MHz. On the other hand, in type-II WPIT systems, power and information are transmitted from different sources, and WPIT requires two transmission phases [22–33]. In the first transmission phase, the RF source transmits a downlink 4

RF signal for powering the IoT devices. Utilizing the harvested energy from the downlink signal, the IoT devices transmit their information signals in the second phase. This is the case shown in Figure 1.3b, in which a tag reader collects information from a RFID tag. It can be seen that by adopting the WPIT technique, the RFID could completely get rid of batteries while still guaranteeing proper circuit operation and reliable wireless communication [7].

1.3

Beamforming Design

Unfortunately, it can be seen from Figures 1.3a and 1.3b that the maximum transmission distance supported by WPIT systems is short. This reveals the issue of low transmission efficiency in the current WPIT systems. In order to improve the transmission efficiency, a promising solution is the multiple-input multipleoutput (MIMO) technique with an appropriate beamforming design [15–20]. In particular, as shown in Figure 1.4, the beamforming design is to intelligently determine the complex coefficients {cn } (where the amplitude of cn represents the power being allocated to antenna n and the phase of cn represents the transmission direction of antenna n) so as to fully exploit the multi-path channels {hn,k } and to concentrate the RF energy to a narrow beam towards the target users. This h11 c1

c1 x

User 1

c2 x

User 2

Information Decoder

c2

x

cN x

«...

«...

cN

hNK

User K

Energy Harvester

«...

Figure 1.4: Illustration of beamforming design in WPIT.

5

technology has already demonstrated its effectiveness for wireless information transmission in the fourth and fifth generation mobile networks [34]. Furthermore, recent performance analysis and prototypes also show that beamforming could achieve efficient transmission of power [35]. In type-I WPIT systems, the beamforming design needs to balance the tradeoff between the data-rate received at information decoders and the harvested power at energy harvesters [15–20]. This is nontrivial since the sensitivity of energy harvesters is significantly lower than the sensitivity of information decoders [8]. Consequently, the RF power required at energy harvesters is orders of magnitude higher than that required at information decoders. For instance, the minimum RF energy to activate the energy harvesting circuit is from −30 dBm to −10 dBm, while the minimum RF energy to activate the information receiver could be smaller than −50 dBm. In type-II WPIT systems, the beamforming design becomes even more challenging. This is because the IoT device is operated under the harvest-thentransmit protocol and thus acts as a dual role: the power receiver and the information transmitter [22–27]. As a result, the uplink information transmission and the downlink power transmission are coupled in type-II WPIT systems, and the beamforming design needs to resolve this uplink-downlink coupling.

1.4 1.4.1

Contributions of This Thesis Energy Harvesting Model

No matter which type of WPIT systems is considered, the effectiveness of beamforming design depends on the accuracy of energy harvesting model. As the output harvested power Pout of the energy harvesting circuit is a monotonically increasing function of the input incident power Pin , exiting works mostly adopt a linear model Pout = ηPin [15–19] for beamforming design, where η is a constant 6

representing the energy harvesting efficiency. Unfortunately, since the energy harvester contains non-linear elements such as diodes [36–47], linear energy harvesting model cannot accurately reflect the operation of energy conversion process. This mismatch between model and circuit would significantly degrade the performance of WPIT beamforming design. In cases of low input power regime (e.g., Pin ≤ −10 dBm) and high input power regime (e.g., Pin ≥ 10 dBm), the beamforming design based on linear model might completely break down [45, 46]. In order to address the above problem, a practical nonlinear energy harvesting model, which matches the experimental data of energy harvesting products very well, is proposed in Chapter 2. To reveal the impact of nonlinear model on WPIT, the beamforming deisgns based on linear and nonlinear models are compared in a type-I WPIT system. It is shown that the beamforming design based on the proposed model significantly improves the WPIT transmission efficiency compared to traditional designs based on linear model.

1.4.2

Beamforming Algorithms

With the proposed nonlinear energy harvesting model, this thesis further contributes two beamforming algorithms for type-II WPIT systems in Chapter 3 and Chapter 4. Beamforming Algorithm For WPIT With Mobile Charger In Chapter 3, the beamforming design of a mobile charger is investigated. With such a scheme, the moving charger could vary its location at the cost of motion energy, but has the flexibility of being close to different energy harvesting users at different times [49–52]. However, as the charger is capable of moving, its beamforming design is dependent on the charger moving trajectory and the stopping time at each point along the trajectory. Furthermore, since the har7

vested energy at users can be stored in batteries, the wireless power transmission at previous locations might have a long-term impact on subsequent locations. Consequently, compared to the fixed charger case, the dimension of beamforming design is significantly increased and the resultant problem involves an additional nonlinear coupling between time allocation and beamformers. To address the above challenges, a convergence guaranteed iterative algorithm based on difference of convex (DC) programming is proposed, which achieves a data-rate close to the upper bound. Moreover, it is found that the proposed beamforming design under an appropriate moving trajectory significantly outperforms the fixed charger case. Beamforming Algorithm For WPIT With Massive MIMO In Chapter 4, the beamforming design under massive antenna arrays at transmitter is considered, since scaling up the number of antennas would facilitate the concentration of the energy beams towards target directions [53–56]. However, in large-scale settings, since the inverse of Hessian matrices becomes extremely time consuming, traditional algorithms such as DC programming cannot be used in practice if the number of antennas is in the range of hundreds or more [57–63]. While existing methods assuming infinite number of antennas and exploiting the law of large numbers could lead to simple beamforming designs [54–56], such beamforming designs [54–56] suffer from significant performance loss compared to DC programming when applied in scenarios with large but finite number of antennas. In view of the apparent research gap, Chapter 4 proposes a method that reduces the computation time by orders of magnitude compared to DC programming while still guaranteeing the same performance. In particular, the beamforming design problem in WPIT is first transformed into a two-stage optimization problem, representing the uplink and downlink problems, respectively. While the

8

uplink problem is a challenging multi-objective problem that usually has multiple Pareto solutions, it is shown that there exists a unique Parato solution and this solution can be obtained by simply computing a series of conjugate gradients (CG), with the iteration number of the proposed CG method no larger than the number of users. Based on the obtained solution of the uplink problem, the downlink problem can be transformed into a large-scale homogeneous quadratically constrained quadratic programming (QCQP) problem. By applying an accelerated first-order method to this problem, an iterative algorithm is proposed and it is proved to converge to a Karush-Kuhn-Tucker (KKT) solution, which is the best solution one can obtain in polynomial time for homogeneous QCQP problems. Furthermore, the iteration complexity of the proposed accelerated first-order method is proved to touch the lower bound derived for any first-order method. Therefore, the proposed method is among the fastest algorithms for solving large-scale homogeneous QCQP problems. Since both procedures for solving the uplink and downlink problems only require the computation of inner products between channel vectors, the total computational complexity for solving the two-stage problem is linear in terms of the number of antennas at access point. With the proposed algorithms for uplink and downlink problems capable of running in parallel for all the users, its computation time can be further reduced in practice. Finally, numerical results validate the low-complexity nature of the proposed method, and show that the optimization performance of the proposed scheme is equivalent to that of DC programming.

9

Chapter 2 Nonlinear Energy Harvesting Model In order to improve the transmission efficiency of WPIT systems, beamforming with MIMO is a promising technique. However, the beamforming design in WPIT systems is highly dependent on the accuracy of the adopted energy harvesting model. As the output harvested power Pout of the energy harvesting circuit is a monotonically increasing function of the input incident power Pin , a straightforward way is to adopt a linear model Pout = ηPin , where η is a constant representing the energy harvesting efficiency. Unfortunately, since the energy harvester contains nonlinear elements such as diodes, linear energy harvesting model cannot accurately reflect the actual energy conversion process. This mismatch between model and circuit would significantly degrade the performance of WPIT beamforming design. In cases of low input power regime (e.g., Pin ≤ −10 dBm) and high input power regime (e.g., Pin ≥ 10 dBm), the beamforming design based on linear model might completely break down. In order to address the above problem, this chapter would present a novel nonlinear model.

10

2.1

Proposed Nonlinear Model

As shown in Figure 2.1, an energy harvesting circuit contains eight basic elements: radio frequency source, antenna resistance RD , input filter, matching impedance, diodes, microstrip line, capacitor C, and load resistance RL . It can be seen from this architecture that the energy harvester is a nonlinear device. In order to determine the circuit model Pout = Θ(Pin ) of energy harvesters, we set up a prototype experiment at the University of Hong Kong. As shown in Figure 2.2, the experiment is based on the transmitter Powercast TX91501-3WID and the receiver Powercast P2110. The transmit power at TX91501-3W-ID is 3 W at 915 MHz, and the antenna gain at P2110 is 6 dBi. Based on the above prototype, we measure the voltages and currents using a multimeter with the transmitter-receiver distance ranging from 1m to 15m. By computing the input incident power and the output harvested power from the voltages and currents, we obtain the results listed in Table 2.1. It can be observed from Table 2.1 that Θ should satisfy the following properties. (i) Θ(Pin ) is a monotonically increasing function of Pin .

Impedance

Input Filter

Microstrip Line

Diodes

RD

C

RL

RF

Figure 2.1: Circuit diagram of a typical energy harvester.

11

Power Transmitter 8 meters

Energy Harvester

Figure 2.2: The prototype experiment at the University of Hong Kong. The transmitter is Powercast TX91501-3W-ID. The energy harvester is implemented by using Powercast P2110.

(ii) When Pin is smaller than the harvester’s sensitivity threshold P0 , we must have Pout = Θ(Pin ) = 0. (iii) As Pin increases, the energy harvesting efficiency Θ(Pin )/Pin would increase to a maximum value and then decrease. (iv) Θ(Pin ) ≤ Pmax for all Pin , where Pmax is the maximum harvested power when the energy harvesting circuit is saturated. Based on Property (i), a straightforward way is to adopt a linear model Θlinear (Pin ) = ηPin ,

(2.1)

where η is a constant representing the energy harvesting efficiency. However, it can be seen from Property (iii) that the efficiency η is not a constant. Furthermore, if Pin < P0 , we must have Pout = Θ(Pin ) = 0 according to Property (ii), 12

Table 2.1: Output Power versus Input Power

Distance (m)

Input power (mW) Output power (mW)

Efficiency

1

8.119

4.186

51.6%

2

2.030

1.097

54.1%

3

0.902

0.413

45.8%

4

0.507

0.238

46.9%

5

0.325

0.159

49%

6

0.226

0.107

47.7%

7

0.166

0.07

42.3%

8

0.127

0.045

35.4%

9

0.1

0.026

25.5%

10

0.081

0.011

12.9%

11

0.067

0.001

1.7%

12

0.056

0

0

13

0.048

0

0

14

0.041

0

0

15

0.036

0

0

which contradicts to Θlinear . Therefore, Θlinear cannot accurately represent the practical conversion process. To accurately represent the energy conversion process, a nonlinear model is needed. In particular, according to Property (i)-(iv), Θ should have an “S” shape, and to represent this shape, a logistic model is proposed in [39]. However, the model in [39] does not satisfy the sensitivity property (ii). In order to address the sensitivity issue, we modify the logistic model into the following form: [ ) ]+ ( Pmax 1 + exp(−τ P0 + ν) , (2.2) Θnl (Pin ) = −1 exp(−τ P0 + ν) 1 + exp(−τ Pin + ν) where the parameter P0 is the harvester’s sensitivity threshold defined in Prop13

10 5

Output Power (dBm)

0 -5 -10 -15 -20 -25

Experimental data from Powercast The nonlinear energy harvesting model The linear energy harvesting model The logistic model [39]

-30 -35 -40 -10

-5

0

5

10

Input Power (dBm)

Figure 2.3: Comparison between the experimental data and the nonlinear energy harvesting model. The parameters in the model are given by τ = 330, ν = 0.62, Pmax = 0.0049 W, and P0 = 0.000064 W.

erty (ii) and Pmax is the maximum harvested power defined in Property (iv). The parameters τ and ν are used to capture the nonlinear dynamics defined in Property (iii). It can be verified that (2.2) satisfies all the properties (i)-(iv). To evaluate the models Θlinear and Θnl , we fit the models to the experimental data in Table 2.1 based on least squares criterion. It can be seen from Figure 2.3 that with the choice of η = 0.29, the linear energy harvesting model in (2.1) cannot represent the experimental data. In addition, with the choice of aj = 600, bj = 0.0032, and Mj = 0.004927 W, the logistic model in [39] cannot represent the experimental data in the low input power regime. On the other hand, with the choice of τ = 274, ν = 0.29, Pmax = 0.004927 W, and P0 = 0.000064 W, the nonlinear energy harvesting model in (2.2) matches the experimental data very well. To further demonstrate the versatility of the model, we also fit the model in

14

30

Output Power (dBm)

20

10

0 Avago HSMS-286x at Vbr =12V

-10

Nonlinear model at V br =12V Avago HSMS-286x at Vbr =7V

-20

Nonlinear model at V br =7V Avago HSMS-286x at Vbr =3V

-30

Nonlinear model at V br =3V -10

-5

0

5

10

15

20

25

Input Power (dBm)

Figure 2.4: Comparison between the Avago HSMS-286x data and the nonlinear energy harvesting model. The parameters in the model are given as follows. (i) Vbr = 3V: τ = 411, ν = 2.2, Pmax = 0.0082 W, and P0 = 0.00008 W; (ii) Vbr = 7V: τ = 116, ν = 2.3, Pmax = 0.0375 W, and P0 = 0.00008 W; (iii) Vbr = 12V: τ = 47, ν = 2.4, Pmax = 0.11 W, and P0 = 0.00008 W.

(2.2) to the experimental data of Avago HSMS-286x diode, which is obtained from Figure 6b in [38]. It can be observed from Figure 2.4 that the nonlinear energy harvesting model matches the data of this circuit very well for all values of breakdown voltage Vbr .

2.2

Impact of Nonlinear Model on WPIT

To reveal the impact of the proposed model on the performance of WPIT, we consider a type-I WPIT system consisting of an access point equipped with N antennas, K single-antenna energy harvesters (EHs), and L single-antenna in-

15

formation decoders (IDs)1 . During the downlink transmission, the access point delivers power to EHs and information to IDs within time duration T . To model the time varying feature of wireless channels, each time duration is further divided into M slots, and the channels are independent in different slots. At the mth time slot (1 ≤ m ≤ M ), the access point transmits a symbol xm ∈ CN ×1 ( ) N ×N with covariance Xm = E xm xH and power Tr (Xm ). m ∈ C Based on the above system model, the received signal at the lth ID (1 ≤ H 1×N is the downlink l ≤ L) is given by rl,m = hH l,m xm + nl,m , where hl,m ∈ C

channel vector from the access point to the lth ID and nl,m is the noise at the lth ID. Based on the expression of rl,m , the data-rate achievable at the lth ID is ( ( ) 2) 2 2 log2 1 + Tr hl,m hH l,m Xm /σ , with σ = E (|nl,m | ) being the noise power. On the other hand, the received power at the k th EH (1 ≤ k ≤ K) is ( H ) ( ) H E |gk,m xm |2 = Tr gk,m gk,m Xm provided that the thermal noise at the k th H EH is negligible compared with that of xm , where gk,m ∈ C1×N is the downlink

channel vector from access point to the k th EH. Accordingly, the harvested power ( ( )) H is denoted by Θ Tr gk,m gk,m Xm , where Θ is the function representing the energy conversion process.

2.2.1

Problem Formulation

In WPIT systems, a main task is to guarantee sufficient harvested power at EHs and provide reliable communications for the IDs. Having the energy harvesting and data-rate requirements satisfied, it is then crucial to minimize the power consumption at the access point, and an optimization problem can be formulated as: P2.1 1

min

{Xm ≽0}

M 1 ∑ Tr(Xm ) M m=1

The energy harvesters and information decoders are separately located as in [15] and [39].

Furthermore, the derivations in this section can be readily extended to multi-antenna users.

16

s.t.

M )) 1 ∑ ( ( H Θ Tr gk,m gk,m Xm ≥ γk , ∀k = 1, ..., K, (2.3a) M m=1 ( ) M ( ) 1 1 ∑ H log 1 + 2 Tr hl,m hl,m Xm ≥ θl , ∀l = 1, ..., L. (2.3b) M m=1 2 σ

Before we solve P2.1, we need to specify the energy conversion function Θ, which unfortunately does not have a closed-form expression and can only be obtained numerically, e.g., using look-up tables [38]. If Θ is approximated as Θlinear in (2.1), the constraint defined in (2.3a) becomes M ( ) 1 ∑ H ηTr gk,m gk,m Xm ≥ γ k . M m=1

Based on the above procedure, P2.1 becomes a convex problem which can be solved by CVX Mosek, a popular software package for solving convex problems [64]. Since {Xm } are involved in M semi-definite constraints of dimension N ×N , K linear constraints, and M L linear terms, the complexity for solving P1 under ( )] [√ the linear model is O M N (K + M L) (K + M L)2 + (K + M L)M N 2 + M N 3 [65]. On the other hand, if Θ is approximated as Θnl in (2.2), fractional constraints are involved in (2.3a). In such a case, traditional methods like convex optimization in [15] and sum-of-ratios programming in [39] are not applicable. To tackle this challenge, in the following, we first provide an optimal solution in the case of M = 1, and then provide an iterative algorithm for the general case of M ≥ 2.

2.2.2

Optimal Solution for the Case of M = 1

In the case of M = 1, the subscript m can be dropped to simplify the notation ( ( )) ≥ γk . From this equivalent and constraint (2.3a) reduces to Θnl Tr gk gkH X constraint, we consider two cases as follows. (i) If γk ≥ Pmax , the constraint (2.3a) would always be infeasible, since the function value of Θnl is upper bounded by Pmax . In such a case, we simply write (2.3a) as Tr(gk gkH X) → ∞. 17

(ii) If 0 < γk < Pmax , the operator [·]+ in (2.3a) can be dropped. By further solving for Tr(gk gkH X), we get Tr(gk gkH X)

) ν 1 ( 1 + e−τ P0 +ν ≥ − ln −1 . −1 e−τ P0 +ν γ τ τ 1 + Pmax k {z } | :=A(γk )

Combining (i)-(ii), problem P2.1 can be reformulated as P2.2 min

{X≽0}

Tr(X)

s.t. Tr(gk gkH X) ≥ Θ†nl (γk ), ∀k, ( θ ) 2 l Tr(hl hH l X) ≥ 2 − 1 σ , ∀l, where Θ†nl (γk ) = +∞ if γk ≥ Pmax and Θ†nl (γk ) = A(γk ) if 0 < γk < Pmax , and the second constraint of P2.2 is obtained from (2.3b). Now P2.2 is a semi-definite programming (SDP) problem and can be optimally solved using CVX with a [√ ( )] complexity of O N (K + L)3 + (K + L)2 N 2 + (K + L)3 N 3 [65].

2.2.3

Iterative Algorithm for the Case of M ≥ 2

If M ≥ 2, solving P1 is difficult due to the sum of fractional functions. However, by observing that the denominator of Θnl involves exponential functions, we may H introduce slack variables {λ } such that λ = eτ Tr(gk,m gk,m Xm ) for all k, m. k,m

k,m

With the above slack variables, the constraint (2.3a) can be re-written as [ ( ) ]+ M 1 ∑ Pmax 1 + e−τ P0 +ν −1 ≥ γk , (2.4) M m=1 e−τ P0 +ν 1 + eν /λk,m | {z } :=Φ(λk,m )

and Φ(λk,m ) can be reformulated as Φ(λk,m ) =

Pmax (1 + e−τ P0 +ν ) Pmax Pmax (1 + e−τ P0 +ν ) eν − − · . e−τ P0 +ν e−τ P0 +ν e−τ P0 +ν λk,m + eν eν λk,m +eν

ν

is convex, − λk,me +eν is concave. As a consequence, Φ(λk,m ) H is concave. On the other hand, we can relax λk,m = eτ Tr(gk,m gk,m Xm ) into Since the term

λk,m ≤ eτ Tr(gk,m gk,m Xm ) . H

18

(2.5)

As proved in Appendix 2.4.1, the solution of the relaxed problem is optimal to the original problem, and thus the relaxation can be safely applied. After the above procedure, the remaining obstacle in (2.4) is the operators [·]+ . Fortunately, since [x]+ = max(x, 0) ≥ x, dropping the operators [·]+ would make the feasible set of (2.4) smaller and the solution of such a problem would be feasible for P2.1. To this end, P2.1 becomes P2.3

min

{Xm ≽0,λk,m ≥1}

s.t.

M 1 ∑ Tr(Xm ) M m=1 H τ Tr(gk,m gk,m Xm ) ≥ ln(λk,m ), ∀k, m,

(2.6a)

M 1 ∑ Φ(λk,m ) ≥ γk , ∀k, M m=1 ( ) M Tr(hl,m hH X ) 1 ∑ m l,m log 1 + ≥ θl , ∀l. M m=1 2 σ2

(2.6b)

(2.6c)

where the first constraint is obtained from (2.5). Now the only nonconvex part in P2.3 is Ξ(λk,m ) := ln(λk,m ) in (2.6a). But since ln(λk,m ) is concave, we can apply the inner approximation method [57–63] to replace Ξ(λk,m ) with its first-order approximation around a feasible point. In particular, given any feasible point {λ⋆k,m } of P2.3, we define a surrogate function ˜ as Ξ ˜ k,m |λ⋆ ) = ln(λ⋆ ) + λk,m − 1. Ξ(λ k,m k,m λ⋆k,m

(2.7)

˜ k,m |λ⋆ ) ≥ Due to the concave property of Ξ(λk,m ), we immediately have Ξ(λ k,m ˜ expanded around {λ⋆ }, Ξ(λk,m ). Therefore, if we replace the function Ξ by Ξ k,m the solution of the surrogate problem is also feasible for P2.3. By treating the obtained solution as another feasible point and continue to construct the next round surrogate functions, we can improve the solution iteratively. In particu[n]

[n]

lar, assuming that the solution at the nth iteration is given by {Xm , λk,m }, the

19

following problem is considered at the (n + 1)th iteration: P2.3[n + 1]

min

{Xm ≽0,λk,m ≥1}

s.t.

M 1 ∑ Tr(Xm ) M m=1 H ˜ k,m |λ[n] ), ∀k, m, τ Tr(gk,m gk,m Xm ) ≥ Ξ(λ k,m

(2.6b) − (2.6c). Problem P2.3[n + 1] is a convex problem because (i) the first line of constraints are linear; (ii) (2.6b) is convex due to Φ being concave; (iii) (2.6c) is convex due to the logarithm function being concave. Therefore, P2.3[n + 1] can be optimally { } solved via existing software. Denoting its optimal solution as X∗m , λ∗k,m , then { } [n+1] [n+1] ∗ ∗ we can set Xm = Xm , λk,m = λk,m , and the process repeats with solving the problem P2.3[n + 2]. The entire procedure is summarized as Algorithm 2.1 and the following property (proved in Appendix 2.4.2) can be established. Property 2.1 Algorithm 2.1 converges to a Karush-Kuhn-Tucker solution of P2.3. Property 2.1 indicates that the converged point {X⋄m } generated by Algorithm 2.1 is only a KKT solution of P2.3. As a result, the proposed Algorithm 2.1 is a suboptimal method for solving P2.3 and P2.1. However, as shown later in the simulations, the performance of Algorithm 2.1 is very close to the lower bound of P2.1, showing that Algorithm 2.1 provides a good solution in practice. In terms of computational complexity, Algorithm 2.1 requires a complexity of ( )] [ √ O Q M N (K + M L) (K + M L)2 + (K + M L)M N 2 + M N 3 , where Q is the number of iterations for the algorithm to converge.

2.2.4

Performance Lower Bound

To evaluate the optimization performance brought by adopting Θlinear and Θnl , we need a performance lower bound for P2.1. Accordingly, we propose to use a piecewise linear function min(βx, Pmax ) to bound Θ from above. To choose β such

20

Algorithm 2.1: Iterative Algorithm for M ≥ 2. { } [0] [0] 1: Initialize a feasible Xm , λk,m of P2.3 and set n = 0. 2: Repeat 3:

} { [n+1] [n+1] Solve P2.3[n + 1] and update Xm , λk,m .

4:

Set n = n + 1.

5: Until convergence. The converged point is

{ ⋄ ⋄ } Xm , λk,m .

that min(βx, Pmax ) is as tight as possible, the following problem is considered: min β β

s.t. min(βx, Pmax ) ≥ Θ(x), ∀x ≥ 0.

(2.8)

Due to Pout ≥ 0 and Pout ≤ Pin , β is bounded within 0 ≤ β ≤ 1. Moreover, it can be seen that the above problem only has one scalar variable β. Therefore, a bisection search algorithm can be applied within the interval [0, 1], and ( ( )) H the optimal solution is denoted as β ∗ . Replacing Θ Tr gk,m gk,m Xm with ( ∗ ) H min β gk,m gk,m Xm , Pmax in P2.1, the modified problem is convex and can be optimally solved by CVX Mosek.

2.2.5

Simulation Results and Discussions

This section provides simulation results to compare the linear and nonlinear energy harvesting models. In particular, the number of antennas is set to N = 16 and the noise powers are σ 2 = −50 dBm (corresponding to power spectral density −100 dBm/Hz with 100 kHz bandwidth). The data-rate requirements are set to θ1 = · · · = θL = 5 in bps/Hz, and the look-up table function Θ is obtained from Table 2.1. Based on Θ, the parameters in the nonlinear model are obtained by fitting the experimental data of Θ to the model, and they are given by τ = 274, ν = 0.29, Pmax = 4.927 mW, and P0 = 0.064 mW. Since the received powers at EHs are not known a prior, the energy harvesting efficiency η in the linear model is also not known, and in this thesis we choose η ∈ {0.3, 0.5, 0.7}. 21

Furthermore, using the method in Section 2.3.3, we can compute the lower bound parameter β ∗ = 0.5518. Notice that a feasible solution under the model Θlinear or Θnl may not be feasible under the real conversion process Θ [38]. As a result, the obtained solutions need to be scaled such that all the EH requirements are satisfied under the function Θ. On the other hand, the path-loss model ρk = ρ0 · ( ddu0 )−α is adopted, where dk is the distance from the k th EH to the access point, ρ0 = 10−3 , d0 = 1 m, and α is the path-loss exponent set to 2.7 [33]. The distances are set to dk ∼ U(1, 5) in the unit of meters for EHs, where U denotes the uniform distribution. Based on the path-loss model, gk,m is generated according to CN (0, ρk I). Following a similar procedure and by setting the distances as U(10, 50) for IDs, channels {hl,m } can be generated. Each point in the figures is obtained by averaging over 100 simulation runs, with independent channels between consecutive runs. With the energy harvesting and path-loss models, we first consider the case of M = 1 with K = L = 2. It can be observed from Figure 2.5a that with η properly chosen for the linear model, the power consumptions based on linear and nonlinear models are similar. This is because the received powers at EHs are stable and the linear model can be locally accurate. However, since the appropriate η may vary under different system setups, we need to try different values of η for the linear model and this searching procedure would increase the complexity of beamforming optimization. In contrast, the beamforming designs under the nonlinear model can automatically adjust the efficiency and do not require such a searching procedure. Next, we consider the case of M = 10 with K = L = 2. It can be observed from Figure 2.5b that the nonlinear model achieves a transmit power close to the lower bound and brings significant performance gain compared to the linear model. This is because the received powers at EHs vary in a wide range during different time slots and a constant η in the linear model cannot represent the

22

50

Power Consumption (dBm)

48

Linear model η=0.3 Linear model η=0.5 Linear model η=0.7 Nonlinear model Lower bound

46

44

42

40

38 0.6

0.65

0.7

0.75

0.8

Energy Harvesting Requirement (mW)

(a)

Power Consumption (dBm)

55

50

Linear model η=0.3 Linear model η=0.5 Linear model η=0.7 Nonlinear model Lower bound

45

40

35 0.6

0.65

0.7

0.75

0.8

Energy Harvesting Requirement (mW)

(b)

Figure 2.5: Power consumption versus energy harvesting requirement in the case of (a) M = 1 and K = L = 2; (b) M = 10 and K = L = 2. The vertical lines indicate the 95% confidence intervals.

23

time-varying energy harvesting efficiency. Moreover, it can be seen from Figure 2.5b that the variances of transmit powers based on nonlinear model are significantly smaller, which indicates that the nonlinear model leads to more stable performance in time-varying channels. Therefore, a nonlinear model is necessary when the fluctuation of received power is significant.

2.3

Conclusions

This chapter proposed a nonlinear energy harvesting model. It was shown that the proposed model matches the experimental data of energy harvesting circuits very well. Based on the nonlinear model, we further studied the beamforming design problem in a type-I WPIT system, and proposed efficient optimization algorithms under both linear and nonlinear models. Adopting the developed algorithms, it was found that the nonlinear model results in a significantly lower power consumption than the linear model if the received powers vary in a wide range.

2.4 2.4.1

Appendix Proof of Equation (2.5)

To prove that the relaxation would not change the solution, we first compute ∇Φ(λk,m ) =

Pmax (1 + e−τ P0 +ν ) eν · ≥ 0. e−τ P0 +ν (λk,m + eν )2

(2.9)

It can be seen from the above equation that Φ(λk,m ) is a monotonically increasing function. Therefore, we can always increase λk,m to activate the constraint (2.5) without violating the constraint (2.4). This indicates that there always exist an optimal λ∗k,m of the relaxed problem that activates the constraint (2.5), which completes the proof. 24

2.4.2

Proof of Property 2.1

To prove the property, we first compute ˜ ⋄ |λ⋄ ) = ln(λ⋄ ) = Ξ(λ⋄ ), Ξ(λ k,m k,m k,m k,m ˜ k,m |λ⋄ ) ∂ Ξ(λ 1 ∂Ξ(λk,m ) k,m = = |λk,m =λ⋄k,m . ⋄ ∂λk,m λk,m ∂λk,m λk,m =λ⋄k,m

(2.10a) (2.10b)

Using the above results and based on [57, Theorem 1], Algorithm 2.1 must converge to a KKT solution of P2.3.

25

Chapter 3 Beamforming Algorithm For WPIT With Mobile Charger With the nonlinear energy harvesting model proposed in Chapter 2, the beamforming algorithm of a mobile charger is investigated in this chapter. In particular, by endowing the charger with mobility, the distances between the charger and users can be varied, thus providing a potential solution to combat path-loss at the cost of motion energy. However, as the charger is capable of moving, its beamforming design is dependent on the charger moving trajectory and the stopping time at each point along the trajectory, which leads to a more complicated situation than the fixed charger case. To tackle the above challenge, this chapter proposes a convergence guaranteed iterative algorithm based on DC programming to obtain efficient beamforming designs at all the stopping points.

3.1 3.1.1

System Model Charger Movement

We consider a two-way relay channel (TWRC) consisting of a mobile charger (i.e., the relay) with N antennas, and 2 single-antenna users. As shown in Figure 26

3.1, the two users intend to exchange messages with each other through the relay while the relay is moving along a pre-defined trajectory within a time duration T (for the special case of fixed relay, the length of trajectory is simply zero). This type of system can be found in various IoT applications, where low-cost sensors need to exchange data to jointly complete tasks such as localization and anomaly detection. In particular, from the starting point m = 1, the relay stops for a duration 2t1 and then it moves to point m = 2, and stops for a duration 2t2 . The relay keeps on moving and stopping along the trajectory until it reaches the destination m = M . Based on such a model, the moving time sm from the mth to the (m+1)th point is sm = Dm /v with Dm being the distance between the two points and v being the constant velocity. As the relay has to move from point m = 1 to point ∑ m = M within the duration T , we have M m=1 (2tm + sm ) = T , where sM = 0 due to the M th point being the destination. Furthermore, since the total motion energy EG of the relay is proportional to the total motion time [49–51], it can be expressed in the form of EG = (λ1 + λ2 v)

M ∑

sm ,

(3.1)

m=1

where λ1 and λ2 are parameters of the model (e.g., for a Pioneer 3DX robot, λ1 = 0.29 and λ2 = 7.4 [50, Sec. IV-C]). At each stopping point, lattice code based compute-and-forward network coding is used to accomplish the two-way communication, since the lattice-based twoway relaying achieves a larger data-rate than the amplify-and-forward scheme, at the expense of slightly higher complexity [72, 73]. Furthermore, the system is assumed to operate in a half-duplex mode, i.e., the uplink and downlink each takes tm for transmission1 . Below, we give the details of the uplink and downlink signal models when the relay is at the mth stopping point. 1

The derivations in this paper can be extended to the full-duplex case by taking the self-

interference cancelation into account.

27

Uplink Signal

pC for circuit consumption

Energy

Downlink Signal

h1,1

q1,m for uplink transmission Energy Harvester Information Decoder

E1,m from battery

Trajectory

Zoom in

N Antennas «...

g2,1

g1,1 h2,m

h1,m

Power splitter

User 1 Received signal

h2,1

g1,m h1,M

Data

Location m

User 2

g2,m

h2,M g2,M

g1,M Location M

Figure 3.1: System model of wirelessly powered TWRC with a mobile charger.

3.1.2

Uplink Signal Model

For the uplink transmission, user i transmits a vector xi,m ∈ CLm ×1 to the relay, with i ∈ {1, 2}, m ∈ {1, ..., M }, and Lm being the number of symbols. The xi,m is generated from the pre-determined lattice code (details documented in [33, Appendix A]) and the user transmit power is qi,m =

1 E[||xi,m ||2 ]. tm

Then,

the received signal Ym ∈ CN ×Lm at the relay is given by Ym = h1,m xT1,m + h2,m xT2,m + Nm , where hi,m ∈ CN ×1 denotes the uplink channel vector2 of user i, and Nm ∈ CN ×Lm is the Gaussian noise with E[vec(Nm )vec(Nm )H ] = σr2 IN Lm . H Applying a receive beamformer wm (with ||wm || = 1) to Ym , the uplink signal/ H to-noise ratio (SNR) of user i is qi,m |wm hi,m |2 σr2 . Using the uplink SNR and the UL results from [72, Theorem 3], the uplink achievable rate Ri,m from user i to the

relay can be computed to be [ ( UL = log2 Ri,m

3.1.3

H H hi,m |2 qi,m |wm hi,m |2 qi,m |wm + ∑2 2 H σr2 j=1 qj,m |wm hj,m |

) ]+ .

(3.2)

Downlink Signal Model

H Ym to generate a lattice symFor the downlink transmission, the relay uses wm

bol sm ∈ CLm ×1 with power pm = 2

1 ||sm ||2 tm

(the lattice decoding and encod-

The channels can be pre-determined according to [52].

28

ing procedure at relay can be found in [33, Appendix A]). After the above procedure, the relay transmits sm through the transmit beamforming vector vm ∈ CN ×1 with ||vm || = 1. Therefore, the received signal rTi,m ∈ C1×Lm at H H user i is rTi,m = gi,m vm sTm + nTi,m , where gi,m ∈ C1×N is the downlink channel

vector from the relay to user i, and nTi,m ∈ C1×Lm is the Gaussian noise at user 2 i with E[ni,m nH i,m ] = σu ILm . To implement WPIT, the received signal at user

i in the downlink is further splitted into two branches, one for the information decoder and the other for the energy harvester. At the information decoder side, the signal is given by e rTi,m =

√ H βi,m gi,m vm sTm +

√ βi,m nTi,m + zTi,m , where 0 ≤ βi,m ≤ 1 is the power splitting factor3 , and zTi,m ∈ C1×Lm is Gaussian noise introduced by the power splitter, with E[zi,m zH i,m ] = σz2 ILm . Based on the expression of e rTi,m , the downlink SNR for user i is / H βi,m pm |gi,m vm |2 (σu2 βi,m + σz2 ), DL and the downlink achievable rate Ri,m at user i is expressed as ) ( H 2 β p |g v | i,m m m i,m DL . Ri,m = log2 1 + σu2 βi,m + σz2

(3.3)

(3.4)

On the other hand, at the energy harvester of user i, the input power can be expressed as (1 − βi,m )E[||ri,m ||2 ]/tm . Based on the expression of ri,m , it can be H further expressed as (1 − βi,m )pm |gi,m vm |2 .

3.2

Problem Formulation

In TWRC, the beamforming design needs to maximize the data-rate pair (R1 , R2 ) supported by the system, where Ri denotes the average data-rate from user i to 3

The considered framework also includes the special case where only one user needs to

harvest energy. For example, one user might be a local access point while the other user is a low-cost sensor. In this case, we simply set βi,m = 1 for the local access point.

29

its paired user. Since the end-to-end transmission consists of uplink and downlink phases, Ri needs to satisfy [72, Theorem 1] : M ( UL DL ) 1 ∑ Ri ≤ tm min Ri,m , R3−i,m , ∀i = 1, 2, T m=1

(3.5)

where the index 3−i is used to represent the paired user of user i due to i ∈ {1, 2}. Having the data-rate constraints satisfied, the users and relay also need to meet their energy requirements. Specifically, as the user transmit power qi,m is harvested from the previous downlink wireless signal, we must have m ) ( ∑ H Ei + tl Θnl (1 − βi,l )pl |gi,l vl |2 ≥ l=1

m [ ] ∑ tl qi,l + (2tl + sl )pc , ∀i = 1, 2, m = 1, ..., M,

(3.6)

l=1

where Ei is the initial local energy at user i, and pc is the users’ circuit power consumption. On the other hand, since the energy consumption at relay includes motion energy EG and transmission energy, we must have EG + Ec +

M ∑

tm pm ≤ Er ,

(3.7)

m=1

where Er is the energy available for the relay to use and Ec is the circuit energy consumption at relay. Finally, the variables {tm , vm , wm , pm , qi,m , βi,m } need to satisfy the following basic constraints M ∑

(2tm + sm ) = T,

m=1

tm ≥ 0, ||vm || = ||wm || = 1, pm ≥ 0, m = 1, ..., M, qi,m ≥ 0, βi,m ∈ [0, 1], i = 1, 2, m = 1, ..., M.

(3.8)

Under the constraints (3.5)-(3.8), the beamforming design problem that maximizes the data-rates (R1 , R2 ) subject to the energy budget constraints can be written as: } { max (R1 , R2 ) ∈ R2+ : (3.5) − (3.8) . 30

For the above multi-objective problem, we need to find all the Pareto optimal solutions that maximize the data-rate pair (R1 , R2 ). To this end, the vector (R1 , R2 ) can be transformed into (µ1 Rsum , µ2 Rsum ), where Rsum := R1 + R2 and µi := Ri /Rsum satisfying µ1 + µ2 = 1 [71, Sec. III-B]. After this transformation, ∗ ∗ ∗ a point on the Pareto front can be obtained as (µ1 Rsum , µ2 Rsum ), where Rsum is

the maximum sum-rate by solving the following problem with a given µ1 : P3.1 :

max { [

Rsum ,{tm ,vm ,wm ,pm ,qi,m ,βi,m }

Rsum

( )]+ M H H qi,m |wm hi,m |2 1 ∑ qi,m |wm hi,m |2 min tm log2 ∑2 s.t. + , H 2 T m=1 σr2 j=1 qj,m |wm hj,m | ( )} H β3−i,m pm |g3−i,m vm |2 tm log2 1 + ≥ µi Rsum , ∀i = 1, 2, (3.9a) σu2 β3−i,m + σz2 m [ m ( ) ∑ ∑ H tl qi,l tl Θnl (1 − βi,l )pl |gi,l vl |2 ≥ Ei + l=1

l=1

]

+ (2tl + sl )pc , ∀i = 1, 2, m = 1, ..., M, EG + Ec +

M ∑ m=1

tm pm ≤ Er ,

M ∑

(2tm + sm ) = T,

(3.9b) (3.9c)

m=1

tm ≥ 0, pm ≥ 0, ||vm || = ||wm || = 1, ∀m = 1, ..., M,

(3.9d)

βi,m ∈ [0, 1], qi,m ≥ 0, ∀i = 1, 2, m = 1, ..., M.

(3.9e)

Then by varying µ1 within the interval (0, 1) and solving P3.1 repeatedly, we can obtain all the Pareto solutions of the multi-objective problem.

3.3

Optimal Solution for Fixed Relay Case

To solve P3.1, we first consider its special case of M = 1, i.e., wirelessly powered TWRC with a fixed relay. This special case is very important for subsequent comparison with the general case of mobile relay. In particular, since the trajectory length is zero when M = 1, we have s1 = 0 ∑M and EG = 0, and the constraint m=1 (2tm + sm ) = T reduces to 2t1 = T . 31

Furthermore, due to M = 1, the subscript m can be dropped to simplify the notation. Therefore, problem P3.1 becomes F1 :

max (

Rsum ,v,w,p,{qi ,βi }

[

Rsum

) ]+ qi |wH hi |2 qi |wH hi |2 s.t. log2 ∑2 + ≥ 2µi Rsum , ∀i, H 2 σr2 j=1 qj |w hj | ) ( βi p|giH v|2 ≥ 2µ3−i Rsum , ∀i, log2 1 + 2 σu βi + σz2 ( ) 2Ei + Θnl (1 − βi )p|giH v|2 ≥ qi + 2pc , ∀i, T ] [ 2Er − 2Ec , qi ≥ 0, βi ∈ [0, 1], ∀i, p ∈ 0, T ||v|| = 1, ||w|| = 1.

3.3.1

Recasting F1 as a Feasibility Problem

∗ Given a fixed µ1 (and thus fixed µ2 ), a simple way to determine Rsum in F1 is to

increase Rsum from 0 until the problem becomes infeasible, where the feasibility of F1 with a fixed Rsum is discussed in the next paragraph. On the other hand, it can be seen that a smaller Rsum would always loosen the first two constraints ′ of F1, leading to a larger feasible set. Therefore, if F1 with a fixed Rsum = Rsum ′ is infeasible, then F1 with Rsum ≥ Rsum would be infeasible. This indicates that ∗ a more efficient way to search for Rsum is to use bisection algorithm. Specifically,

given an upper bound for the searching interval, say κmax , and a lower bound κmin , the trial point is set to κ = (κmax + κmin )/2. If F1 with Rsum = κ is feasible, the lower bound is updated as κmin = κ; otherwise, the upper bound is updated as κmax = κ. The process is repeated until |κmax − κ| < ϵ, where ϵ is a small positive constant to control the accuracy. As Rsum ≥ 0, an initial κmin can be chosen as 0. On the other hand, a valid initial κmax can be found as follows. From

32

the second line of constraints of F1, we have [ ] ( ) ( ) 1 β1 p|g1H v|2 β2 p|g2H v|2 Rsum ≤ min log2 1 + 2 /µ2 , log2 1 + 2 /µ1 . 2 σu β1 + σz2 σu β2 + σz2

(3.10)

Using the constraints in the last line of F1, it can be observed that log2 (1 + βi p|giH v|2 2 β +σ 2 ) σu i z

is maximized when v = gi /||gi ||, p = (2Er − 2Ec )/T , and βi = 1.

Therefore, a valid upper bound of Rsum is given by [ ( ) 1 (2Er − 2Ec )||g1 ||2 Rsum ≤ min log2 1 + /µ2 , 2 T (σu2 + σz2 ) ] ( ) (2Er − 2Ec )||g2 ||2 log2 1 + /µ1 . T (σu2 + σz2 )

(3.11)

With the obtained upper bound, we are now ready to apply the bisection ∗ algorithm to find Rsum in F1. An efficient way to provide a feasibility check of

F1 given Rsum = κ is to first minimize the transmit power p via the following problem F2 :

min

v,w,p≥0,{qi ≥0,βi }

p

qi |wH hi |2 qi |wH hi |2 + ≥ 22µi κ , ∀i, s.t. ∑2 2 H 2 σr j=1 qj |w hj | βi p|giH v|2 ≥ 22µ3−i κ , ∀i, σu2 βi + σz2 ( ) 2E i H 2 ≥ qi + 2pc , ∀i, Θnl (1 − βi )p|gi v| + T 1+

βi ∈ [0, 1], ∀i,

||v|| = 1, ||w|| = 1,

(3.12a) (3.12b) (3.12c) (3.12d)

and then check whether the optimal p∗ satisfies p∗ ≤ (2Er − 2Ec )/T . If so, problem F1 with Rsum = κ is feasible; otherwise, the transmit power or energy at relay cannot support sum-rate κ and F1 with Rsum = κ is infeasible. Notice that (3.12a) comes from the first constraint of F1 with the operator [·]+ dropped. However, dropping the operator would not change the constraint due to 22µi κ ≥ 1. On the other hand, the constraint (3.12b) comes from the second constraint of F1. 33

3.3.2

Dimension Reduction

Since F2 is a high dimensional problem, we will first reduce its dimension based on the following procedure. More specifically, applying [33, Property 1] to the first constraint (3.12a) and defining αi = 22µi κ − 22µi κ /(22µ1 κ + 22µ2 κ ), it can be shown that (3.12a) could be replaced by qi = αi σr2 /|wH hi |2 without changing the optimal objective value of F2. Putting this result into (3.12c), constraint (3.12c) is equivalent to ( ) Θnl (1 − βi )|giH v|2 p ≥

αi σr2 2Ei + 2pc − , ∀i. H 2 |w hi | T

(3.13)

On the other hand, constraint (3.12b) can be reformulated as βi |giH v|2 p ≥ θi (σu2 βi + σz2 ), ∀i,

(3.14)

with θi = 22µ3−i κ − 1. Based on the above manipulations, problem F2 is equivalently transformed into min

p,v,w,{βi }

p

(3.15)

s.t. (3.13), (3.14), βi ∈ [0, 1], ∀i,

||v|| = 1, ||w|| = 1.

To reduce the dimension of problem (3.15), we apply [33, Property 2] to obtain the optimal w∗ ∈ span{h1 , h2 }. Moreover, due to property (i) of Θnl (x) and by using [33, Property 2], the optimal v∗ ∈ span{g1 , g2 }. Therefore, the Schmidt orthogonal basis for v∗ is given by g1 and z = g2 − g1H g2 /||g1 ||2 · g1 . Using this basis and the norm constraint of v, we can express the transmit beamformer v as √ g1 z √ + 1 − ρ · exp(jϕ2 ) , (3.16) ρ · exp(jϕ1 ) ||g1 || ||z|| √ where the scalar variable 0 ≤ ρ ≤ 1 and j = −1. To determine ϕ1 and ϕ2 , we √ √ √ first compute |g1H v| = ρ||g1 || and |g2H v| = ρejϕ1 g2H g1 /||g1 || + 1 − ρejϕ2 ||z|| v(ρ, ϕ1 , ϕ2 ) =

34

using (3.16). Then according to [74, Lemma 5-6], we have ϕ2 − ϕ1 = ∠(g2H g1 ). As a common phase between ϕ1 and ϕ2 does not affect |giH v|, without loss of generality, we set ϕ1 = 0. Putting this result into (3.16), we obtain v(ρ) =

√ g1 z √ + 1 − ρ · exp[j∠(g2H g1 )] . ρ· ||g1 || ||z||

(3.17)

With a similar procedure to v, it can be shown that the receive beamformer w can be expressed as w(γ) =

√ h1 u √ γ· + 1 − γ · exp[j∠(hH , 2 h1 )] ||h1 || ||u||

(3.18)

2 where the scalar variable 0 ≤ γ ≤ 1, and u = h2 − hH 1 h2 /||h1 || · h1 . By putting

(3.17) and (3.18) into problem (3.15), the following equivalent problem is obtained F3 :

min

p,ρ,γ,{βi }

p

s.t. (1 − βi )Ai (ρ)p ≥ Bi (γ), ∀i = 1, 2,

(3.19a)

βi Ai (ρ)p ≥ θi (σu2 βi + σz2 ), ∀i = 1, 2,

(3.19b)

ρ ∈ [0, 1], γ ∈ [0, 1], βi ∈ [0, 1], ∀i = 1, 2,

(3.19c)

where A1 (ρ) = ρ||g1 ||2 , ( )2 √ |g2H g1 | √ A2 (ρ) = ρ + 1 − ρ||z|| , ||g1 || ( ) α1 σr2 2E1 † B1 (γ) = Θnl + 2pc − , γ||h1 ||2 T   2 α2 σ 2E2  , B2 (γ) = Θ†nl  √ |hH h | √r + 2pc − 2 1 T 2 + ( γ ||h 1 − γ||u||) 1 ||    + ∞, if x ≥ Pmax    ν ) ( 1 1 + exp(−τ P0 + ν) − 1 , if 0 < x < Pmax . (3.20) − ln Θ†nl (x) = −1 exp(−τ P + ν)x  τ τ 1 + Pmax 0      0, if x ≤ 0 35

Now, problem F3 only has four independent scalar variables, and is of much lower dimension than problem (3.15). A naive way to solve F3 is to perform a four-dimensional exhaustive search over ρ, γ, β1 and β2 . However, this approach has a very high computational complexity. Below, we will show that F3 has a closed-form solution for a fixed γ. Therefore, F3 can be reduced to a onedimensional search problem over γ.

3.3.3

Closed-Form Solution of F3 Given γ

In this subsection, we derive the solution {p⋄ , ρ⋄ , βi⋄ } of F3 given a fixed γ. To simplify the notation, variables that are functions of γ only are written as constants, e.g., Bi (γ) is written as Bi . To begin with, we will determine the optimal solution of β1 and β2 . Since the constraints (3.19a) may be redundant depending on the values of B1 and B2 , we divide the discussion into two cases below. (i) If Bi = 0, then (3.19a) will always be satisfied. Furthermore, the constraint (3.19b) can be rewritten as p ≥

θi (σu2 Ai (ρ)

+

σz2 ). βi

From this equivalent con-

straint, the feasibility space of p is enlarged by using the maximum value of βi in [0, 1] [74]. This gives βi = 1. (ii) If Bi > 0, from (3.19a) we immediately have βi = 1 is not feasible. On the other hand, due to θi = 22µ3−i κ − 1 defined under (3.14) and µ3−i > 0, we must have θi > 0 and therefore from (3.19b) βi ̸= 0. Then, the range of βi becomes (0, 1), and (3.19a) and (3.19b) can be rewritten as p ≥ and p ≥

θi (σu2 Ai (ρ)

+

σz2 ), βi

Bi (1−βi )Ai (ρ)

respectively. Taking the intersection of the above

inequalities, they can be combined as ( )) ( Bi θi σz2 2 p ≥ max , . σu + (1 − βi )Ai (ρ) Ai (ρ) βi

(3.21)

Inside the max function of (3.21), the first term is an increasing function of βi while the second term is a decreasing function of βi . Therefore, the 36

minimum of p is obtained when Bi θi = (1 − βi )Ai (ρ) Ai (ρ)

( ) σz2 2 σu + . βi

(3.22)

Solving (3.22) for βi leads to βi⋄ = σz2 − σu2 +

Bi θi

√( +

2σz2 σz2 − σu2 +

Bi θi

.

)2

(3.23)

+ 4σu2 σz2

Combining the two cases above, we have the following result. Property 3.1 The optimal βi⋄ to F3 with γ fixed is given by (3.23) when Bi > 0; otherwise βi⋄ = 1. Putting the result of Property 3.1 into the constraints (3.19a) and (3.19b), i they can be combined into p ≥ Aiξ(ρ) for i = 1, 2, where  ( √( ) )2   θ B B i i i  2 2  σz + σu + + σz2 − σu2 + + 4σu2 σz2 , if Bi > 0 2 θ θ i i ξi = . (3.24)     θi (σ 2 + σ 2 ), if Bi = 0 u z

By further taking the intersection of both users’ constraints, (3.19a) and (3.19b) 2 ). As a result, F3 with a fixed γ is reduced into become p ≥ max( A1ξ1(ρ) , A2ξ(ρ) )} { ( ξ1 ξ2 F4 : min , . (3.25) p : p ≥ max p,ρ∈[0,1] A1 (ρ) A2 (ρ)

With the definition of A1 (ρ) in (3.20), it is clear that the term

ξ1 A1 (ρ)

in (3.25)

is a decreasing function of ρ, and its minimum is obtained when ρ = 1. On the other hand, taking the derivative of A2ξ2(ρ) with respect to ρ, we have ( ) ( )−3 ∂ ξ2 /A2 (ρ) √ |g2H g1 | √ = −2ξ2 ρ + 1 − ρ||z|| ∂ρ ||g1 || ( ) 1 1 |g2H g1 | − √ × ||z|| . (3.26) √ 2 ρ ||g1 || 2 1−ρ ( ) ∂ ξ2 /A2 (ρ) |gH g |2 |gH g |2 Setting = 0, we get ρ = ||g1 ||22 ||g1 2 ||2 . Notice that when ρ < ||g1 ||22 ||g1 2 ||2 , ∂ρ ( ) ( ) ∂ ξ2 /A2 (ρ) ∂ ξ2 /A2 (ρ) |g2H g1 |2 < 0, and when ρ > ||g1 ||2 ||g2 ||2 , we have > 0. As a we have ∂ρ ∂ρ result, ρ =

|g2H g1 |2 ||g1 ||2 ||g2 ||2

is the minimum point of 37

ξ2 . A2 (ρ)

[i Ai ( U )

[1

A1 ( U )

[2

A2 ( U )

| gH2 g1 |2

0

|| g1 ||2|| g2 ||2

Figure 3.2: Illustration of

U int

ξ1 A1 (ρ)

1

and

U

ξ2 A2 (ρ) .

Based on the above analysis, we can plot the two functions

ξ1 A1 (ρ)

and

ξ2 A2 (ρ)

as

in Figure 3.2. It is observed from Figure 3.2 that there are three possibilities for ) ( 2 , and the optimal ρ⋄ must be within the set minρ∈[0,1] max A1ξ1(ρ) , A2ξ(ρ) { G=

} |g2H g1 |2 int 1, ,ρ , ||g1 ||2 ||g2 ||2

(3.27)

2 where ρint is the intersection point of A1ξ1(ρ) and A2ξ(ρ) , which can be calculated (√ H ) 2 √ |g g | using A1ξ1(ρ) = A2ξ2(ρ) , i.e., ξ1 ρ ||g2 1 ||1 + 1 − ρ||z|| = ξ2 ρ||g1 ||2 . Taking square

root on both sides gives ξ1 ||z||2 . ρint = ( ) √ √ |g2H g1 | 2 ξ2 ||g1 || − ξ1 + ξ1 ||z||2 ||g1 ||

(3.28)

With the optimal ρ⋄ being in the set G and since p is a variable to be minimized, the optimal p⋄ to problem F4 (thus to problem F3 with γ fixed) is given by ( ) ξ2 ξ1 ⋄ , p = min max . (3.29) ρ∈G A1 (ρ) A2 (ρ) In conclusion, the procedure for computing the optimal solution when M = 1 is summarized in Algorithm 3.1.

38

Algorithm 3.1: Computing the optimal solution with fixed relay position. 1: Given (Er , E1 , E2 ), {hi,m , gi,m } and parameters T, pc , µ1 . 2: Initialize κmax using (3.11) and κmin = 0. 3: Repeat 4:

Update κ = (κmin + κmax )/2.

5:

Solve p∗ = minγ∈[0,1] p⋄ (γ), where p⋄ (γ) is obtained from (3.29).

6:

Update κmin = κ if p∗ ≤ (2Er − 2Ec )/T ; κmax = κ if p∗ > (2Er − 2Ec )/T .

7: Until |κmax − κ| < ϵ, where ϵ is a small positive constant to control the

accuracy. 8: Vary µ1 from 0 to 1 and repeat the above steps.

3.4

Iterative Algorithm For Mobile Relay Case

In order to analyze the performance gain brought by the moving relay, in this section, we consider the case of M ≥ 2 for which the relay is moving along a pre-defined trajectory and has multiple stopping points. While exactly solving P3.1 in such a case is generally difficult, we can derive an iterative algorithm based on DC programming. To apply DC programming, the first challenge is to resolve the nonlinear coupling between the stopping time {tm } and the variables {wm , qi,m , vm , pm , βi,m } in the logarithm of data-rates, which is apparent in the first constraint of P3.1. To this end, we introduce substitution variables {Qi,m , Vm , ωi,m } to replace {qi,m , pm , vm , βi,m } as 1 Qi,m , ∀i = 1, 2, m = 1, ..., M, · H tm |wm hi,m |2 Vm H = pm vm vm ≽ 0, Rank(Vm ) ≤ 1, ∀m = 1, ..., M, tm 1 βi,m = ωi,m · , ∀i = 1, 2, m = 1, ..., M. H Tr(gi,m gi,m Vm ) qi,m =

Based on the above variable substitution, the constraint (3.9a) of P3.1 is rewritten

39

as { [ ( ) ]+ M ∑ 1 Qi,m Qi,m min tm log2 + 2 , T m=1 Q1,m + Q2,m σr tm ( )} H β3−i,m Tr(g3−i,m g3−i,m Vm ) tm log2 1 + ≥ µi Rsum , (β3−i,m σu2 + σz2 )tm which can be further reformulated as {[ ( ) ( ) ]+ M 1 ∑ Qi,m σr2 tm min tm log2 + tm log2 1 + , T m=1 σr2 tm Q1,m + Q2,m }  1  ≥ µi Rsum , tm log2 1 + 2 σz2 ( Tr(g3−i,mσguH Vm ) + ω3−i,m )tm

(3.30)

3−i,m

Q

Q

/t

Q

/t

m m i,m where the first line of (3.30) is obtained by using Q1,m +Q + i,m = i,m · σr2 σr2 2,m ( ) 2t m 1+ Q1,mσr+Q , while the second line of (3.30) is obtained by dividing the numer2,m

ator and the denominator of

H β3−i,m Tr(g3−i,m g3−i,m Vm ) 2 (β3−i,m σu +σz2 )tm

H by β3−i,m Tr(g3−i,m g3−i,m Vm ). 2

2

1 z = Tr(gi,mσguH Vm ) + ωσi,m } With further introduction of slack variables {ai,m : ai,m i,m ([ ( )] ) + 2t Q a m and {ri,m : ri,m = min tm log2 ( σ2i,m )+tm log2 1+ Q1,mσr+Q , tm log2 (1+ ti,m ) } tm m 2,m r

for the constraint (3.30), the constraint (3.30) can be significantly simplified. Moreover, since the newly added equalities can be relaxed into inequality constraints and such a relaxation would not change the problem (as the optimal slack variables always activate the inequalities), problem P3.1 can be equivalently reformulated as P3.2 :

max

Rsum ,{tm ,Vm ,wm ,Qi,m ,ωi,m }, {ri,m ,ai,m ,bi,m ,ci,m }

Rsum

M 1 ∑ s.t. µi Rsum − ri,m ≤ 0, ∀i T m=1 [ ) ( Qi,m ri,m + − tm log2 σr2 tm ( ) ]− σr2 tm − tm log2 1 + ≤ 0, ∀i, m Q1,m + Q2,m

40

(3.31a)

(3.31b)

( ) ai,m ≤ 0, ∀i, m r3−i,m − tm log2 1 + tm σu2 σz2 1 + − ≤ 0, ∀i, m H Tr(gi,m gi,m Vm ) ωi,m ai,m [ ] ( ( )) m ∑ ci,l 1 bi,l − tl Θnl ln + (2tl + sl )pc ≤ Ei , ∀i, m τ tl l=1 H |wm hi,m |2 ≤ 0, ∀i, m bi,m Qi,m ) ( tm ci,m H Tr(gi,m gi,m Vm ) − ωi,m ≥ ln , ∀i, m τ tm M M ∑ ∑ EG + Ec + Tr(Vm ) ≤ Er , (2tm + sm ) = T

1

−

m=1

(3.31c) (3.31d) (3.31e) (3.31f) (3.31g) (3.31h)

m=1

tm ≥ 0, Vm ≽ 0, ||wm || ≤ 1, ∀m

(3.31i)

Qi,m ≥ 0, ωi,m ≥ 0, ci,m ≥ tm ∀i, m

(3.31j)

Rank(Vm ) ≤ 1, ∀m.

(3.31k)

Notice that the constraint (3.30) becomes (3.31a)-(3.31d), while (3.31e)-(3.31g) come from the constraint (3.9b) of P3.1 by introducing slack variables {bi,m : Qi,m Hh 2 |wm i,m |

≤ bi,m } and {ci,m :

H V )−ω Tr(gi,m gi,m m i,m tm

≥

c 1 ln( ti,m )}. τ m

Furthermore,

the constraints (3.9c)-(3.9e) of P3.1 become (3.31h)-(3.31k) and the constraint ||wm || = 1 is relaxed into ||wm || ≤ 1 which would not change the problem due to ∗ the optimal ||wm || = 1.

To proceed to solve P3.2, the second challenge is the rank constraints in (3.31k). To this end, we will apply the rank relaxation to drop the rank constraints [66], and a property (proved in Appendix 3.8.1) can be established to show that the relaxation does not affect the optimality. Property 3.2 The rank relaxed problem of P3.2 has at least one optimal solution ∗ ∗ ) ≤ 1. with Rank(Vm Vm

Property 3.2 guarantees that an optimal rank-one or rank-zero solution to the rank relaxed problem of P3.2 exists. However, there may be an alternative solution with a higher rank. If we obtain such a solution, the rank reduction 41

procedure in [67] can be applied to obtain the rank-one solution. Therefore, the problem P3.2 is equivalent to its rank relaxed problem, and we shall focus on the rank relaxed problem in the subsequent part. With the rank constraints being relaxed, the next challenge in P3.2 comes ( ) ci,l 1 − from the operator [·] in (3.31b) and the nonlinear function −tl Θnl τ ln( tl ) in ) ( c (3.31e). However, based on the expression of Θnl in (2.2), −tl Θnl τ1 ln( ti,ll ) in (3.31e) can be expanded as ( )) ( 1 ci,l ln − tl Θnl τ tl [ ( ) ]+ Pmax 1 + exp(−τ P0 + ν) = −tl −1 c exp(−τ P0 + ν) 1 + exp(ν)( ti,ll )−1 [ ( ) ]− Pmax 1 + exp(−τ P0 + ν) = −tl −1 , c exp(−τ P0 + ν) 1 + exp(ν)( ti,ll )−1 | {z }

(3.32)

:=H(ci,l ,tl )

and it is proved in Appendix 3.8.2 that H(ci,l , tl ) is convex. So, we only need to focus on the operators [·]− in (3.31b) and (3.32). But since [x]− = min(x, 0) ≤ x, dropping the operators [·]− would make the feasible set of P3.2 smaller and the solution of such a problem would be feasible for P3.2. After the above procedures, the only nonconvex parts in P3.2 are ( (i) Φm (Q1,m , Q2,m , tm ) := −tm log2 1 +

σr2 tm Q1,m +Q2,m

) in (3.31b);

1 (ii) Υi,m (ai,m ) := − ai,m in (3.31d);

(iii) Ξi,m (wm , Qi,m ) := − (iv) Ψi,m (ci,m , tm ) =

Hh 2 |wm i,m | Qi,m

c tm ln( ti,m ) τ m

in (3.31f);

in (3.31g).

Fortunately, we have the following property (proved in Appendix 3.8.3) for those terms. Property 3.3 The functions Φm , Υi,m , Ξi,m , and Ψi,m are concave.

42

Since Φm , Υi,m , Ξi,m , and Ψi,m are concave, we can apply the DC programming technique [57–63] to replace them with convex upper bounds, which further reduce the feasible set of P3.2. More specifically, given any feasible solution ⋆ ˜ m, Υ ˜ i,m , Ξ ˜ i,m , Ψ ˜ i,m {t⋆m , wm , Q⋆i,m , a⋆i,m , c⋆i,m } of P3.2, we define surrogate functions Φ

as ( ⋆ ⋆ ⋆ ⋆ ˜ Φm (Q1,m , Q2,m , tm |Q1,m , Q2,m , tm ) := −tm log2 1 +

σr2 t⋆m Q⋆1,m + Q⋆2,m

)

( ) (t⋆m )2 /ln2 ⋆ ⋆ Q + Q − Q − Q 1,m 2,m 1,m 2,m ⋆ ⋆ ⋆ ⋆ t⋆m (Q1,m + Q2,m ) + (Q1,m + Q2,m )2 /σr2 ] [ ( ) ( ) σr2 t⋆m σr2 t⋆m /ln2 ⋆ + − log2 1 + ⋆ − 2⋆ tm − tm , (3.33) Q1,m + Q⋆2,m σr tm + Q⋆1,m + Q⋆2,m +

˜ i,m (ai,m |a⋆ ) := − 2 + 1 ai,m , Υ i,m a⋆i,m (a⋆i,m )2 ⋆ e i,m (wm , Qi,m |wm Ξ , Q⋆i,m ) := ( ) ⋆ H ⋆ H (wm ) hi,m hH w |(wm ) hi,m |2 m i,m − 2Re + Qi,m , Q⋆i,m (Q⋆i,m )2

e i,m (ci,m , tm |c⋆ , t⋆ ) := Ψ i,m m ( ⋆ ) [ 1 ( c⋆ ) 1 ] ⋆ c tm t⋆m i,m i,m ⋆ ln + ⋆ (cm − cm ) + ln − (tm − t⋆m ). ⋆ τ tm τ cm τ t⋆m τ

(3.34)

(3.35)

(3.36)

By applying the result of Property 3.3 and the first-order condition for concave functions, we immediately have ˜ m (Q1,m , Q2,m , tm |Q⋆ , Q⋆ , t⋆ ) ≥ Φm (Q1,m , Q2,m , tm ), Φ 1,m 2,m m ˜ i,m (ai,m |a⋆ ) ≥ Υi,m (ai,m ), Υ i,m e i,m (wm , Qi,m |w⋆ , Q⋆ ) ≥ Ξi,m (wm , Qi,m ), Ξ i,m m e i,m (ci,m , tm |c⋆i,m , t⋆m ) ≥ Ψi,m (ci,m , tm ). Ψ

(3.37)

With the above observation, a feasible solution can be directly obtained if we ˜ m, Υ ˜ i,m , Ξ ˜ i,m , Ψ e i,m } expanded replace the functions {Φm , Υi,m , Ξi,m , Ψi,m } by {Φ around a feasible point. However, a better solution can be achieved if we treat the obtained solution as another feasible point and continue to construct the next 43

round surrogate functions. In particular, assuming that the solution at the nth [n]

[n]

[n]

[n]

[n]

iteration is given by {tm , wm , Qi,m , ai,m , ci,m }, the following problem is considered at the (n + 1)th iteration P3.2[n + 1] :

max

Rsum ,{tm ,Vm ,wm ,Qi,m ,ωi,m }, {ri,m ,ai,m ,bi,m ,ci,m }

Rsum

M 1 ∑ s.t. µi Rsum − ri,m ≤ 0, ∀i, T m=1 ) ( Qi,m ri,m − tm log2 σr2 tm [n]

(3.38a)

[n]

˜ m (Q1,m , Q2,m , tm |Q1,m , Q2,m , t[n] ) ≤ 0, ∀i, m, +Φ m ( ) ai,m r3−i,m − tm log2 1 + ≤ 0, ∀i, m, tm σu2 σz2 ˜ i,m (ai,m |a[n] ) ≤ 0, ∀i, m, + +Υ i,m H Tr(gi,m gi,m Vm ) ωi,m m [ ] ∑ bi,l + H(ci,l , tl ) + (2tl + sl )pc ≤ Ei , ∀i, m,

(3.38b) (3.38c) (3.38d) (3.38e)

l=1

1 bi,m

e i,m (wm , Qi,m |w[n] , Q ) ≤ 0, ∀i, m, +Ξ m i,m [n]

(3.38f)

H e i,m (ci,m , tm |c[n] , t[n] ), ∀i, m, Tr(gi,m gi,m Vm ) − ωi,m ≥ Ψ i,m m

(3.38g)

(3.31h) − (3.31j).

(3.38h)

Now, except for the second and third constraints, all the constraints in P3.2[n+1] ( ) Q are obviously convex. However, it can be seen that −tm log2 σ2i,m is the perspecr ( ) ( )tm Q a tive of the convex function −log2 σi,m and −tm log2 1 + ti,m is the perspective 2 m r ) ( of the convex function −log2 1 + ai,m . Since the perspective operation preserves convexity [64], problem P3.2[n + 1] is convex and can be solved by CVX Mosek [64], a Matlab software package for solving convex problems. Denoting its ∗ , Q∗i,m , a∗i,m , c∗i,m }, then we can set {tm optimal solution as {t∗m , wm

[n+1]

n+1 = = t∗m , wm

∗ , Qi,m = Q∗i,m , ai,m = a∗i,m , ci,m = c∗i,m }, and the process repeats with solvwm [n+1]

[n+1]

[n+1]

ing the problem P3.2[n + 2]. According to [57, Theorem 1] and condition (3.37), this iterative algorithm is guaranteed to converge monotonically. 44

Finally, for the above algorithm, we need a feasible starting point. By apply[0]

[0]

[0]

[0]

[0]

ing the penalty method [61, Sec. 3], a feasible {tm , wm , Qi,m , ai,m , ci,m } can be found and the detailed procedure is given in Appendix 3.8.4. The entire iterative procedure for computing the beamforming design when M ≥ 2 is summarized in Algorithm 3.2. Algorithm 3.2: Iterative algorithm for the mobile relay case. 1: Given (Er , E1 , E2 ), {hi,m , gi,m } and parameters T, pc , {sm }, EG , Ec , µ1 . [0]

[0]

[0]

[0]

[0]

2: Initialize {tm , wm , Qi,m , ai,m , ci,m } according to Appendix 3.8.4 and set

n = 0. 3: Repeat 4:

[n+1]

Update {tm

[n+1]

, wm

[n+1]

[n+1]

[n+1]

, Qi,m , ai,m , ci,m } by solving P3.2[n + 1]. Set

n = n + 1. ⋄ 5: Until convergence and the converged point is Rsum .

6: Vary µ1 from 0 to 1 and repeat the above steps.

In terms of computational effort, solving P3.2[n + 1] is dominated by {Vm }, which has M semi-definite cones of dimension N and 2M linear terms. Therefore, its dual form has 2M variables and one semi-definite constraint consisting of M blocks with sizes N × N [64]. By applying the result from [65, Sec. 6.63], the [ complexity for solving P3.2[n + 1] can be computed to be O (1 + M N )1/2 (8M 3 + ] 4M 3 N 2 + 2M 2 N 3 ) . As a consequence, the total complexity for solving P3.2 is [ ] 1/2 3 3 2 2 3 O Q(1 + M N ) (8M + 4M N + 2M N ) , where Q is the number of iterations needed for the DC programming method to converge. Remark 3.1: If N is so large that the semi-definite programming is not possible to be carried out, we can apply the method in [33, Sec. IV-A] to reduce the dimension of Vm from N × N to 2 × 2.

45

6

Output Power (mW)

5

4

3

2 Λ(x) Θnl(x)

1

0 0

5

10

15

20

25

Input Power (mW)

Figure 3.3: Illustration of Θnl and Λ.

3.5

Performance Upper Bound

Besides the proposed iterative algorithm, we also need a performance upper bound for comparison. To achieve this goal, the major challenge is the nonlinear function Θnl in (3.31e) of P3.2, and we propose to use a piecewise linear function Λ(x) = min(ζx, Pmax ) to bound Θnl from above as shown in Figure 3.3. In order to choose ζ such that Λ is as tight as possible, the following problem is considered: min ζ ζ≥0

s.t. min(ζx, Pmax ) ≥ Θnl (x), ∀x ≥ 0.

(3.39)

Since the above problem only has one scalar variable ζ, a bisection search algorithm similar to that of Section 3.3.1 can be applied, and the optimal solution is denoted as ζ ∗ . With the function Θnl replaced by Λ, we will further apply four relaxations to enlarge the feasible set of P3.2, and such relaxations would provide an upper bound for P3.2: (i) Using

σr2 tm Q1,m +Q2,m

≤

σr2 tm , Qi,m

the constraint (3.31b) can be relaxed to ri,m + 46

[−tm log2 (1 +

Qi,m − )] σr2 tm

≤ 0. Furthermore, due to log2 (1 +

Qi,m ) σr2 tm

≥ 0, the

operator [·]− can be dropped without changing the constraint; H (ii) Using Tr(gi,m gi,m Vm ) ≥ ωi,m due to (3.31g), (3.31d) is relaxed into

σu2 + σz2 1 ; ≤ H ai,m Tr(gi,m gi,m Vm ) H (iii) Applying |wm hi,m |2 ≤ ||hi,m ||2 (due to ||wm || = 1) to (3.31f), the relaxed

constraint becomes Qi,m ≤ bi,m ||hi,m ||2 ; H (iv) Since ωi,m ≥ 0, the constraint Tr(gi,m gi,m Vm ) − ωi,m ≥ H can be relaxed into Tr(gi,m gi,m Vm )/tm ≥ τ1 ln(

ci,m ). tm

c tm ln( ti,m ) τ m

in (3.31g)

Combining this con-

straint and (3.31e) into one constraint, we obtain m { ( ) } ∑ H bi,l − tl Λ Tr(gi,m gi,m Vm )/tm + (2tl + sl )pc ≤ Ei . l=1

After the above relaxations, the following upper bound problem of P3.2 is obtained: P3.3 :

max

Rsum ,{tm ,Vm ,Qi,m },{ri,m }

Rsum

M 1 ∑ ri,m ≤ 0, ∀i, s.t. µi Rsum − T m=1 ( ) Qi,m ri,m − tm log2 1 + 2 ≤ 0, ∀i, m, σr tm ) ( H Tr(gi,m gi,m Vm ) ≤ 0, ∀i, m, r3−i,m − tm log2 1 + (σu2 + σz2 )tm m ( ∑ Qi,l H + max[−ζ ∗ Tr(gi,l gi,l Vl ), −tl Pmax ] 2 ||hi,l || l=1 ) + (2tl + sl )pc ≤ Ei , ∀i, m,

EG + Ec +

M ∑

Tr(Vm ) ≤ Er ,

m=1

Vm ≽ 0, tm ≥ 0, ∀m,

M ∑

(2tm + sm ) = T,

(3.40a) (3.40b) (3.40c)

(3.40d) (3.40e)

m=1

Qi,m ≥ 0, ∀i, m. 47

(3.40f)

Because P3.3 only contains linear, semi-definite, relative entropy, and maximum functions, problem P3.3 is convex and can be solved by the existing software. Interestingly, it can be seen from the constraints (3.40b) and (3.40c) that P3.3 also provides an upper bound for the amplify-and-forward scheme and the decodeand-forward scheme. The procedure to compute the upper bound is summarized in Algorithm 3.3. Algorithm 3.3: Computing the upper bound for P3.2. 1: Given (Er , E1 , E2 ), {hi,m , gi,m } and parameters T, pc , {sm }, EG , Ec , µ1 . ′ . 2: Solve P3.3 and the optimal value is given by Rsum

3: Vary µ1 from 0 to 1 and repeat the above steps.

3.6

Numerical Results and Discussions

This section provides numerical results to illustrate the performance of the proposed algorithm. In particular, the distance-dependent path-loss model of user i ϱi,m = ϱ0 · (

di,m −α ) d0

is adopted, where di,m is the distance from user i to the

mth stopping point of the relay, ϱ0 = 10−3 , d0 = 1 m is the reference distance, and α is the path-loss exponent set to be 2.7 [33]. Based on the path-loss model, channels gi,m and hi,m are generated according to CN (0, ϱi,m IN ). The parameters in the energy harvesting model are obtained by fitting the experimental data from the Powercast energy harvester P2110 [37] to the model, and they are given by τ = 274, ν = 0.29, Pmax = 0.004927 W, and P0 = 0.000064 W. By using these parameters, we can further compute ζ ∗ = 0.5732. It is assumed that the circuit power pc = 10 dBm and the relay circuit energy Ec = 100 J. The receiver noise powers are set to σr2 = σu2 = −70 dBm (corresponding to power spectral density −120 dBm/Hz with 100 kHz bandwidth), which include thermal noise, intermodulation noise, crosstalk and impulse noise [68, pp. 109]. The thermal noise at the power splitter is set to σz2 = −104 dBm (corresponding to power 48

spectral density −154 dBm/Hz with 100 kHz bandwidth). Furthermore, within the duration T = 50 s, the energy available at the relay is Er = 300 J while the energies available at users are Ei ∼ U (400, 800) in mJ, where U represents the uniform distribution. Each point in the figures is obtained by averaging over 100 simulation runs, with independent channels and realizations of Ei in each run. We consider the moving relay case of M = 4 and N = 8 with the setting shown in Figure 3.4. The distance between the two users is 8 meters, while the initial position of the relay is 4 meters away from both users. We consider two trajectories. Starting from the initial position, the relay either moves along a horizontal or vertical trajectory, with each trajectory having M = 4 stopping points marked with squares. Since the length of different sections are (3, 6, 3) m, the moving time would be (s1 , s2 , s3 ) = (1.5, 3, 1.5) s with a relay velocity of v = 2 m/s [49]. Furthermore, based on the power model for Pioneer 3DX robot [50, Sec. IV-C], the motion power can be computed to be 0.29 + 7.4v = 15.09 W, and the required motion energy is therefore EG = 15.09 × 6 = 90.54 J. Under the trajectories defined above, the convergence of the Algorithm 3.2 in Section 3.4 is shown in Figure 3.5 when µ1 = µ2 = 0.5. It can be seen that the iterative Algorithm 3.2 converges after 20 iterations. Therefore, the number of iterations is set to be 20 for the rest of this section. On the other hand, the data-rate of fixed relay using Algorithm 3.1 and the data-rates of mobile relay using Algorithms 3.2 and 3.3 are compared in Figure 3.6. It can be seen that the data-rate of the proposed Algorithm 3.2 is very close to the corresponding upper bound no matter which trajectory the relay takes. Furthermore, of the two trajectories, the data-rate of trajectory 1 outperforms the data-rate of fixed relay significantly. This shows the potential advantage of enabling mobility in wirelessly powered TWRC. However, when the relay is moving along the second trajectory, the data-rate is smaller than that of the fixed relay. This indicates that moving is not always beneficial. In fact, it highly

49

Trajectory 1

4

m=2

Trajectory 2 1

User 1

5

3

3

m=2

m=3

m=1,4

User 2

Unit: meter

m=3

Figure 3.4: Two moving trajectories of relay with M = 4. 13 12

Sum-Rate (bps/Hz)

11 10 Trajectory 1 with Algorithm 3.3 Trajectory 1 with Algorithm 3.2 Trajectory 2 with Algorithm 3.3 Trajectory 2 with Algorithm 3.2

9 8 7 6 5 0

5

10

15

20

25

30

Number of Iterations

Figure 3.5: Sum-rate versus number of iterations for Algorithm 3.2.

depends on the trajectory and the required motion energy relative to the available energy at relay. In order to illustrate the last point, the sum-rate versus the available relay energy with µ1 = µ2 = 0.5 is shown in Figure 3.7. It can be observed that when the relay energy is small, even moving along trajectory 1 deteriorates the performance compared to the fixed relay case since the additional motion energy 50

12 Trajectory 1 with Algorithm 3.3 Trajectory 1with Algorithm 3.2 Fixed relay with Algorithm 3.1 Trajectory 2 with Algorithm 3.3 Trajectory 2 with Algorithm 3.2

10

6

2

R (bps/Hz)

8

4

2

0 0

2

4

6

8

10

12

R (bps/Hz) 1

Figure 3.6: Data-rates of wirelessly powered TWRC for fixed relay and mobile relay.

consumption depletes the energy for transmission. However, when the available relay energy exceeds the sum of circuit energy and motion energy, a slight increase of relay energy would enable the sum-rate to exceed that of the fixed relay, which increases much slower as shown in Figure 3.7. This result implies that there exists a trade-off between spending energy on moving and on transmission. To get more insights into the trade-off, it is proved in Appendix 3.8.5 that a closed-form upper bound for Rsum in P3.1 is [ ( ) 1 2hmax [Ei − T pc + ζ ∗ gimax (Er − EG − Ec )] i Rsum ≤ min min log2 1 + , i=1,2 2µi σr2 T )] ( max (Er − EG − Ec ) 2g3−i , (3.41) log2 1 + (σu2 + σz2 )T = maxm=1,...,M ||hi,m ||2 and gimax = maxm=1,...,M ||gi,m ||2 . From where hmax i (3.41), it is observed that if moving (i.e., increasing EG ) does not lead to improved and gimax ), the bound would decrease, thus moving in channels (in terms of hmax i this way might not be beneficial. This corroborates with the result of trajectory and gimax do not change but EG increases from 0 to 90.54J. 2, in which hmax i 51

14

Sum-Rate (bps/Hz)

12

10

Trajectory 1 with Algorithm 3.3 Trajectory 1 with Algorithm 3.2 Fixed relay with Algorithm 3.1 Trajectory 2 with Algorithm 3.3 Trajectory 2 with Algorithm 3.2

8

6

4

2

0 100

150

200

250

Relay Energy (J)

Figure 3.7: Sum-rate versus available relay energy for fixed relay and mobile relay.

On the other hand, if moving leads to improved channels such that there is a net increase in gimax (Er − EG − Ec ) for both i = 1, 2, the bound would increase, and the system might be benefited from the moving relay. This is the case for trajectory 1. While the above discussion provides a rough guideline to determine which trajectory is a good one, Algorithms 3.1-3.3 allow us to precisely figure out how much gain can be obtained from mobile relay along a particular trajectory.

3.7

Conclusions

This chapter studied the beamforming design of a mobile relay in wirelessly powered TWRC. For the case of fixed relay, the optimal beamforming design was obtained using bisection algorithm, with closed-form solution in each iteration. For the case of moving relay, given a moving trajectory, an iterative algorithm was proposed which converges to a data-rate close to the upper bound. Numerical results showed that by adopting the proposed beamforming algorithm and mov-

52

ing the relay along an appropriate trajectory, the data-rate can be significantly increased compared to the fixed relay case.

3.8 3.8.1

Appendix Proof of Property 3.2

To prove the property, we only need to focus on the constraints of P3.2 that are related to Vm , which are given by (3.31d), (3.31g), and (3.31h). In particular, the σz2 −1 ) and ωi,m c tm ln( ti,m ) + ωi,m . m

1 H constraint (3.31d) can be rearranged as Tr(gi,m gi,m Vm ) ≥ σu2 ( ai,m − H the constraint (3.31g) can be rewritten as Tr(gi,m gi,m Vm ) ≥

Now, consider the following semi-definite programming problem find {Vm ≽ 0} s.t.

[

H Tr(gi,m gi,m Vm )

( t∗m ln

c∗i,m t∗m

]

)

EG + Ec +

≥ max

( σu2

1 a∗i,m

σz2 − ∗ ωi,m

)−1 ,

∗ + ωi,m , ∀i, m, M ∑

Tr(Vm ) ≤ Er ,

(3.42)

m=1 ∗ where {t∗m , ωi,m , a∗i,m , c∗i,m } is the optimal solution to the rank relaxed problem of

P3.2, and the constraints of (3.42) are obtained by taking intersection of (3.31d), (3.31g), and (3.31h). Since problem (3.42) is equivalent to the rank relaxed problem of P3.2 with ∗ , ai,m = a∗i,m , ci,m = c∗i,m }, problem (3.42) and the rank {tm = t∗m , ωi,m = ωi,m ∗ }. On the relaxed problem of P3.2 must have the same optimal solution {Vm

other hand, problem (3.42) has all together three constraints on each Vm . Ac∗ ∗ ) ≤ 3, which yields with Rank2 (Vm cording to [67, Theorem 3.2], there exists Vm √ ∗ ∗ ) ≤ 3, and Rank(Vm Rank(Vm ) ≤ 1 holds.

53

3.8.2

Proof of Convexity for H(x, t)

Based on the definition of H in (3.32), H(x, t) is the perspective transformation of I(x) in terms of t, where Pmax I(x) = − exp(−τ P0 + ν)

(

) 1 + exp(−τ P0 + ν) −1 . 1 + exp(ν)x−1

(3.43)

Therefore, H(x, t) and I(x) have the save convex property [64], and we only need to focus on I(x). In particular, I(x) can be reformulated as Pmax [1 + exp(−τ P0 + ν)] Pmax − exp(−τ P0 + ν) exp(−τ P0 + ν) Pmax [1 + exp(−τ P0 + ν)] exp(ν) + · . exp(−τ P0 + ν) x + exp(ν)

I(x) =

Since the term

exp(ν) x+exp(ν)

(3.44)

is convex in x for x ≥ 0, I(x) is a convex function. As a

consequence, H(x, t) is convex.

3.8.3

Proof of Property 3.3

To prove the property, we first show that Φm is concave. More specifically, since ) ( 2 r , which is the composition with Φm is the perspective of −log2 1 + Q1,mσ+Q 2,m ( ) 2 an affine mapping x = Q1,m + Q2,m from the function f (x) = −log2 1 + σxr with x > 0, Φm has the same convex property as f (x). Furthermore, due to 2

σr ∇2 f (x) = − ln2 (x2 + σr2 x)−2 (2x + σr2 ) < 0, f (x) is a concave function. As a result,

Φm is a concave function with respect to Q1,m , Q2,m , tm . Secondly, we will show that Υi,m is concave. By taking the second order derivative of Υi,m with respect to ai,m , we have ∇2 Υi,m = −2a−3 i,m . Since ai,m defined above P3.2 satisfies

1 ai,m

≥

2 σu H V ) Tr(gi,m gi,m m

+

σz2 , ωi,m

we have ai,m ≥ 0 (due to

H Vm ) ≥ 0 and ωi,m ≥ 0). Therefore, ∇2 Υi,m ≤ 0 and Υi,m is concave. Tr(gi,m gi,m

Thirdly, we will show that Ξi,m is concave. In particular, since the term −Ξi,m =

Hh 2 |wm i,m | Qi,m

H hi,m |2 [64, is the perspective of a convex quadratic function |wm

Ch. 3.2.6], −Ξi,m is convex. Using the fact that a concave function is the negative of a convex function [64, Ch. 3], Ξi,m is concave. 54

Finally, Ψi,m is concave because the term Ψi,m (ci,m , tm ) =

c tm ln( ti,m ) τ m

is the

perspective of a concave function τ1 ln(ci,m ). This completes the proof.

3.8.4

Initialization for Algorithm 3.2

The penalty method adopts an iterative procedure to find a feasible point for the DC programming algorithm. In particular, at the nth iteration, the penalty method solves the following problem: max

Rsum ,{tm ,Vm ,wm ,Qi,m ,ωi,m }, {ri,m ,ai,m ,bi,m ,ci,m },{φi,m }

Rsum − ϵ

2 ∑ M ∑

φi,m

i=1 m=1

s.t. (3.38a) − (3.38d), (3.38f) − (3.38h) m [ ] ∑ bi,l + H(ci,l , tl ) + (2tl + sl )pc ≤ Ei + φi,m , ∀i, m, l=1

where the penalty ϵ is sufficiently large (e.g., ϵ = 106 ). The above iterative ′ procedure is started by any {t′m , wm , Q′i,m , a′i,m , c′i,m } satisfying (3.31h)-(3.31j), ∑ ∑ −10 and is terminated until 2i=1 M . m=1 φi,m is sufficiently small, e.g., less than 10 [0]

[0]

[0]

[0]

[0]

The last round solution is then used as the initial point {tm , wm , Qi,m , ai,m , ci,m } for P3.2[1].

3.8.5

Proof of Equation (3.41)

We prove (3.41) by applying a series of relaxations to the constraints of P3.1. First, using

q |wH hi,m |2 ∑2 i,m m H q |wm hj,m |2 j,m j=1

≤ 1,

H β3−i,m pm |g3−i,m vm |2 2 2 σu β3−i,m +σz

≤

pm ||g3−i,m ||2 2 +σ 2 σu z

(due to β3−i,m ≤

H hi,m |2 ≤ ||hi,m ||2 (due to ||wm || = 1), the first con1 and ||vm || = 1), and |wm

straint of P3.1 is relaxed into { )} ) ( ( M 1 ∑ pm ||g3−i,m ||2 qi,m ||hi,m ||2 ≥ µi Rsum . , tm log2 1 + min tm log2 1 + T m=1 σr2 σu2 + σz2

55

Now, exchanging the operators

∑M m=1

and min, and by applying Jensen’s Inequal-

ity, the above inequality is further relaxed into ) { ( ∑M ∑M 2 t t q ||h || m i,m i,m m=1 m , min log2 1 + m=1 ∑M 2 T σr m=1 tm ( )} ∑M 2 t p ||g || m m 3−i,m tm log2 1 + m=1 ≥ µi Rsum . ∑M 2 2 (σu + σz ) m=1 tm

(3.45)

On the other hand, from the second constraint of P3.1 and using Λ(x) ≥ Θnl (x) for ∑ ∑ 2 max 2 all x ≥ 0, we have M (Ei −T pc +ζ ∗ M m=1 tm qi,m ||hi,m || ≤ hi m=1 tm pm ||gi,m || ), ∑ 2 max and from the third constraint of P3.1, we have M (Er − m=1 tm pm ||gi,m || ≤ gi ∑M EG − Ec ). Putting the above results and m=1 2tm ≤ T into (3.45), we immediately obtain (3.41).

56

Chapter 4 Beamforming Algorithm For WPIT With Massive MIMO In the last chapter, the beamforming design of a mobile charger is investigated. In this chapter, we further investigate a WPIT network with massive MIMO [82–85], since scaling up the number of antennas would concentrate the energy beam towards target users [53–56]. Unfortunately, in the large-scale settings, the inverse of Hessian matrices would involve high computational complexity. Consequently, traditional DC programming method becomes prohibitively time consuming, and cannot be used in practice if the number of antennas is in the range of hundreds or more. On the other extreme, by using the assumption of infinite number of antennas and applying the law of large numbers, simple beamforming solution is possible. However, when applied to scenarios with finite number of antennas, the performance of such asymptotic solution is far from that of DC programming. To resolve this apparent complexity-performance dilemma, this chapter develops an algorithm which reduces the computation time by orders of magnitude while still guaranteeing the same performance compared to the DC programming. In particular, the proposed algorithm consists of two fast-convergent iterative procedures and

57

pc for circuit

qk for uplink

consumption

transmission Energy

battery

Data

Information Decoder

User K

Received Signal

N Antennas

User k Power Splitter

Zoom in

.. « .

Ek from

Energy Harvester

Downlink Energy and Information Flow

« .. .

Uplink Information Flow

User 1

Access Point

Figure 4.1: System model of WPIT network with large number of antennas.

is guaranteed to obtain a Karush-Kuhn-Tucker solution. Furthermore, in each iteration, the algorithm only requires the computation of inner products between channel vectors, and can be run in parallel for all the users. Thus the complexity of the proposed algorithm scales only linearly with the number of antennas at access point.

4.1

System Model

We consider a WPIT network consisting of an access point equipped with N antennas, and K single-antenna users. As shown in Figure 4.1, the access point intends to multicast a piece of common information to K users, and the K users intend to send individual information to the access point. This type of systems can be found in many IoT applications. For example, in smart warehouses, the access point needs to collect data from sensors and RFIDs using the WPIT technique. Moreover, in such systems, the transmission involves two phases, i.e., downlink multicasting phase and uplink multiple access phase. Below, we give the details of each transmission phase. In the first downlink phase, the access point multicasts a signal s with E[|s|2 ] = 1 to all the users through the transmit beamforming vector v ∈ CN ×1 with power

58

||v||2 . Accordingly, the received signal rk ∈ C at the user k is rk = gkH vs + nk , where gkH ∈ C1×N is the downlink channel vector from the access point to the k th user, and nk ∈ C is the Gaussian noise at the k th user with power σu2 . The received signal rk is further split into two branches, one for the information decoder and the other for the energy harvester. At the information decoder side, the signal is given by r˜k =

√ βk gkH vs +

√ βk nk + zk , where βk ∈ [0, 1] is the splitting factor, and zk ∈ C is Gaussian noise introduced by the power splitter, with E[|zk |2 ] = σz2 . Based on the expression of r˜k , the downlink signal-to-interference-plus-noise ratio (SINR) of the k th user is ΓDL k =

βk |gkH v|2 . βk σu2 + σz2

(4.1)

On the other hand, at the energy harvester of user k, the input power can be expressed as (1 − βk )E[|rk |2 ]. Based on the expression of rk , the input power can be further expressed as (1 − βk )|gkH v|2 , provided that the noise power σu2 is negligible compared to the signal power of gkH vs. Accordingly, the harvested ( ) power is denoted by Θ (1 − βk )|gkH v|2 , where Θ is the function representing the energy conversion process [36–47]. In this chapter, we adopt Θ = Θnl , where Θnl is defined in (2.2). In the second uplink phase, all the users transmit data symbols to the access point simultaneously via spatial division multiple access (SDMA), with the k th user symbol being xk ∈ C with E[|xk |2 ] = 1 and the user transmit power being qk . ∑ √ The received signal y ∈ CN ×1 at the access point is given by y = K k=1 hk qk xk + m, where hk ∈ CN ×1 is the uplink channel vector, and m ∈ CN ×1 is the Gaussian noise at access point with E[mmH ] = σa2 IN . In order to detect the signal of user k while limiting the array processing complexity, a linear receive beamforming vector wkH ∈ C1×N is applied to y, and we can express the uplink SINR of the k th user as ΓUL k = ∑

qk |wkH hk |2 . H 2 2 2 l̸=k ql |wk hl | + σa ||wk || 59

(4.2)

Remark 4.1: Since the antenna number is large, the downlink channels can be estimated via channel reciprocity. Notice that the WPIT system considered in this chapter can be employed in scenarios beyond traditional cellular networks. For example, the wirelessly powered system could be employed in IoT and millimeter wave networks, where the pilot contamination effect can be safely ignored. In particular, in IoT networks, since the coverage of the access point is small [77, Table III], the access points are sparsely distributed and the intercell interference is weak. On the other hand, in millimeter wave networks, since broadband frequency spectrum is used, such systems would be operated in noise limited regime according to the measurements in [78, Fig. 1]. In both cases, the pilot contamination effect is not the utmost issue. Remark 4.2: While time division multiple access (TDMA) may outperform SDMA if there are no local powers at the users [79], the result cannot be directly applied to the case when users have local powers. Moreover, SDMA is an appealing technology for delay sensitive applications and for user fairness provisioning, since SDMA is a simultaneous transmission scheme [79]. As a result, it is important to study SDMA for wirelessly powered networks. Notice that even under the TDMA scheme, the proposed Algorithm 4.2 in Section 4.3.2 can still be applied to design the downlink energy beamformer.

4.2

Problem Formulation

In the considered network, the design variables that can be controlled are the transmit-receive beamformers {v, wk }, users’ transmit powers {qk }, and users’ power splitting ratios {βk }. Since a fundamental quality-of-service (QoS) requirement in a communication system is the guaranteed SINR [80,81], our aim is to provide reliable communication for all the users at their requested SINR targets, with the uplink and downlink SINR requirements for the k th user denoted

60

by αk and θk , respectively. On the other hand, since the uplink SINR ΓUL in k (4.2) depends on the user transmit power qk , which in part was harvested from ( ) 1 H 2 the downlink wireless signal, we must have 2 Θnl (1 − βk )|gk v| + Ek ≥ 12 qk + pc , where Ek is the local power per symbol-time available at the k th user in duration T , and pc is the circuit power consumption per symbol-time at user terminals (the coefficient 1/2 is due to the two transmission phases with equal duration). Having the QoS and energy harvesting requirements satisfied, it is crucial to minimize the total transmit power at access point and users1 because saving power translates to cost reduction and environmental benefits. As a result, by accounting for all the factors mentioned above, an optimization problem can be formulated as: P4.1 :

min

{v,wk ,qk ,βk }

||v||2 +

K ∑

qk

k=1

s.t. qk |wkH hk |2 ≥ αk (

(∑

) ql |wkH hl |2 + σa2 ||wk ||2 , ∀k = 1, ..., K,

l̸=k

) βk |gkH v|2 ≥ θk βk σu2 + σz2 , ∀k = 1, ..., K, ) 1 ( 1 Θnl (1 − βk )|gkH v|2 + Ek ≥ qk + pc , ∀k = 1, ..., K, 2 2 qk ≥ 0, βk ∈ [0, 1], ∀k = 1, ..., K, where the first and second constraints are the uplink and downlink SINR QoS requirements, respectively. Unfortunately, problem P4.1 is non-convex due the coupled terms of {qk , wk } and {βk , v}, as well as the nonlinear function Θnl . In fact, P4.1 is NP-hard in general since its special case of {αk = 0} is NP-hard according to [82, Claim 1]. Remark 4.3: Since a large number of antennas at the access point may incur significant antenna circuit power consumption, it is also interesting to consider the energy efficiency maximization problem at access point. In such a case, the 1

The user transmit powers are minimized to save energy for the battery.

61

objective of P4.1 becomes

( log 2 1+ k=1

∑K max

∑

qk |wkH hk |2 H 2 2 2 q |w l̸=k l k hl | +σa ||wk ||

||v||2 + P0

{v,wk ,qk ,βk }

) ,

where P0 is the antenna circuit power consumption at access point. By using the Dinkelbach method, the above fractional objective function can be transformed into a subtractive form, and the resultant problem can be solved using a similar approach to that of P4.1.

4.3

Proposed First-Order Method for P4.1

To resolve the non-convexity, a traditional way is to reformulate P4.1 into a DC programming problem (by introducing substitution variables χk =

1 qk

to

replace qk ) [57–63]. However, DC programming needs to solve a convex quadratic problem in each iteration, and each problem has at least KN + N + 2K variables and K 2 second-order cone (SOC) constraints of dimension 2. Therefore, DC programming requires a complexity of O(K 4 N 3 ) in each iteration [64], and such a method would be extremely slow when N is large. To overcome the drawback of DC programming, an accelerated first-order method is presented below. To provide a fast algorithm for solving P4.1, the first step is to decompose the problem P4.1 into smaller pieces. To achieve this goal, it is first observed from the third constraint that smaller (q1 , ..., qK ) loosens the constraints on v since Θnl is a monotonically increasing function. As smaller (q1 , ..., qK ) also helps in ∗ ) must therefore be within the reducing the objective value, the optimal (q1∗ , ..., qK

set Q, which is the Pareto optimal set for the following multi-objective uplink problem

{ U:

min

{bk ≥0,wk }

≥ αk

( ∑

(b1 , ..., bK ) : bk |wkH hk |2 ) bl |wkH hl |2 + σa2 ||wk ||2

l̸=k

62

} , ∀k = 1, ..., K .

(4.3)

By defining the feasible set of problem U as Y, Q can be expressed as { } Q=

(b1 , ..., bK ) ∈ Y :̸ ∃(x1 , ..., xK ) ∈ Y with (x1 , ..., xK ) ≺ (b1 , ..., bK ) , (4.4)

where ≺ represents “Pareto dominate” (for minimization problems) [86]. Based on the above observation, the feasible set of (q1 , ..., qK ) can be restricted into the Pareto set Q and P4.1 can be equivalently transformed into the following problem: P4.2 : s.t.

min

{v,βk ∈[0,1],qk }

βk |gkH v|2 (

||v|| + 2

≥

(

K ∑

qk

k=1

θk βk σu2

+ )

σz2

)

, ∀k,

1 1 Θnl (1 − βk )|gkH v|2 + Ek ≥ qk + pc , ∀k, 2 (2 ) q1 , ..., qK ∈ Q. By inspection, the problems U and P4.2 appear to be inseparable. This is because the uplink problem U is a multi-objective problem, which may have many Pareto solutions [86]. Since different Pareto solutions (of the uplink problem) would impose different constraints on downlink beamformer v (observing from the second constraint of P4.2) and lead to different objective values of the entire problem P4.2, we have to choose a solution among all the Pareto solutions of the uplink problem such that the total power of P4.2 is minimized. In fact, such uplink-downlink coupling is an inevitable issue in wirelessly powered system [22–24]. But fortunately, we have the following property to decouple U and P4.2. ∑ 1 Property 4.1 If Rank([h1 , ..., hK ]) + K k=1 1+αk > K, then |Q| = 1. ∑ 1 Proof: If Rank([h1 , ..., hK ]) + K k=1 1+αk > K, we have Q ̸= ∅ based on [80, ) ( Theorem III.1] and there must exist some points q1 , ..., qK in Q. Furthermore, ) ( according to the fixed point equation in [80, Appendix A], the points q1 , ..., qK in Q must satisfy the following: hH k

(∑

αk H 2 l̸=k ql hl hl + σa IN

63

)−1

− qk = 0. hk

(4.5)

Since the left hand side of (4.5) is strictly radially quasi-concave (see [87] for definition of “strictly radially quasi-concave”) for all k, there is at most one solution to (4.5) according to [87, Corollary 1]. Therefore, |Q| = 1. Property 4.1 indicates that there exists a unique Pareto solution for the uplink problem under certain condition. Notice that if the access point is equipped with large number of antennas, we must have N ≫ K and Rank([h1 , ..., hK ]) = K. Together with the fact that αk is the QoS requirement of the k th user and ∑ 1 must be positive, Rank([h1 , ..., hK ]) + K k=1 1+αk > K is always satisfied, implying that the unique solution is guaranteed. Therefore, we only need to find the unique point in Q and put it back into P4.2 for subsequent derivation.

4.3.1

Solving the Uplink Problem

To find the unique Pareto point in Q, an outer approximation algorithm can be [0]

[0]

[1]

[1]

adopted, which generates a sequence of lower bounds {(γ1 , ..., γK ), (γ1 , ..., γK ), ...} [0]

[0]

to approach the solution. As a valid lower bound, we can set (γ1 , ..., γK ) = 0. Furthermore, consider the following update of {zk , γk } at the nth iteration: [n+1]

zk

=

(∑

[n]

2 γl hl hH l + σ a IN

)−1

hk ,

(4.6)

l̸=k [n+1]

γk

αk =

(∑

[n] [n+1] H ) hl |2 l̸=k γl |(zk [n+1] H ) h

|(zk

+

2 k|

[n+1] σa2 ||zk ||2

) ,

(4.7)

and the following property, which is proved in Appendix 4.7.1 based on the fixedpoint iteration, can be established. Property 4.2 If Rank([h1 , ..., hK ]) +

∑K

1 k=1 1+αk

[0]

> K, then with γk = 0 for all

⋄ ) ∈ Q. k, the sequence γk converges to the unique limit point (γ1⋄ , ..., γK [n]

For the above alternating minimization (AM) procedure (4.6)-(4.7), while [0]

the first iteration is straightforward (due to γk = 0), the computational complexity for n ≥ 1 is dominated by the inverse operation of the N × N matrix ∑ [n] 3 H 2 l̸=k γl hl hl + σa IN . This leads to a cubic complexity O(KN ) in terms of the 64

antenna numbers [81], and thus direct implementation of (4.6) is not suitable for systems with large number of antennas. To get around the matrix inversion, we will apply the first-order method (i.e., gradient-based method) for the implementation of (4.6), which has linear complexity in terms of antenna numbers [88]. In particular, we first reformulate (4.6) as an unconstrained least squares problem ) (∑ ( H ) [n+1] [n] 2 I + σ zk = argmin xH γl hl hH l a N x − 2Re hk x . x

|

l̸=k

{z :=Φ[n] (x)

(4.8)

} (∑

[n]

H Then it can be observed from (4.8) that the Hessian matrix l̸=k γl hl hl + ) σa2 IN of Φ[n] (x) is a sum of (K − 1) rank-one Hermitian matrices and an identity (∑ ) [n] H 2 matrix. Thus the matrix has at most K distinct eigenl̸=k γl hl hl + σa IN

values, and problem (4.8) can be solved by the conjugate gradient (CG) method within K iterations [88, Theorem 5.4]. More specifically, CG method is an iterative algorithm which generates a set of searching directions that are orthogonal (∑ ) [n] H 2 to each other in the Euclidean space weighted by l̸=k γl hl hl + σa IN . By doing so, CG method is guaranteed to converge much faster than the gradient descent method while still maintaining a low per-iteration complexity. In order to apply CG method to problem (4.8), the warm-start initialization [n]

d0 = zk is used. Then the initial searching direction can be set to c0 = ∇Φ[n] (d0 ) (∑ ) [n] H 2 (notice that ∇ := ∂/∂ conj(x)), where ∇Φ[n] (x) = l̸=k γl hl hl +σa IN x−hk . At the mth iteration, dm+1 and cm+1 are updated as follows [89]: dm+1 = dm − δm cm , cm+1 = ∇Φ[n] (dm+1 ) − ηm cm .

(4.9) (4.10)

In equation (4.9), the term δm is the line search step-size, and the optimal stepsize can be found by exact minimization, which is given by 2 ∑ [n] 2 d − h d + σ l̸=k γl hl hH k m a m l . δm = ∑ [n] H H 2 cH c h h c + σ γ c m l m a m m l l̸=k l 65

(4.11)

On the other hand, in equation (4.10), the factor ηm is a carefully chosen coef(∑ ) [n] H 2 ficient such that cH γ h h + σ I l l i a N cj = 0 holds for all i ̸= j. In order l̸=k l to satisfy such orthogonal property, one possibility is to use Fletcher-Rieves factor [89]: ||∇Φ[n] (dm+1 )||2 ||∇Φ[n] (dm )||2 2 ∑ [n] 2 H l̸=k γl hl hl dm+1 + σa dm+1 − hk = − ∑ 2 . [n] 2 d − h d + σ l̸=k γl hl hH m k a m l

ηm = −

(4.12)

Following a similar proof to that of [88, Theorem 5.3], we have cH i

(∑

) [n] 2 γl hl hH + σ I l a N cj = 0

l̸=k

holds for all i ̸= j. Using the above orthogonal property and according to [88, Theorem 5.1], the sequence {d0 , d1 , ...} is guaranteed to converge and the converged point d⋄ is the optimal solution of (4.8). More importantly, according to [88, Theorem 5.4], the number of iterations for CG method to converge must be less than or equal to K. Based on the CG method, the entire procedure to compute the optimal solution of the uplink problem U is summarized in Algorithm 4.1.

4.3.2

Solving the Downlink Problem

By substituting the optimal solution of the uplink problem {qk = γk⋄ } into P4.2, the problem P4.2 is equivalently transformed into D:

min

{v,βk ∈[0,1]}

||v||2

s.t. |gkH v|2 ≥ θk (σu2 + |gkH v|2 ≥

σz2 ), ∀k = 1, ..., K βk

µk , ∀k = 1, ..., K, 1 − βk

66

(4.13a) (4.13b)

Algorithm 4.1: Computing the optimal solution for the uplink problem U. 1: Input {hk , αk }, σa2 . [0]

2: Initialize {γk = 0, ∀k} and set n = 0. [1]

[1]

3: Update zk = hk /σa2 and γk = αk σa2 /||hk ||2 for all k. Update n = 1. 4: Repeat 5:

For k = 1 : K [n]

7:

Initialize d0 = zk . ) (∑ [n] 2 H Initialize c0 = γ h h + σ I z0 − hk . N l a l l̸=k l

8:

Set m = 0.

9:

Repeat

6:

10:

Update dm+1 using (4.9).

11:

Update cm+1 using (4.10).

12:

Set m := m + 1.

13:

Until m = K and the converged point is d⋄ = dK . Set zk

[n+1]

14:

End

15:

Update γk

[n+1]

= d⋄ .

using (4.7) for all k.

Set n := n + 1. √ ∑ [n] [n−1] 2 17: Until ) < 10−6 . The converged point is {γk⋄ , z⋄k }. k (γk − γk 16:

18: Output {qk∗ = γk⋄ , wk∗ = z⋄k }.

where µk = Θ†nl (γk⋄ + 2pc − 2Ek ) and    + ∞, if x ≥ Pmax    ν ) ( 1 1 + exp(−τ P0 + ν) − 1 , if 0 < x < Pmax − ln Θ†nl (x) = −1 exp(−τ P + ν)x  τ τ 1 + Pmax 0      0, if x ≤ 0 can be considered as pesudo-inverse of Θnl (x). To solve the downlink problem D, we will first determine the optimal solution of βk . In particular, if θk = 0, then the first constraint of D always holds. From the second constraint of D, the feasibility set of v is maximized by using the 67

minimum value of βk , which leads to βk∗ = 0. On the other hand, if θk > 0, we need to divide the discussion into three cases, since Θ†nl (x) is a piecewise function consisting of three cases. (i) If γk⋄ + 2pc − 2Ek ≥ Pmax , then µk → +∞ and the problem D is infeasible. (ii) If γk⋄ + 2pc − 2Ek ≤ 0, then µk = 0 and the second constraint of D always holds. On the other hand, from the first constraint of D, the feasibility set of v is maximized by using the maximum value of βk , which leads to βk∗ = 1. (iii) If 0 < γk⋄ + 2pc − 2Ek < Pmax , then βk and v would be involved in both constraints in D. Taking the intersection of the two inequalities, they can be combined as |gkH v|2

[ σz2 µk ] 2 ≥ max θk (σu + ), . βk 1 − βk

(4.14)

Inside the max function of (4.14), the first term is a decreasing function of βk while the second term is an increasing function of βk . Therefore, the minimum of pk is obtained when θk (σu2 +

σz2 µk )= . βk 1 − βk

(4.15)

Solving (4.15) for βk leads to βk∗

= σz2

−

σu2

µk + + θk

√(

2σz2 σz2

−

σu2

µk )2 + + 4σu2 σz2 θk

.

(4.16)

Since putting µk = 0 into (4.16) also leads to βk∗ = 1, we can combine the cases of (ii)-(iii) into one, and the solution of βk∗ for the case of θk > 0 is given by (4.16). Furthermore, by putting βk∗ for the cases of θk = 0 and θk > 0 into the constraints of D and defining   if θk = 0  µk , √ ) , ξk = θ ( µ µ   k σz2 + σu2 + k + (σz2 − σu2 + k )2 + 4σu2 σz2 , if θk > 0 2 θk θk 68

(4.17)

problem D becomes D1 : min ||v||2 v

s.t. |gkH v|2 ≥ ξk , k = 1, ..., K. Now, due to the nonconvexity of the term |gkH v|2 , problem D1 is generally NP-hard [82, Claim 1]. To resolve the nonconvexity, one traditional way is to apply semi-definite relaxation (SDR) [82], which results in an SDP problem with K variables and one semi-definite constraint of dimension N × N . However, (√ ) the SDR method requires a complexity O N (K 3 + K 2 N 2 + KN 3 ) [65], and could only provide a lower bound to the objective function of D1. A slightly better way to resolve the nonconvexity is applying the successive linear approximation (SLA) method [83] to the constraints of D1. This would lead to an iterative algorithm with the number of iterations being MSLA . In each iteration, we need to solve a second-order cone programming problem with N variables and K linear constraints. Therefore, the SLA method requires a complexity of ) ( √ O MSLA K(N 3 + 2N K) , which is still too large for massive MIMO applications. To overcome the drawbacks of SDR and SLA methods, we propose an accelerated primal-dual gradient (APDG) algorithm for D1, which is a parallel first-order algorithm for solving large-scale homogeneous QCQP problems. In particular, starting from a feasible point v[0] (one possibility is to choose v[0] = √ (∑K √ gl ) p0 ξl ||gl ||2 , where p0 = maxk ∑ √ξk gH gl 2 ), consider the following l=1 K k 2 l=1 ξl ||gl ||

iteration of {φk } and v: (

[n+1] [n+1] φ1 , ..., φK

) = argmax {λk ≥0}

K ∑

λk (ξk +

|

|gkH v[n] |2 )

K 2 ∑ H [n] λk gk gk v , −

{z

k=1

k=1

}

:=Ξ[n] ({λ1 ,...,λK })

(4.18) v[n+1] =

K ∑

[n+1]

φk

gk gkH v[n] ,

(4.19)

k=1

69

and the following theorem (proved in Appendix 4.7.2) can be established. Theorem 4.1 With a feasible v[0] , the sequence {v[0] , v[1] , ...} is convergent, and the converged point v⋄ is a Karush-Kuhn-Tucker solution of D1. For the above iterative procedure, equations (4.18) and (4.19) are the dual and primal updates, respectively. Interestingly, the iteration (4.18)-(4.19) can also be interpreted as alternating optimization, where (4.18) optimizes the combining coefficients and (4.19) optimizes the beamforming vector. Notice that a Karush-Kuhn-Tucker solution of D1 may not be the optimal solution of D1. However, the optimal solution of D1 must satisfy the Karush-Kuhn-Tucker solution of D1 according to [90, Proposition 3.3.1], provided that problem D1 is regular. Furthermore, the Karush-Kuhn-Tucker solution can result in satisfying performance empirically. As a consequence, finding a Karush-Kuhn-Tucker solution is very meaningful both in theory and in practice. The goal of applying Theorem 1 is to transform the non-smooth optimization problem D1 (as the constraints of D1 can be viewed as indicator functions) into successive smooth optimization problems (4.18)-(4.19). Such a smoothing procedure is very important because the first-order methods converge very slow for non-smooth problems [91]. In contrast, since the gradients of smooth problems are Lipschitz continuous, it is possible to use the Lipschitz condition for acceleration [92]. More specifically, observing that the objective function in (4.18) is a smooth concave function, we will propose an accelerated projected gradient method to solve problem (4.18) based on Nesterov’s acceleration [92]. In particular, starting [0]

[0]

[n]

K from {λk = 0}K k=1 (for n ≥ 1, a warm start of {λk = φk }k=1 is applied), the [m+1]

following accelerated projected gradient method can be applied to update λk at the mth iteration [92, 93]: [m+1]

λk

) ( 1 [m] = ΠR+ ρk + ∇λk Ξ[n] ({λ1 , ..., λK }) [m] L {λk =ρk }K k=1

70

K [ ] )}+ { ∑ 1( [m] [m] H [n] 2 = ρk + ξk + |gk v | − 2 Re ρl (v[n] )H gk gkH gl glH v[n] , (4.20) L l=1

where ΠR+ is the projection onto the set R+ and L is the Lipschitz constant of the ( ∑ ) [m] K H [n] 2 gradient ∇ || k=1 λk gk gk v || [92]. Moreover, ρk is a linear combination of [m]

λk

[m−1]

and λk

as [m]

[m]

ρ k = λk +

c[m−1] − 1 [m] [m−1] (λk − λk ), [m] c

(4.21)

and c[m] is a particularly tuned parameter satisfying √ 1 + 1 + 4(c[m−1] )2 c[0] = 1, c[m] = . 2 The key insight of the accelerated gradient method is that the traditional gradient method is too conservative, and we need to add some overshoots (the [m]

quantity λk

[m−1]

− λk [m]

in equation (4.21)) as shown in Figure 4.2 (if we drop

[m−1]

the quantity λk − λk

, then the accelerated gradient method reduces to the

gradient method). However, we also need to guarantee that the overshoots would not be too “confident” such that we miss the optimal point. Therefore, we need to design a monotonically increasing sequence {c[m] }, which can be viewed as a damping system [94], to represent how much we trust in the overshoots. In particular, at the beginning, c[m] is small and over-damping is used to push the solution points forward. As c[m] becomes larger, under-damping is used to pull the solution points back to the optimal point. Notice that the step size 1/L in (4.20) is the inverse of L, which can be computed as follows. In particular, by computing the Hessian matrix G[n] = ) ( ∑ H [n] 2 , it can be shown that the (k, l)th element of G[n] is λ g g v || ∇2 || K k=1 k k k ] [ equal to 2Re (v[n] )H gk gkH gl glH v[n] . Then, due to G[n] ≼ λmax (G[n] )IK , where λmax (G[n] ) takes the largest eigenvalue of G[n] , the Lipschitz constant of the gradient is L = λmax (G[n] ). [m]

With the obtained L, λk

computed using (4.20) would converge to the opti71

U k[ m1] U k[ m ]

Ok[ m2]

Ok[ m1]

O

After acceleration

O

Direction of acceleration

[m] k

U k[ m1]

Before acceleration

[ m 1] k

Figure 4.2: Illustration of acceleration.

mal solution of (4.18) with an iteration complexity of [89]: v  u K ( )2 1 u ∑ [0] λ⋆k − λk · √ , O tL ϵ k=1 √ where

∑

⋆ k (λk

[0]

− λk )2 is the Euclidean distance between the converged point [0]

{λ⋆k } and the initial point {λk } of (4.20), L = λmax (G[n] ) is the Lipchitz constant of gradients, and ϵ is the target solution accuracy. This iteration complexity is significantly smaller than that of the subgradient method and the alternating direction method of multipliers (ADMM) method2 . In fact, the iteration complexity touches the lower bound derived in [89, Theorem 2.1.6] for any first-order method, meaning that the proposed algorithm is among the lowest complexity in the class of first-order methods. Since solving the problem D is equivalent to solving for {βk } using (4.16) and solving for v using D1, the entire procedure for computing the solution of problem D using the APDG algorithm is summarized in Algorithm 4.2. 2

The iteration complexity of the subgradient method is O( ϵ12 ) [95] and that of a standard

ADMM method is O( 1ϵ ) [96].

72

Algorithm 4.2: Computing a KKT solution for the problem D 1: Input {gk , θk , }, σu , σz , and {qk∗ = γk⋄ }. 2: Compute {βk∗ } using (4.16) and {ξk } using (4.17). 3: Initialize v[0] =

√

p0

(∑

K l=1

√ gl ) ξl ||gl ||2 with

p0 = max

k=1,...,K

ξk ∑ √ H 2 . K g g l=1 ξl ||gkl ||2l

[0]

4: Set {φk = 0} and n = 0. 5: Repeat 6:

[1]

[0]

[n]

Initialize λk = λk = φk for all k. Compute |gkH v[n] |2 and Re[(v[n] )H gk gkH gl glH v[n] ].

7:

Set c[0] = 1 and m = 1.

8:

Repeat

9:

For k = 1 : K

10:

Update c

[m]

=

11:

Update ρk

[m]

using (4.21).

√

[m+1] λk

1+

12:

Update

13:

Set m := m + 1.

14: 15: 16:

1+4(c[m−1] )2 . 2

using (4.20).

End [n] −10 Until |Ξ[n] ({λk }K }K and set {φk k=1 ) − Ξ ({λk k=1 )| < 10 ∑ [n+1] Set v[n+1] = K gk gkH v[n] . k=1 φk [m]

[m−1]

[n+1]

[m]

= λk }.

Set n := n + 1. 18: Until ||v[n] ||2 − ||v[n−1] ||2 < 10−6 and the converged point is v⋄ . 17:

19: Output v = v⋄ , {βk = βk∗ }.

4.3.3

Overall Algorithm and Complexity Analysis

Since the problem P4.1 can be decoupled into two subproblems according to Property 4.1, and each subproblem can be further solved by Algorithms 4.1 and 4.2, respectively, the overall algorithm is to execute Algorithm 4.1 and Algorithm 73

Table 4.1: Summary of Complexity

Scheme CG-APDG DC programming AM-SDR AM-SLA

Complexity ( ) 1 3 √ O M1 K N + M2 (KN + K ϵ ) O (M2 K 4 N 3 ) ( ) √ O M1 KN 3 + N (K 3 + K 2 N 2 + KN 3 ) ( ) √ O M1 KN 3 + M2 K(N 3 + 2N K)

4.2 sequentially. As Algorithm 4.1 is based on conjugate gradient (CG), and Algorithm 4.2 is based on accelerated primal-dual gradient (APDG), we called the overall algorithm CG-ADPG. Furthermore, according to Property 4.2 and Theorem 4.1, the CG-APDG method is guaranteed to converge to a KKT solution of problem P4.2. Since we have shown that P4.1 can be equivalently transformed into P4.2, the obtained solution is also a KKT solution for P4.1. For complexity analysis, it can be seen from Algorithm 4.1 that the computational complexity is dominated by (4.9) and (4.10), which require O(KN ) floating-point operations. Therefore, with K iterations for the CG procedure and K users for computation, the total complexity is O(M1 K 3 N ), where M1 is the number of outer iterations for Algorithm 4.1 to converge. On the other hand, in each iteration of Algorithm 4.2, we need to first compute |gkH v[n] |2 and Re[(v[n] )H gk gkH gl glH v[n] ], which requires O(N ) operations for each user. After ( ) that we need to perform O √1ϵ scalar operations3 to update the beamforming coefficients for each user, where ϵ is the target solution accuracy. Therefore, the ) ( 1 √ complexity of Algorithm 4.2 is O M2 (KN + K ϵ ) , where M2 is the number 3

Before v[n] falls into the convergence region, it is not necessary to accurately solve (4.18),

and the inner loop can be terminated after a fixed number of iterations [62, Sec. II-D]. As a result, the complexity of the APDG method is dominated by the part after v[n] falls into the √ ∑ [0] convergence region. In such a case, L k (λ∗k − λk )2 is a small constant, and the complexity ( ) becomes O √1ϵ .

74

of outer iterations for Algorithm 4.2 to converge. Based on the complexities of Algorithms 4.1 and 4.2, the proposed CG-APDG method requires a complexity of ( ) 1 3 √ O M1 K N + M2 (KN + K ϵ ) . Notice that with ϵ = 10−4 , the term √1ϵ = 100 would be in the same order as N . Based on the above complexity analysis and the complexity results of DCprogramming, AM, SDR, and SLA, the complexities of the proposed methods and alternative solutions are summarized in Table 4.14 . It can be seen from Table 4.1 that if N = 100 and K = 20, the proposed CG-APDG method requires O(106 ) operations (the complexity is dominated by O(M1 K 3 N )). On the other hand, the AM-SDR requires O(108 ) operations, and the AM-SLA requires O(107 ) operations. This indicates that the proposed method reduces the computational complexity by at least 90% compared to alternative methods. Moreover, it can be seen from (4.6)-(4.7) and (4.20) that the proposed algorithms for uplink and downlink problems are capable of running in parallel for all the users. Therefore, its computation time can be further reduced in practice.

4.4 4.4.1

Practical Considerations Hybrid Beamforming

In massive MIMO systems, a common assumption is that the RF chains are limited compared to the large number of antennas. In such a case, a hybrid beamforming design, which consists of an analog beamformer and a digital beamformer, is required. There are two ways to handle the design of hybrid beamformer (the hybrid receiver can be designed similarly). The first one is to design a traditional 4

The number of outer iterations for Algorithm 4.1 to converge is equal to that for AM [80],

i.e., M1 = MAM . On the other hand, the number of outer iterations for Algorithm 4.2 to converge is equal to that for DC programming and SLA [58], i.e., M2 = MDC = MSLA . As shown by simulations, M1 and M2 are much smaller than N .

75

beamformer v⋄ and then factorize it into the multiplication of VRF and vBB , where VRF ∈ CN ×NRF is the analog beamformer controlling the phase shifts, NRF is the number of RF chains, and vBB ∈ CNRF ×1 is the digital beamfomer. Mathematically, the above factorization can be written as [97, 98]: min

{VRF ,vBB }

||v⋄ − VRF vBB ||2 RF | = 1, ∀i, j, ||VRF vBB || = ||v⋄ ||, |Vi,j

s.t.

(4.22)

where v⋄ is the traditional beamformer. In this context, the proposed algorithm represents an efficient design for v⋄ . Once v⋄ is obtained, problem (4.22) can be solved efficiently by alternating minimization [97, 98]. The second way to design a hybrid beamformer is to apply a two-stage approach [99]. In particular, the first stage designs the analog beamformer VRF based on beamsteering codebooks [99] or Fourier matrix (which is asymptotically optimal if N → ∞ [99]). With the result of VRF in the first stage and by replacing v with VRF vBB in problem P4.1, the second stage problem is given by min

{vBB ,wk ,qk ,βk }

s.t.

||V

RF

v

|| +

BB 2

K ∑

qk

k=1

qk |wkH hk |2

≥ αk

(∑ l̸=k

βk |gkH VRF vBB |2 (

≥

ql |wkH hl |2 (

θk βk σu2

+ )

+

σz2

σa2 ||wk ||2

)

)

, ∀k,

, ∀k,

(4.23a) (4.23b)

1 1 Θnl (1 − βk )|gkH VRF vBB |2 + Ek ≥ qk + pc , ∀k, 2 2

(4.23c)

qk ≥ 0, βk ∈ [0, 1], ∀k.

(4.23d)

It can be seen that the above problem has exactly the same structure as P4.1, and we can execute the proposed algorithm in Section 4.3 to solve this problem. In conclusion, no matter which method to employ for hybrid beamforming design, the proposed algorithm in Section 4.3 constitutes an indispensable part.

76

4.4.2

Time Allocation

In practice, the users may not need to receive information in the downlink phase. In such a case, users could operate in an idle state and it is beneficial to reduce the duration of uplink transmission. To address this problem, we can introduce a scalar variable 0 ≤ z ≤ 1 to represent the proportion of the downlink phase. Moreover, since the downlink and uplink phases are unequal, we need to minimize the weighted average transmit power and the problem P4.1 is modified as Q:

min

{v,wk ,qk ,βk },z

z||v|| + (1 − z) 2

(

K ∑

qk

) qk |wkH hk |2 s.t. (1 − z)log2 1 + ∑ ≥α ek , ∀k = 1, ..., K, H 2 2 2 l̸=k ql |wk hl | + σa ||wk || ( ) βk |gkH v|2 zlog2 1 + ≥ θek , ∀k = 1, ..., K, βk σu2 + σz2 ( ) [ ] zΘnl (1 − βk )|gkH v|2 + Ek ≥ (1 − z)qk + z 1θek + (1 − z) pc , ∀k = 1, ..., K, k=1

qk ≥ 0, βk ∈ [0, 1], ∀k = 1, ..., K, 0 ≤ z ≤ 1. where α ek and θek represent the uplink and downlink data-rate targets, respectively. The constant 1θek = 1 if θek > 0 and 1θek = 0 if θek = 0. This constant 1θek is used to model the idle state during the downlink phase if users do not need to receive signals. It can be seen that problem Q has the same structure with problem P4.1 when the variable z is fixed. Furthermore, since z is a scalar and is bounded by 0 ≤ z ≤ 1, a one-dimensional search can be applied to find the optimal z ∗ , with each iteration running Algorithm 4.1 and Algorithm 4.2 in Section 4.3.

4.5

Simulation Results and Discussions

This section presents simulation results to verify the performance of the proposed scheme. The distance-dependent pathloss model of the k th user ϱk = ϱ0 · ( ddk0 )−2.7 77

is adopted [33], where ϱ0 = 10−3 , dk is the distance from the k th user to the access point, and d0 = 1m is the reference distance [33]. In the simulations, dk ∼ U (1, 10) in meter, where U represents the uniform distribution, and hk is generated according to CN (0, ϱk IN ). Moreover, due to channel reciprocity in time division duplex (TDD) systems5 , we have gk = hk . The parameters in the energy harvesting model are obtained by fitting the experimental data from the Powercast energy harvester P2110 to the model (2.2), and they are given by τ = 274, ν = 0.29, Pmax = 0.004927 W, and P0 = 0.000064 W [45, 46]. It is assumed that noise power σa2 = σu2 = σz2 = −40 dBm (corresponding to power spectral density −90 dBm/Hz with 100 kHz bandwidth), which includes thermal noise, intermodulation noise, crosstalk and impulse noise. The circuit power consumption at users is set to pc = 5 dBm; the available energy at users is Ek ∼ U (2, 8) in dBm; and the users’ SINR requirements are θk = αk = 3 dB. Each point in the figures is obtained by averaging over 100 simulation runs, with independent channels between consecutive runs. All problem instances are solved by Matlab R2015b on a desktop with Intel Core i5-4570 CPU at 3.2 GHz and 8 GB RAM.

4.5.1

Convergence Behavior

To verify the convergence of Algorithm 4.1 in Section 4.3.1, Figure 4.3 shows the transmit powers of different users versus number of outer iterations (the number of inner iterations is smaller than or equal to K) when K = 20 and N = 256. To avoid too much clutter in the figure, we only plot the results for four of the users, as the results for the other users are similar. It can be seen that all the transmit powers converge fast and stabilize after 3 iterations. This result verifies 5

The results for frequency division duplex (FDD) systems, where gk and hk are indepen-

dently generated, are similar to that of TDD systems. Therefore, the results for FDD systems are not reported here.

78

User Transmit Power (mW)

0.2

0.15

0.1

User 1 User 2 User 3 User 4

0.05

0 0

2

4

6

8

10

Number of Outer Iterations

Figure 4.3: User transmit powers versus the number of outer iterations in Algorithm 4.1 for the case of K = 20 and N = 256.

Function Value Ξ

[0]

15

10

5

Accelerated projected gradient Projected gradient 0 0

2000

4000

6000

8000

Number of Inner Iterations

Figure 4.4: Function value Ξ[0] of (4.18) versus number of inner loop iterations in Algorithm 4.2 when K = 20 and N = 256.

the convergence property in Property 4.2 and also indicates that the number of outer iterations M1 for Algorithm 4.1 to converge is small.

79

Transmit Power at Access Point (dBm)

45

44

43

42

41

40

39 0

10

20

30

40

50

Number of Outer Iterations

Figure 4.5: Transmit power at access point versus number of outer loop iterations in Algorithm 4.2 when K = 20 and N = 256.

Next, in order to verify the convergence of the inner loop (i.e., line 8 to line 15 in Algorithm 4.2) of the proposed APDG method in Section 4.3.2, Figure [m]

4.4 shows the function value of Ξ[0] ({λk }K k=1 ) in (4.18) versus number of inner iterations (m = 0, ..., 8000) when K = 20 and N = 256. It can be seen that the accelerated projected gradient method and the projected gradient method converge to the same value. However, the accelerated projected gradient method converges significantly faster than the projected gradient method, which reveals the improved convergence rate brought by the acceleration. Notice that since the [m]

[m]

[2] K convergence behavior for {Ξ[1] ({λk }K k=1 ), Ξ ({λk }k=1 ), ...} would be similar,

they are not repeated here. On the other hand, to verify the convergence of the outer loop (i.e., line 5 to line 18 in Algorithm 4.2) of the APDG method in Section 4.3.2, Figure 4.5 shows the transmit power at access point versus number of outer iterations (n = 0, ..., 50) when K = 20 and N = 256. As observed from the figure, the proposed algorithm converges after 30 iterations, which corroborates with the convergence result of Theorem 4.1 and indicates that the number of 80

outer iterations M2 for Algorithm 4.2 to converge is small.

4.5.2

Performance and Running Time

Now, in order to verify the performance and complexity of the overall algorithm CG-APDG, we focus on the case of K = 20 with N ∈ {64, 128, 256, 512, 1024}. Besides the methods in Table 4.1, we also simulate the scaled asymptotic solution6 . It can be observed from Figure 4.6 that the proposed CG-APDG achieves the same transmit power as the AM-SLA and the DC programming method, and they are very close to the lower bound (see the description below problem D1). Moreover, all these algorithms significantly outperform the scaled asymptotic solution. Notice that the performance gap between the proposed CG-APDG and the scaled asymptotic solution is diminishing as N becomes larger. This is because larger N results in weaker interference, which approaches the zero interference assumption in the scaled asymptotic solution. On the other hand, Figure 4.7 shows the average execution time versus the number of antennas at access point. Compared to the DC programming method, the proposed CG-APDG algorithm reduces the computation time by orders of magnitude, which demonstrates the effectiveness of the proposed two-stage optimization. Furthermore, compared to AM-SDR and AM-SLA, the CG-APDG algorithm saves at least 90% of their computation times at N = 128, which corroborates with the complexity analysis in Section 4.3.3. Under all the simulated values of N , the CG-APDG algorithm saves at least 80% of the computation times of AM-SLA, revealing the low-complexity nature of the CG-APDG algo6

For scaled asymptotic solution, it is assumed that N → ∞. In such a case, based on the

law of large numbers, all the user channels would be asymptotically orthogonal (i.e., hH i hj = 0 and giH gj = 0 for any i ̸= j). However, since the number of antennas is finite in practice, the obtained solution is not feasible for P4.1, and the transmit power needs to be scaled up until all the constraints are satisfied. If scaling up the power still cannot make the constraints satisfied, this setting is removed from the simulation.

81

65 Scaled asymptotic solution DC programming Proposed CG-APDG AM-SLA Lower bound

Total Transmit Power (dBm)

60 55 43.46

50

43.44 43.42

45

128

40 35 30 0

200

400

600

800

1000

Number of Antennas

Figure 4.6: Total transmit power versus number of antennas for the case of K = 20. 10 3

Average Execution Time (s)

too long time if N≥512 10

2

10 1

10 0

10

DC programming AM-SDR AM-SLA Proposed CG-APDG Scaled asymptotic solution

-1

10 -2

0

200

400

600

800

1000

Number of Antennas

Figure 4.7: Average execution time versus number of antennas for the case of K = 20.

rithm. Notice that for DC-programming and AM-SDR, their simulation times are so large that the results cannot be obtained in a reasonable time when N ≥ 512. To study the relationship between the total transmit power and the number of users, we consider the case of N = 256 with K ∈ {10, 20, 40, 80}. Since 82

Total Transmit Power (dBm)

70

65

Scaled asymptotic solution Proposed CG-APDG

60

55

50

45

40 10

20

30

40

50

60

70

80

Number of Users

Figure 4.8: Total transmit power versus number of users for the case of N = 256.

CG-APDG, AM-SDR, AM-SLA, and DC programming have the same transmit power, we only simulate the proposed CG-APDG algorithm, and compare it to the scaled asymptotic solution. It can be observed from Figure 4.8 that CGAPDG outperforms the scaled asymptotic solution under all the settings of K. Furthermore, the performance gap between CG-APDG and the scaled asymptotic solution grows with the number of users K. This is because larger K results in stronger interference, which departs more from the zero interference assumption in the scaled asymptotic solution.

4.6

Conclusions

This chapter studied the large-scale beamforming design problem in type-II WPIT systems. By reformulating the original problem into a two-stage optimization problem, a fast parallel iterative algorithm was proposed and it was proved to converge to a KKT solution. As each iteration of the proposed algorithm only involves computation of inner products between channel vectors, its complexity

83

scales linearly in terms of the number of antennas at the access point. Simulation results showed that the proposed algorithm reduces the execution time by orders of magnitude compared to the DC programming method while guaranteeing the same performance.

4.7

Appendix

4.7.1

Proof of Property 4.2 [n]

[0]

First we prove γk is monotonically increasing by induction. Since putting γk = 0 [1]

[0]

[1]

into (4.6) and (4.7) leads to γk = αk σr2 /||hk ||2 , we have γk ≤ γk . Now we [n−1]

assume that γk

[n] γk

= ≤ ≤

[n]

≤ γk for all k and some n ≥ 1. From (4.7), we have ∑ [n−1] [n] [n] |(zk )H hl |2 + σr2 ||zk ||2 ) αk ( l̸=k γl [n]

|(zk )H hk |2

∑ [n−1] [n+1] [n+1] αk ( l̸=k γl |(zk )H hl |2 + σr2 ||zk ||2 ) [n+1] H ) hk |2 ∑ [n] [n+1] [n+1] αk ( l̸=k γl |(zk )H hl |2 + σr2 ||zk ||2 ) [n+1] |(zk )H hk |2

|(zk

[n+1]

= γk

,

(4.24)

where the inequality in the second line is due to [n]

zk =

(∑

[n−1]

γl

2 hl hH l + σr I

l̸=k

= argmin

αk (

∑

)−1

hk

[n−1]

l̸=k

γl

x

|xH hl |2 + σr2 ||x||2 ) |xH hk |2

(4.25)

which can be obtained from (4.6) and [81, Corollary 2]. The inequality in the [n−1]

third line is due to γk

[n]

≤ γk , and the equality in the last line is due to (4.7).

[n]

Thus γk is monotonically increasing. [n]

[n]

Then we prove γk is upper bounded. Specifically, we will show that γk ≤ bk by induction, where {bk , wk } is an arbitrary feasible solution of problem U. Since [0]

[n]

γk = 0 ≤ bk , we assume that γk ≤ bk holds for some n ≥ 0. Using (4.7), we 84

have [n+1] γk

= ≤ ≤

∑ [n] [n+1] [n+1] αk ( l̸=k γl |(zk )H hl |2 + σr2 ||zk ||2 ) [n+1] H ) hk |2

|(zk

∑ [n] 2 2 2 αk ( l̸=k γl |wH k hl | + σr ||wk || ) 2 |wH k hk | ∑ 2 2 2 αk ( l̸=k bl |wH k hl | + σr ||wk || ) 2 |wH k hk |

≤ bk ,

(4.26)

where the inequality in the second line is due to (4.25) (replacing n with n + 1), [n]

the inequality in the third line is due to γk ≤ bk , and the inequality in the last line is due to the constraint in problem U. Therefore as long as P4.2 has a feasible [n]

solution, the sequence γk is upper bounded. [0]

[1]

Lastly, using the above two results, we immediately have {γk , γk , ...} is [n+1]

convergent. As seen from (4.6), {zk [1]

[n]

} is only dependent on {γk }, thus the

[2]

sequence {zk , zk , ...} also converges. Denoting the limit solution of iteration [∞]

[∞]

(4.6)-(4.7) as {γk , zk }, we have the following two properties: [∞]

(i) γk

≤ bk for all k, with bk being any feasible solution in the problem U.

[∞]

[∞]

(ii) {γk , zk } is a feasible solution for problem U. This can be easily verified [n+1]

by putting γk

[n]

[∞]

= γk = γk [∞]

into (4.7). [∞]

Combining (i)-(ii), {bk = γk , wk = zk } is optimal for the problem U.

4.7.2

Proof of Theorem 4.1

Defining the set { } Vn = x : ξk + |gkH v[n] |2 − 2Re[(v[n] )H gk gkH x] ≤ 0, ∀k , we have the following two lemmas. Lemma 4.1 v[n+1] = argminx∈Vn ||x||2 for all n ≥ 0.

85

Proof: Since K K K ∑ 2 ∑ ∑ H [n] [n] H H λk gk gk v = λk Re[(v ) gk gk ( λl gl glH v[n] )], k=1

k=1

(4.27)

l=1

the function inside argmax of (4.18) is K ∑

λk (ξk +

k=1 K ∑



|gkH v[n] |2 )

K ∑ 2 − λk gk gkH v[n] = k=1

2 λk gk gkH v[n]

+

k=1

(

K ∑

λk ξk +

|gkH v[n] |2

− 2Re[(v )

[n] H

) λl gl glH v[n] )] .

K ∑

gk gkH (

k=1

l=1

(4.28) Observing that x =

∑K k=1

min ||x|| + 2

x

λk gk gkH v[n] is the optimal solution to the problem

K ∑

( λk ξk +

|gkH v[n] |2

− 2Re[(v )

[n] H

)

gk gkH x]

,

k=1

which can be proved by setting the derivative of the above objective function to zero, we have max

K ∑

{λk ≥0}

K 2 ∑ λk gk gkH v[n] λk (ξk + |gkH v[n] |2 ) − k=1

k=1

= max min ||x||2 + {λk ≥0}

x

K ∑

(

) λk ξk + |gkH v[n] |2 − 2Re[(v[n] )H gk gkH x] .

(4.29)

k=1

Since the function in (4.29) is convex in both λk and x, strong duality holds, and we can exchange the operators max and min [64]. Further noticing that max ||x||2 +

{λk ≥0}

=

K ∑

( ) λk ξk + |gkH v[n] |2 − 2Re[(v[n] )H gk gkH x]

k=1

  ||x||2 , if ξk + |gkH v[n] |2 − 2Re[(v[n] )H gk gkH x] ≤ 0 

∞,

,

(4.30)

otherwise

taking the minx on both sides of (4.30), it is obvious that x cannot be chosen so that the second case in (4.30) holds. Therefore, the right hand side of (4.29) is equal to minx∈Vn ||x||2 , and we have the following holds: max

{λk ≥0}

K ∑ k=1

K 2 ∑ λk gk gkH v[n] = min ||x||2 . λk (ξk + |gkH v[n] |2 ) − k=1

86

x∈Vn

(4.31)

Now, it can be observed from (4.31) that x and {λk } are a pair of primal and dual variables, with the Lagrangian given by L = ||x||2 +

K ∑

λk (ξk + |gkH v[n] |2 − 2Re[(v[n] )H gk gkH x]).

k=1

Therefore, the primal optimal solution x∗ and the dual optimal solution {λ∗k } should together satisfy the KKT condition ∂L/∂ conj(x) = 0, which leads to x∗ = ∑ ∑K ∗ [n+1] [n+1] ∗ H [n] from (4.18) and v[n+1] = K gk gkH v[n] k=1 φk k=1 λk gk gk v . Finally, as λk = φk from (4.19), we immediately have v[n+1] = x∗ . Lemma 4.2 v[n] ∈ Vn for all n ≥ 0. Proof: We prove the lemma by considering two cases. (i) n = 0. Since v[0] is feasible for D1, we have ξk − |gkH v[0] |2 ≤ 0, which is equivalent to ξk + |gkH v[0] |2 − 2Re[(v[0] )H gk gkH v[0] ] ≤ 0. This indicates v[0] ∈ V0 . (ii) n ≥ 1. Based on Lemma 4.1, v[n] = argminx∈Vn−1 ||x||2 , and from the definition of the set Vn−1 , we have ξk + |gkH v[n−1] |2 − 2Re[(v[n−1] )H gk gkH v[n] ] ≤ 0.

(4.32)

Moreover, due to |gkH (v[n−1] − v[n] )|2 ≥ 0, expanding the square gives |gkH v[n−1] |2 − 2Re[(v[n−1] )H gk gkH v[n] ] ≥ −|gkH v[n] |2 .

(4.33)

Subtracting (4.33) from (4.32), we obtain ξk − |gkH v[n] |2 ≤ 0, which can be rewritten as ξk + |gkH v[n] |2 − 2Re[(v[n] )H gk gkH v[n] ] ≤ 0. Therefore, v[n] ∈ Vn . Combining (i)-(ii), the proof is completed. Using v[n+1] = argminx∈Vn ||x||2 according to Lemma 4.1 and v[n] ∈ Vn according to Lemma 4.2, we immediately have ||v[n+1] || ≤ ||v[n] ||. Therefore, the sequence {||v[0] ||, ||v[1] ||, ...} is monotonically decreasing. As the norm is lower bounded by zero, the sequence {||v[0] ||, ||v[1] ||, ...} must converge. 87

Finally, we will show that the converged point v⋄ satisfies the KKT condition of D1. According to Lemma 4.1, v⋄ is the optimal solution to { } 2 H ⋄ 2 ⋄ H H min ||x|| : ξk + |gk v | − 2Re[(v ) gk gk x] ≤ 0, ∀k . x

(4.34)

Based on (4.34), we can obtain two results. Firstly, since v⋄ must satisfy the constraint in (4.34), putting it into the constraint leads to ξk − |gkH v⋄ |2 ≤ 0, indicating that v⋄ is feasible for D1. Secondly, due to v⋄ being the optimal solution of (4.34), there always exists Lagrange multipliers {ζk ≥ 0} such that the following two equations hold:  K ( ) ∑  ∂  ⋄ H ⋄ 2 ⋄ H H  v + ζk ξk + |gk v | − 2Re[(v ) gk gk x] =0 ∂ conj(x) x=v⋄ k=1  ( )    ζk ξk + |gkH v⋄ |2 − 2Re[(v⋄ )H gk gkH v⋄ ] = 0, ∀k

. (4.35)

By simple calculations, it can be proved that ( ) ( ) ∂ ∂ ξk + |gkH v⋄ |2 − 2Re[(v⋄ )H gk gkH x] ξk − |gkH x|2 = ∂ conj(x) ∂ conj(x) x=v⋄ x=v⋄ and ( ) ζk ξk + |gkH v⋄ |2 − 2Re[(v⋄ )H gk gkH v⋄ ] = ζk (ξk − |gkH v⋄ |2 ). Therefore, (4.35) becomes  K ∑   ⋄  v + ζk

( ) ∂ ξk − |gkH x|2 =0 ∂ conj(x) x=v⋄ k=1  ) (    ζk ξk − |gkH v⋄ |2 = 0, ∀k

.

(4.36)

Based on (4.36) and since we have shown that v⋄ is feasible for D1, v⋄ is a KKT solution of D1.

88

Chapter 5 Conclusions and Future Research 5.1

Conclusions

In this thesis, the nonlinear model and the beamforming algorithms for WPIT systems were investigated. In Chapter 2, the mismatch between the existing linear energy harvesting model and the RF energy harvesting circuit was first illustrated. Then, a nonlinear energy harvesting model, which matches the experimental data very well, was proposed. Based on the nonlinear model, a convergence guaranteed iterative algorithm was proposed for optimizing the WPIT beamforming design, and the resultant beamforming design was shown to outperform traditional designs based on linear model. With the proposed nonlinear model, the beamforming design of a mobile charger in TWRC was investigated in Chapter 3. In particular, by endowing the relay with mobility, the distances between the relay and users can be varied, thus providing a potential solution to combat pathloss at the expense of energy for transmission. To characterize the consequence brought by such a scheme, the beamforming algorithms for both fixed and mobile relay cases were proposed. By comparing the data-rates for the fixed and mobile relay cases, it is possi-

89

ble to quantify the relative advantage of spending energy on moving versus on transmission in wirelessly powered two-way communication. While the beamforming algorithms in Chapters 2 and 3 could achieve good performance, they need to compute the inverse of Hessian matrices. This operation would be very time consuming and cannot be used in practice if the number of antennas at the access point is in the range of hundreds or more. To this end, Chapter 4 developed an accelerated first-order algorithm, which not only avoids the inverse of Hessian matrices but also guarantees a fast convergence rate. By both complexity analysis and simulation results, it was shown that the proposed method reduces the computation time by orders of magnitude while still guaranteeing the same performance compared to state-of-the-art algorithms.

5.2

Future Research

While Chapter 3 has investigated a WPIT network with mobile charger, the moving trajectory of the charger is fixed. However, since the mobile charger would consume additional motion power, the fixed trajectory design might lead to excessive movement and degrade the performance. Therefore, it is necessary to provide a joint mobility management and beamforming design algorithm for WPIT systems with mobile charger. Unfortunately, due to the practical geometric constraints, the mobile charger needs to avoid obstacles such as walls and objects. This makes the mobility scheduling a discrete optimization problem, which is fundamentally different from the continuous problems considered in this thesis. As a result, it remains an interesting but challenging task to study the joint mobility management and beamforming design in WPIT networks with mobile charger. Another interesting research direction is to study the WPIT systems with channel uncertainty. This is because the IoT devices are small and low-cost in

90

practice, making channel estimation a significant burden for IoT systems. Therefore, the IoT access point usually needs to design beamformers without knowing the channel conditions exactly, which is nontrivial since the beamforming design needs to be robust and computationally efficient to obtain. This results in many research opportunities for future works.

91

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List of Publications Journal Papers 1. Shuai Wang, Minghua Xia, and Yik-Chung Wu, “Backscatter data collection with unmanned ground vehicle: Mobility management and power allocation,” submitted to IEEE Trans. Wireless Commun., May 2018. 2. Shuai Wang, Minghua Xia, and Yik-Chung Wu, “Multicast wirelessly powered network with large number of antennas via first-order method,” IEEE Trans. Wireless Commun., vol. 17, no. 6, Jun. 2018. 3. Shuai Wang, Minghua Xia, and Yik-Chung Wu, “Space-time signal optimization for SWIPT: Linear versus nonlinear energy harvesting model,” IEEE Commun. Lett., vol. 22, no. 2, Feb. 2018. 4. Shuai Wang, Minghua Xia, Kaibin Huang, and Yik-Chung Wu, “Wirelessly powered two-way communication with nonlinear energy harvesting model: Rate regions under fixed and mobile relay,” IEEE Trans. Wireless Commun., vol. 16, no. 12, Dec. 2017. 5. Zhigang Wen, Shuai Wang (Corresponding Author), Xiaoqing Liu, and Junwei Zou, “Joint relay-user beamforming design in full-duplex two-way relay channel,” IEEE Trans. Veh. Techn., vol. 66, no. 3, Mar. 2017. 6. Shuai Wang, Minghua Xia, and Yik-Chung Wu, “Multi-pair two-way relay network with harvest-then-transmit users: resolving pairwise uplink-downlink coupling,” IEEE J. Sel. Topics Signal Process., vol. 10, no. 8, Dec. 2016. Conference Papers 1. Yang Li, Shuai Wang, Edward S. Hui, Di Cui, Hing-Chiu Chang, and Yik-Chung Wu, “Accelerated magnetic resonance fingerprinting reconstruction using majorization-minimization,” Proc. ISMRM’17, Huonululu, Hawaii, Apr. 2017.

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2. Shuai Wang, Minghua Xia, and Yik-Chung Wu, “Quality of service constrained wirelessly powered communication with multiple antennas,” Proc. IEEE GLOBECOM’16 Workshop on WEHCN, Washington, DC, Dec. 2016. 3. Shuai Wang, Minghua Xia, and Yik-Chung Wu, “Achieving global optimality in wirelessly-powered multi-antenna TWRC with lattice codes,” Proc. IEEE ICASSP’16, Shanghai, China, Mar. 2016, pp. 3556-3560.

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