Energy Harvesting and Remotely Powered Wireless Networks

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1 IEEE ISIT 216 Tutorial July 1, 216 Energy Harvesting and Remotely Powered Wireless Networks Ayfer Ozgur (Stanford) Sennur Ulukus (UMD) Aylin Yener(Penn State)

2 Outline of This Tutorial Introduction to energy harvesting (EH) Single-user offline power/rate optimization [Aylin] Single-user online power/rate optimization [Ayfer] Multi-user offline power optimization [Sennur] Multi-user online power optimization [Sennur] Energy cooperation (EC) and optimization [Sennur] Information theory of EH, infinite/zero/unit battery [Aylin] Information theory w/ finite battery, connections to online & offline optimization; IT of EC [Ayfer] 7/1/216 IEEE ISIT 216, Barcelona, Spain 2

3 Prerequisites for the Tutorial Basic command of Optimization Communication Theory Reasonable fluency in Shannon Theory Fairly self-contained otherwise 7/1/216 IEEE ISIT 216, Barcelona, Spain 3

4 IEEE ISIT 216 Tutorial July 1, 216 Energy Harvesting and Remotely Powered Wireless Networks- Part I Aylin Yener yener@ee.psu.edu

5 Acknowledgements NSF support by the following grants: CNS964364, CCF , CNS , ERC(ASSIST) Collaborators: Omur Ozel, Kaya Tutuncuoglu, Sennur Ulukus, Jing Yang. 7/1/216 IEEE ISIT 216, Barcelona, Spain 5

6 Outline Aylin- Part I Introduction to energy harvesting (EH) Communication theory of EH the optimization set up Short term throughput maximization for single link with finite battery Transmission completion time minimization with finite battery Extension to fading channels Transmission policies for nodes with inefficient energy storage 7/1/216 IEEE ISIT 216, Barcelona, Spain 6

7 Introduction Ubiquitous Wireless Communications Mobile / Remote Energy-limited Green Energy Harvesting Wireless Networks Many sources Abundant energy Energy Harvesting 7/1/216 IEEE ISIT 216, Barcelona, Spain 7

8 Energy Harvesting Networks Wireless networking with rechargeable (energy harvesting) nodes: Green, self-sufficient nodes, Extended network lifetime, Smaller nodes with smaller batteries. 7/1/216 IEEE ISIT 216, Barcelona, Spain 8

9 What could EH bring to communications? 7/1/216 IEEE ISIT 216, Barcelona, Spain 9

10 Energy Harvesting Applications Communications satellites Space communications Deep space exploration IEEE ISIT 216, Barcelona, Spain 7/1/216 1

11 Wireless Energy Cooperation 7/1/216 IEEE ISIT 216, Barcelona, Spain 11

12 Energy Harvesting Applications Body area networks Heart sensor Personal access point Wearable Motion sensor IEEE ISIT 216, Barcelona, Spain 7/1/216 12

13 Energy Harvesting Applications KAIST s Solar charged textile battery MC1 s biostamps for medical monitoring, powered wirelessly Image Credits: (top) (bottom) ) 7/1/216 IEEE ISIT 216, Barcelona, Spain 13

14 Energy Harvesting Applications Fujitsu s hybrid device utilizing heat or light. Health tracker built at at the ASSIST Center at North Carolina State University, utilizing solar cells Image Credits: (top) (bottom) 7/1/216 IEEE ISIT 216, Barcelona, Spain 14

15 Energy Harvesting Applications In-body (intravascular) wireless devices Proteus Biomedical pills, powered by stomach acids Image Credits: (top) (middle) (bottom) 7/1/216 IEEE ISIT 216, Barcelona, Spain 15

16 What is in it for us? New: communication theory of EH nodes New: information theory of EH nodes Key new ingredient: A set of energy feasibility constraints based on harvests govern the communication resources. 7/1/216 IEEE ISIT 216, Barcelona, Spain 16

17 Communications New Wireless Network Design Challenge: A set of energy feasibility constraints based on harvests govern the communication resources. Design question: When and at what rate/power should a rechargeable (energy harvesting) node transmit? Optimality? Throughput; Delivery Delay Outcome: Optimal Transmission Schedules 7/1/216 IEEE ISIT 216, Barcelona, Spain 17

18 Two main metrics Short-Term Throughput Maximization (STTM): Given a deadline, maximize the number of bits sent before the end of transmission. Transmission Completion Time Minimization (TCTM): Given a number of bits to send, minimize the time at which all bits have departed the transmitter. 7/1/216 IEEE ISIT 216, Barcelona, Spain 18

19 ST Throughput Maximization [Tutuncuoglu-Yener 12] One Energy harvesting transmitter. Find optimal power allocation/transmission policy that departs maximum number of bits in a given duration T. Energy available intermittently. Up to a certain amount of energy can be stored by the transmitter BATTERY CAPACITY. 7/1/216 IEEE ISIT 216, Barcelona, Spain 19

20 System Model Energy harvesting transmitter: E max E i Data queue Energy queue transmitter receiver Transmitter has backlogged data to send by deadline T Energy arrives intermittently from harvester Stored in a finite battery of capacity E max 7/1/216 IEEE ISIT 216, Barcelona, Spain 2

21 System Model Energy arrivals of energy at times E i si E E 1 E 2 E 3 T s s 1 s 2 s 3 t Arrivals known non-causally by transmitter, Design parameter: power rate r( p). 7/1/216 IEEE ISIT 216, Barcelona, Spain 21

22 Power-Rate Function Transmission with power p yields a rate of r(p) Assumptions on r(p): i. r()=, r(p) as p ii. increases monotonically in p iii. strictly concave iv. r(p) continuously differentiable Rate r( p) Power Example: AWGN Channel, 1 r( p) log 1 2 p N 7/1/216 IEEE ISIT 216, Barcelona, Spain 22

23 Notations and Assumptions Power allocation function: p (t ) Energy consumed: T p ( t ) dt Short-term throughput: T r ( p ( t )) dt Concave rate in power Given a fixed energy, a longer transmission with lower power departs more bits. 7/1/216 IEEE ISIT 216, Barcelona, Spain 23

24 Battery Capacity: Energy Constraints (Energy arrivals of E i at times s i ) Energy Causality: n n n i t i s t s dt t p E ' ) ( 1 1 ' n n n i t i s t s E dt t p E ' ) ( 1 1 ' max n n n i t i s t s n E dt t p E t p ' ) ( ) ( 1 1 ' max,, Set of energy-feasible power allocations 7/1/216 IEEE ISIT 216, Barcelona, Spain 24

25 Energy Tunnel E c Energy Causality E 2 E max E 1 E Battery Capacity s s 1 2 t 7/1/216 IEEE ISIT 216, Barcelona, Spain 25

26 Optimization Problem Maximize total number of transmitted bits by deadline T Convex constraint set, concave maximization problem T t p t p t s dt t p r ) ( ) (.., )) ( ( max n n n i t i s t s n E dt t p E t p ' ) ( ) ( 1 1 ' max,, 7/1/216 IEEE ISIT 216, Barcelona, Spain 26

27 Necessary conditions for optimality of a transmission policy Property 1: Transmission power remains constant between energy arrivals. Let the total consumed energy in epoch s, ] be t [ i s i 1 which is available at.then the power policy p E s i1 total s s i i, t [ si, si 1] Etotal is feasible and better than a variable power transmission; shown easily using concavity of r(p) 7/1/216 IEEE ISIT 216, Barcelona, Spain 27

28 Necessary conditions for optimality Property 2: Battery never overflows. Proof: Assume Define Then T an p( t) energy r(p(t))dt p( t) p( t) T of r(p(t))dt overflows [, else since at ] r(p) time is increasing in p 7/1/216 IEEE ISIT 216, Barcelona, Spain 28

29 Necessary conditions for optimality of a transmission policy Property 3: Power level increases at an energy arrival instant only if battery is depleted. Conversely, power level decreases at an energy arrival instant only if battery is full. r(p (t))dt r(p(t))dt p (t) p*(t) p(t) Policy can be improved Policy cannot be improved 7/1/216 IEEE ISIT 216, Barcelona, Spain 29

30 Necessary conditions for optimality of a transmission policy Property 3: Power level increases at an energy arrival instant only if battery is depleted. Conversely, power level decreases at an energy arrival instant only if battery is full. p(t) p (t) r(p (t))dt r(p(t))dt p*(t) Policy can be improved Policy cannot be improved 7/1/216 IEEE ISIT 216, Barcelona, Spain 3

31 Necessary conditions for optimality of a transmission policy Property 4: Battery is depleted at the end of transmission. Proof: Define Then Assume T p( t) r(p(t))dt an p( t) energy T p( t) r(p(t)) dt of [ T remains, T else since ] r(p) after is p(t) increasing 7/1/216 IEEE ISIT 216, Barcelona, Spain 31

32 Implications of the properties [Tutuncuoglu-Yener 12] Structure of optimal policy is piece-wise linear. p( t) p n i n1 t t T i n, i n { s n }, p n constant For power to increase or decrease, policy must meet the upper or lower boundary of the tunnel respectively. At termination step, battery is depleted. Utilizing this structure, a recursive algorithm emerges to find the unique optimum policy [Tutuncuoglu-Yener 12]. 7/1/216 IEEE ISIT 216, Barcelona, Spain 32

33 Energy Tunnel E c Energy Causality E 2 E max E 1 E Battery Capacity s s 1 2 t 7/1/216 IEEE ISIT 216, Barcelona, Spain 33

34 Shortest Path Interpretation Optimal policy is identical for any concave power-rate function! Let r( p) p 2 1, then the problem solved becomes: max p 2 ( t) 1 dt s. t. p( t) p( t) min p 2 ( t) 1 dt s. t. p( t) p( t) T T length of policy path in energy tunnel The throughput maximizing policy yields the shortest path through the energy tunnel for any concave power-rate function. 7/1/216 IEEE ISIT 216, Barcelona, Spain 34

35 Shortest Path Interpretation Property 1: Constant power is better than any other alternative Shortest path between two points is a line (constant slope) E t 7/1/216 IEEE ISIT 216, Barcelona, Spain 35

36 Alternative Solution (Using Property 1) Transmission power is constant within each epoch: p( t) { pi, t epoch i, i 1,,N } (N: Number of arrivals within [,T]) max p i N i1 L i. r( p i ) (L i : length of epoch i) s. t. n i1 E i L i p i E max n 1,..., N KKT conditions optimum power policy. 7/1/216 IEEE ISIT 216, Barcelona, Spain 36

37 1 max 1 n i i i i n n i i i i n E p L E E p L 's only positive only when battery is full are positive only when battery is empty 's Solution Complementary Slackness Conditions: n E p L E n E p L n i i i i n n i i i i n 1 max 1 n n N n j j j p n decreases with positive increases with positive 1 ) ( 1 * 7/1/216 IEEE ISIT 216, Barcelona, Spain 37

38 Directional Water-Filling [Ozel, Tutuncuoglu, Ulukus, Yener 11] Harvested energies filled into epochs individually E E1 E2 Water levels (v i ) O O O t 7/1/216 IEEE ISIT 216, Barcelona, Spain 38

39 Directional Water-Filling Harvested energies filled into epochs individually Constraints: Energy Causality: water-flow only forward in time E E1 E2 Water levels (v i ) O O O t 7/1/216 IEEE ISIT 216, Barcelona, Spain 39

40 Directional Water-Filling Harvested energies filled into epochs individually Constraints: Energy Causality: water-flow only forward in time Battery Capacity: water-flow limited to E max by taps E E1 E2 Water levels (v i ) O O O t 7/1/216 IEEE ISIT 216, Barcelona, Spain 4

41 Example E 2 E 1 5 E 1 E 9 E E max 1 p s 1 s2 s3 s4 7/1/216 IEEE ISIT 216, Barcelona, Spain 41

42 Directional Water-Filling E E 5 E 4 Energy tunnel E E E2 E1 E 3 E1 E2 E3 E4 E5 t and directional water-filling approaches yield the same policy O O O O O O t 7/1/216 IEEE ISIT 216, Barcelona, Spain 42

43 Directional Water-Filling E Energy tunnel t and directional water-filling approaches yield the same policy O O O O O O t 7/1/216 IEEE ISIT 216, Barcelona, Spain 43

44 Simulation Results Improvement of optimal algorithm over an on-off transmitter in a simulation with truncated Gaussian arrivals. 7/1/216 IEEE ISIT 216, Barcelona, Spain 44

45 Fading Channels [Ozel-Tutuncuoglu-Ulukus-Yener 11] Fading levels h 1 h 2 =h 3 =h 4 h 5 h 6 =h 7 h 8 E E2 E3 E6 E7 E 1,4,5,.. O x O O x x O x O t L 1 L 4 L 7 AWGN Channel with fading h : 1 r( p, h) log(1 hp ) 2 Each epoch defined as the interval between two events. 7/1/216 IEEE ISIT 216, Barcelona, Spain 45

46 Directional Water-Filling for Fading Channels Same directional water filling with base levels adjusted according to channel quality. Directional water flow (Energy causality) Limited water flow (Battery capacity) E E2 E4 Water levels (v i ) Fading levels (1/h i ) O x O x O t 7/1/216 IEEE ISIT 216, Barcelona, Spain 46

47 Transmission Completion Time Minimization (TCTM) [Yang-Ulukus 12] Given the total number of bits to send as B, complete transmission in the shortest time possible. min p ( t ) T s. t. B T r( p( t)) dt, p( t) n1 t' p( t) E k p( t) dt Emax, n, sn 1 t' sn k 7/1/216 IEEE ISIT 216, Barcelona, Spain 47

48 Relationship of STTM and TCTM Lagrangian dual of TCTM problem becomes: max u min p ( t ) P, T T u B T r( p( t)) dt max u min T T ub u.max p(t )P T r(p(t))dt STTM problem for deadline T 7/1/216 IEEE ISIT 216, Barcelona, Spain 48

49 Relationship of STTM and TCTM Optimal allocations are identical: STTM s solution for deadline T departing B bits TCTM s solution for departing B bits in time T STTM solution can be used to solve the TCTM problem IEEE ISIT 216, Barcelona, Spain 7/1/216 49

50 Maximum Service Curve s( T ) max p ( t ) T r( p( t)) dt, s. t. p( t) Maximum Departure (B) Maximum number of bits that can be sent in time T. Each point calculated by solving the corresponding STTM problem. S1 S2 S 3 Deadline (T) 7/1/216 IEEE ISIT 216, Barcelona, Spain 5

51 Maximum Service Curve Continuous, monotone increasing, invertible Maximum Departure (B) B 1 Optimal allocation for TCTM with B 1 bits Optimal allocation for STTM with deadline T 1 S1 S2 T 1 S 3 Deadline (T) 7/1/216 IEEE ISIT 216, Barcelona, Spain 51

52 Maximum Service Curve: Fading Maximum Departure (B) B 1 Continuous, non-decreasing (flat regions when fading is severe) Inverse can be considered as the smallest T that achieves B 1 T 1 Deadline (T) O x x x x x x O O O 7/1/216 IEEE ISIT 216, Barcelona, Spain 52

53 Transmission Policies with Inefficient Energy Storage Energy stored in a battery, supercapacitor,... Real life issues: Storage Loss Leakage Retrieval Loss Degradation [Devillers-Gunduz 12]: Leakage and Degradation [Tutuncuoglu-Yener-Ulukus 15]: Storage and Retrieval Losses 7/1/216 IEEE ISIT 216, Barcelona, Spain 53

54 Battery Degradation [Devillers-Gunduz 12] Degradation Optimal Policy: Shortest path within narrowing tunnel E 7/1/216 IEEE ISIT 216, Barcelona, Spain 54 t

55 Battery Leakage [Devillers-Gunduz 12] Leakage Optimal Policy: When total energy in an epoch is low, deplete energy earlier to reduce leakage. E 7/1/216 IEEE ISIT 216, Barcelona, Spain 55 t

56 Storage/Recovery Losses [Tutuncuoglu-Yener-Ulukus 15] Main Tension: Storage Loss Recovery Loss Concavity of r(p): Use battery to maintain a constant power transmission Battery inefficiency: Storing energy in battery causes energy loss 7/1/216 IEEE ISIT 216, Barcelona, Spain 56

57 Time slotted model E 1 E 2 E 3 E N-1 i 1 i 2... i N 2... ( N 1) N t Time slots of duration 1 s Energy harvests: Size E i at the beginning of time slot i All arrivals known by transmitter beforehand. 7/1/216 IEEE ISIT 216, Barcelona, Spain 57

58 System Model Energy storage (ESD) E max s i η u i Rate: r(p(t)) h i : Harvested power s i : Stored power u i : Retrieved (used) power p i : Transmit power h i Transmitter Receiver p i = h i s i + u i ESD has finite capacity E max and storage efficiency η. Energy Causality: Storage Capacity: i n1 i n1 sn un, i 1,..., N sn un E, i 1,..., N max 7/1/216 IEEE ISIT 216, Barcelona, Spain 58

59 Find optimal energy storage policy that maximizes the average throughput of an energy harvesting transmitter within a deadline of N time slots. Throughput Maximization., 1,,,,,, 1,, ) (.. ) ( max max 1 1, N i u s u s E N i E u s E s t u s E r i i i i i i n i i N i i i i r s i i 7/1/216 IEEE ISIT 216, Barcelona, Spain 59

60 Throughput Maximization., 1,,,,,, 1,, ) (.. ) ( max max 1 1, N i u s u s E N i E u s s t u s E r i i i i i i n i i N i i i i r s i i., 1,,,, 1,, ) (.. ) ( max max 1 1 N i p N i E p E s t p r i i n i i N i i p i Old problem: 7/1/216 IEEE ISIT 216, Barcelona, Spain 6

61 Structure of optimal policy:,,,,,, i u i i u i s i i u i i s i i s i p E p p E p E p E p p Optimal Power Policy p i i u p, i s p, E i Double Threshold Policy 7/1/216 IEEE ISIT 216, Barcelona, Spain 61

62 Optimal Power Policy p i E i * p s,i p i * * p s,i * p u,i * p u,i i 1 i 2 i 3 i 4 i 5 7/1/216 IEEE ISIT 216, Barcelona, Spain 62

63 Optimal Power Policy (Fading channel) p i * p s,i E i * p s,i 1 * p u,i h i * p,i i 1 i 2 i 3 i 4 i 5 7/1/216 IEEE ISIT 216, Barcelona, Spain 63

64 Simulations N E E E ~ i.i.d. U[, 2] J N max i 1 4 τ 1 ms 1 mj h 1 db B 1 MHz 1 time slots 19 W/Hz 7/1/216 IEEE ISIT 216, Barcelona, Spain 64

65 References-Part I [Yang-Ulukus 12] Jing Yang and Sennur Ulukus, Optimal Packet Scheduling in an Energy Harvesting Communication System, IEEE Trans. on Communications, January 212. [Tutuncuoglu-Yener 12] Kaya Tutuncuoglu and Aylin Yener, Optimum Transmission Policies for Battery Limited Energy Harvesting Nodes, IEEE Trans. Wireless Communications, 11(3): , March 212. [Ozel et. al. 11] Omur Ozel, Kaya Tutuncuoglu, Jing Yang, Sennur Ulukus and Aylin Yener, Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies, IEEE Journal on Selected Areas in Communications: Energy-Efficient Wireless Communications, 29(8): ,September 211. [Tutuncuoglu-Yener-Ulukus 15] Kaya Tutuncuoglu, Aylin Yener, and Sennur Ulukus, Optimum Policies for an Energy Harvesting Transmitter Under Energy Storage Losses, IEEE Journal on Sel. Areas in Communications: Wireless Communications Powered by Energy Harvesting and Wireless Energy Transfer, 33(3), pp , Mar [Devillers-Gunduz 12]: Bertrand Devillers and Deniz Gunduz, A general framework for the optimization of energy harvesting communication systems with battery imperfections, Journal of Communications and Networks, 14(2):13 139, April 212. IEEE ISIT 216, Barcelona, Spain 7/1/216 11

66 Online Power Control for Energy Harvesting Nodes Ayfer Özgür Tutorial on Energy Harvesing and Remotely Powered Communication ISIT 216, Barcelona, Spain Ayfer Özgür Online Power Control July 16 1 / 23

67 Offline Power Control for Energy Harvesting Nodes An Alternative Formulation E t E max Battery N t N (, 1) Transmitter X t Y t Receiver E t : i.i.d. energy harvesting process, can be continuous or discrete, its realization is known ahead of time. Power Control Problem: 1 T = sup lim inf g n n E [ n t=1 ] 1 2 log(1 + γg t), where g t : E n R +, t = 1,... n is a power control policy that satisfies : g t b t b t+1 = min(b t g t + e t+1, E max ) Ayfer Özgür Online Power Control July 16 2 / 23

68 Offline Setting Optimal Solution: Ensure battery never overflows. Allocate energy as equally as possible over time. Ayfer Özgür Online Power Control July 16 3 / 23

69 Online Setting Energy arrivals are known causally: g t : E t R +, t = 1,... n Easy to observe that this a Markov Decision Process: state b t state space [, E max ] action g t action space [, b t ] disturbance E t disturbance distribution p(e) or f (e) state evolution b t+1 = min(b t g t + e t+1, E max ) stage reward 1 2 log(1 + γg t) Ayfer Özgür Online Power Control July 16 4 / 23

70 Markov Decision Processes s t+1 = f (s t, u t, w t ) state s t state space S action u t action space U(s t ) disturbance w t disturbance distribution p(w s, u) history h t = (s 1, w 1, w 2,..., w t 1 ) policy π = {µ 1, µ 2,...}, u t = µ t (h t ) reward g(s t, u t ) Goal: maximize average reward 1 J = sup lim inf π n n n E [ g ( S t, µ t (H t ) )] t=1 Ayfer Özgür Online Power Control July 16 5 / 23

71 A dynamic programming approach Bellman Equation If there exists a scalar λ R + and a bounded function h : [, E max ] R + that satisfy { λ + h(b) = sup 1 2 log(1 + γg) + E[h(min{b g + E t, E max })] } g b for all b E max, then the optimal policy is given by g t (E t ) = g (b t (E t )). Limitations: can be computationally demanding; solution depends on the exact statistical model of energy arrivals; no insight on the structure of the optimal policy and the qualitative behavior of the resultant throughput; Ayfer Özgür Online Power Control July 16 6 / 23

72 Heuristic Online Policies Either no or only asymptotic guarantees on performance. Two natural heuristics widely considered: greedy policy and constant policy. Greedy policy: instantenously uses all the incoming energy; ensures no battery overflow; becomes optimal when SNR : 1 n n t=1 1 2 log(1 + γg t) γ 1 2 n n t=1 g t Ayfer Özgür Online Power Control July 16 7 / 23

73 Constant Policy keep power allocation as constant as possible over time; { µ = E[E t ] if b t µ g t = b t if b t < µ. becomes optimal when E max : T = 1 log(1 + γµ). 2 Ayfer Özgür Online Power Control July 16 8 / 23

74 For finite parameter values these schemes can be arbitrarily away from optimality. asymptotic results provide no insights about the gap to optimality. which of the previous two policies is a better choice for a given problem? Ayfer Özgür Online Power Control July 16 9 / 23

75 A constant gap approach Look for policies that are provably close to optimal across all parameter regimes and any distribution of the energy arrivals. Universal near-optimal policies: have minimal dependence on the distribution of the energy arrivals, e.g depend only on the mean. achieve the optimal throughput simultaneously within a constant additive and multiplicative gap for all parameter values and distributions of energy arrivals. Ayfer Özgür Online Power Control July 16 1 / 23

76 Wireless information theory over the last 15 years Degrees of Freedom Generalized Degrees of Freedom Constant Gap Approximations Ayfer Özgür Online Power Control July / 23

77 Starting Point: Bernoulli Arrivals E t E max Battery N t N (, 1) Transmitter X t Y t Receiver Ayfer Özgür Online Power Control July / 23

78 Starting Point: Bernoulli Arrivals E t E max Battery N t N (, 1) Transmitter X t Y t Receiver First, we focus on i.i.d. Bernoulli energy arrival process: { Emax w.p. p E t = w.p. 1 p, Ayfer Özgür Online Power Control July / 23

79 Bernoulli Battery Recharges Law of large numbers for regenerative processes: sup g [ n ] 1 lim inf n n E 1 log(1 + g(t)) = sup 2 g t=1 = max {E i } i=1 : E i i i=1 E i E max i=1 The optimal power control policy: E i = { (N+Emax ) p(1 p) i log(1 + E i) [ L ] 1 EL E 1 2 log(1 + γg(b t)) t=1 p(1 p) i 1 1, i = 1,..., N 1 (1 p) N, i > N where N is the smallest positive integer satisfying 1 > (1 p) N [1 + p(e max + N)]. Ayfer Özgür Online Power Control July / 23

80 Bernoulli Battery Recharges Law of large numbers for regenerative processes: sup g [ n ] 1 lim inf n n E 1 log(1 + g(t)) = sup 2 g t=1 = max {E i } i=1 : E i i i=1 E i E max i=1 The optimal power control policy: E i = { (N+Emax ) p(1 p) i log(1 + E i) [ L ] 1 EL E 1 2 log(1 + γg(b t)) t=1 p(1 p) i 1 1, i = 1,..., N 1 (1 p) N, i > N where N is the smallest positive integer satisfying 1 > (1 p) N [1 + p(e max + N)]. Ayfer Özgür Online Power Control July / 23

81 Exponentially decreasing power allocations L Because rate is a concave function of energy/power, allocate the energy as equally as possible across time. Use p fraction of the available energy at each time slot: g t = pb t Ayfer Özgür Online Power Control July / 23

82 Exponentially decreasing power allocations L Because rate is a concave function of energy/power, allocate the energy as equally as possible across time. Use p fraction of the available energy at each time slot: g t = pb t g t = p(1 p) j E max where j = t max{t t : E t = E max }. Ayfer Özgür Online Power Control July / 23

83 Simplified policy for Bernoulli Arrivals Fixed Fraction Policy: g t = pb t Theorem Let E t be i.i.d Bernoulli(p, E max ) as before. The throughput T FF achieved by the constant fraction policy satisfies T FF 1 2 log(1 + γ pe max).72, and T FF log(1 + γ pe max). Ayfer Özgür Online Power Control July / 23

84 Simulation Throughput Upper Bound 1 2 log(1+γµ) Fixed Fraction gt = qbt Greedy gt = bt Constant gt = µ 1{bt µ} Optimal Θ Gap Fixed Fraction gt = qbt Greedy gt = bt Constant gt = µ 1{bt µ} E max E max E t is Bernoulli(, E max ) with p =.1. Ayfer Özgür Online Power Control July / 23

85 Simulation Throughput Upper Bound 1 2 log(1+γµ) Fixed Fraction gt = qbt Greedy gt = bt Constant gt = µ 1{bt µ} Optimal Θ Gap Fixed Fraction gt = qbt Greedy gt = bt Constant gt = µ 1{bt µ} E max E max E t is Bernoulli(, E max ) with p =.9. Ayfer Özgür Online Power Control July / 23

86 General i.i.d. energy arrivals Fixed Fraction Policy: g t = qb t, where q = µ/e max Theorem The throughput T FF achieved by the constant fraction policy satisfies T FF 1 log(1 + γµ).72, 2 and T FF 1 1 log(1 + γµ). 2 2 Ayfer Özgür Online Power Control July / 23

87 Proof idea Theorem Bernoulli(, E max ) is the worst case among all distributions with the same mean. Previous heuristics: the greedy policy and the constant policy build on the insights from the best case scenario: E t = µ deterministically. The fixed fraction policy builds on the insights from the worst case scenario: Bernoulli arrivals. Ayfer Özgür Online Power Control July / 23

88 Simulation Throughput Upper Bound 1 2 log(1+γµ) Fixed Fraction gt = qbt Greedy gt = bt Constant gt = µ 1{bt µ} Optimal Θ Gap Fixed Fraction gt = qbt Greedy gt = bt Constant gt = µ 1{bt µ} E max E max E t is Exponential(1/.1 E max ). Ayfer Özgür Online Power Control July 16 2 / 23

89 Open Questions and Directions Is Bernoulli the worst case for the optimal policy? Non i.i.d. energy arrivals. Fading Channels. Battery Imperfections. Multi-user Settings (to be discussed in the next part). Ayfer Özgür Online Power Control July / 23

90 Acknowledgement CCF , Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant agreement CCF , and Stanford SystemX. Slides: Dor Shaviv. Ayfer Özgür Online Power Control July / 23

91 References D. Shaviv, A Ozgur, Universally Near Optimal Power Control for Energy Harvesting Nodes, ICC 216; longer version available on ArXiv. A. Kazerouni and A. Ozgur, Optimal online strategies for an energy harvesting system with Bernoulli energy recharges, WiOpt, 215. Y. Dong, F. Farnia, and A. Ozgur, Near Optimal Energy Control and Approximate Capacity of Energy Harvesting Communication, JSAC Special Issue on Wireless Communications Powered by Energy Harvesting and Wireless Energy Transfer, 33(3):54-557, March 215. Ayfer Özgür Online Power Control July / 23

92 Offline Multi-user Energy Harvesting Settings Online Multi-user Energy Harvesting Settings Energy Cooperation in Energy Harvesting Networks Şennur Ulukuş Department of ECE University of Maryland 1

93 So Far, We Learned... Wireless nodes harvesting energy from nature. Single-user communication with an energy harvesting transmitter. Energy arrives (is harvested) during the communication session. A non-trivial shift from the conventional battery powered systems. Transmission policy is adapted to energy arrivals. Objective: maximize throughput, minimize delay. E i data queue Tx Rx 2

94 Single-User Optimal Policy for E max = E E 1 E 2 E 3 E 4 E 5 cumulative energy arrivals i j=1 E j Optimal power level T t Find the tightest curve under the cumulative energy arrival staircase. 3

95 Single-User Optimal Policy for E max < E E 1 E 2 E 3 E 4 E 5 cumulative energy arrivals i j=1 E j E max T t Find the tightest curve in the energy feasibility tunnel. 4

96 Equivalence of Feasibility Tunnel and Directional Water-filling E E 1 E 2 E 3 E 4 E 5 E 6 T T OFF OFF OFF OFF OFF OFF OFF T 5

97 Equivalence of Feasibility Tunnel and Directional Water-filling E E 1 E 2 E 3 E 4 E 5 E 6 T tightest curve ON ON ON ON ON ON ON T causality no overflow T 6

98 Optimal Packet Scheduling: Broadcast Channel E i data queue 1 Rx 1 data queue 2 Tx Rx 2 Energy arrives (is harvested) during the communication session Assume battery has infinite storage capacity: E max = Broadcasting data to two users by adapting to energy arrivals Objective: maximize the data departure region 7

99 Broadcast Channel Model: E max = E i data queue 1 Rx 1 data queue 2 Tx Rx 2 AWGN broadcast channel: where N 1 N(,σ 2 1 ), N 2 N(,σ 2 2 ) Y 1 = X+ N 1, Y 2 = X+ N 2 σ 2 2 > σ2 1 : 2nd user is degraded; we call 1st user stronger and 2nd user weaker 8

100 Broadcast Channel Model R 2 C 2 Broadcast capacity region: r log 2 We work in the (r 1,r 2 ) domain: C 1 ( 1+ αp ) σ 2, r 2 1 ( 1 2 log 2 1+ (1 α)p ) αp+σ 2 2 P=σ (r 1+r 2 ) +(σ 2 2 σ 2 1)2 2r 2 σ 2 2 g(r 1,r 2 ) g(r 1,r 2 ) is the minimum power required to send at rates(r 1,r 2 ) R 1 9

101 Finding the Maximum Departure Region The maximum departure region D(T): union of(b 1,B 2 ) pairs achievable by some rate allocation policy that satisfies the energy causality constraint. T l 1 l 2 l N l N+1 (r 11,r 21 ) (r 12,r 22 ) (r 1N,r 2N ) (r 1(N+1),r 2(N+1) ) Transmission rates, and power, remain constant between energy harvests The energy causality constraint reduces to constraints on (r 1i,r 2i ): k g(r 1i,r 2i )l i i=1 k 1 E i, k=1,...,n+ 1 i= 1

102 Finding the Maximum Departure Region D(T) is a strictly convex region. Characterize D(T) by solving optimization problems for all µ 1,µ 2 : N+1 max µ 1 r 1,r 2 s.t. i=1 N+1 r 1i l i + µ 2 r 2i l i i=1 k g(r 1i,r 2i )l i i=1 k 1 E i, k=1,...,n+ 1 i= B 2 (µ 1, µ 2 ) D(T) B 1 11

103 Structure of the Optimal Policy Total transmit power is the same as the single-user case. The power shares follow a cut-off structure. Cut-off level P c P c =( µ1 σ 2 2 µ 2σ 2 1 µ 2 µ 1 If total power is below P c, then, only transmit to the stronger user. Otherwise, stronger user s power share is P c. P c (share of the stronger user) decreases with µ 2, the priority of the weaker user. ) + 12

104 The Structure of the Optimal Policy for E max = E E 1 E 2 E 3 E 4 E 5 E 6 T i j= E j T P P c T 13

105 The Structure of the Optimal Policy for E max < E E 1 E 2 E 3 E 4 E 5 E 6 T E max E max E max E max E max E max E max E E 1 E 2 E 3 E 4 E 5 E 6 P T P c T 14

106 Conclusions for the Offline Broadcasting Scenario Energy harvesting transmitter with infinite and finite capacity battery Maximize the departure region. Obtain the structure of the solution, such as: the monotonicity of the transmit power the cut-off power property 15

107 Optimal Packet Scheduling: Multiple Access Channel AWGN MAC channel Y = X 1 + X 2 + Z, Z N(,1). The capacity region is a pentagon denoted as C(P 1,P 2 ): R 1 f(p 1 ), R 2 f(p 2 ), R 1 + R 2 f(p 1 + P 2 ) where f(p)= 1 2 log(1+ p). E 1i R 2 C s data queue Tx 1 C 2 E 2i Rx data queue Tx 2 C 1 C s R 1 16

108 Problem Formulation Maximize the departure region D(T) by time T. E 1 E 11 E E 1(K 1) B 1 t s 1 s 2 s 3 s 4 s K 1 s K T E 2 E 23 E E 2K B 2 t s 1 s 2 s 3 s 4 s K 1 s K T B 2 a b c d e f B 1 Each feasible policy gives a pentagon. Union of all feasible policies gives D(T). 17

109 Characterizing D(T) Transmission rate remains constant between energy harvests. For any feasible transmit power sequences p 1, p 2, the departure region is a pentagon B 1 B 2 B 1 + B 2 D(T) is a union of(b 1,B 2 ) and convex. N f(p 1n )l n n=1 N f(p 2n )l n n=1 N f(p 1n + p 2n )l n n=1 The boundary points maximize µ 1 B 1 + µ 2 B 2 for some µ 1,µ 2. 18

110 Point a Single-user power allocation. r 2 a b c d e f r 1 max p 1 s.t. n f(p 1n )l n j j 1 p 1n l n E 1n, j : < j N n=1 n= 19

111 Point b User 2 power is fixed to its single-user power allocation. r 2 a b c d e f r 1 max p 1 s.t. n f(p 1n + p 2n )l n j j 1 p 1n l n E 1n, j : < j N n=1 n= 2

112 Sum-rate: Points between c and d Maximize the sum-rate of the users. r 2 a b c d e f r 1 max p 1,p 2 s.t. n f(p 1n + p 2n )l n j j 1 p 1n l n E 1n, j : < j N n=1 n= j j 1 p 2n l n E 2n, j : < j N n=1 n= 21

113 Sum-Rate: Points between c and d, Equivalent Problem Equivalent problem: max p 1 +p 2 s.t. n f(p 1n + p 2n )l n j j 1 p 1n l n + p 2n l n E 1n + E 2n, j : < j N n=1 n= Power can be divided back to p 1,p 2 in infinite number of ways. r 2 a b c d e f r 1 22

114 Points between b and c: Arbitrary µ 1, µ 2 Each boundary point corresponds to a corner point of some pentagon. µ 2 > µ 1 achieving points between point b and point c: r 2 a b c d e f r 1 max (µ 2 µ 1 p 1,p 2 ) n s.t. f(p 2n )l n + µ 1 n f(p 1n + p 2n )l n j j 1 p 1n l n E 1n, j : < j N n=1 n= j j 1 p 2n l n E 2n, j : < j N n=1 n= 23

115 Generalized Iterative Backward Waterfilling Solve the problem via generalized iterative backward waterfilling: Given p 1, solve for p 2: max (µ 2 µ 1 ) p 2 s.t. N n=1 f(p 2n )l n + µ 1 N n=1 j j 1 p 2n l n E 2n, < j N n=1 n= f(p 1n + p 2n)l n Once p 2 is obtained, we do a backward waterfilling for the second user. We perform the optimization for both users in an alternating way. The iterative algorithm converges to the global optimal solution. 24

116 Conclusions for the Offline Multiple Access Scenario Energy harvesting transmitters sending messages to a single access point. The problem: maximization of the departure region. Obtain the structure using generalized iterative waterfilling. 25

117 So Far, We Learned... E 1 E E 3 E 2 only causally known cumulative energy arrivals cumulative energy expenditure t t So far, mostly: dynamic programming, learning algorithms, heuristics. 26

118 A Unique Approach: Online Power Scheduling Feel the Bernoulli. Steps of the approach: Study Bernoulli energy arrivals with{,b} support. Propose a simple sub-optimal policy for Bernoulli arrivals. Bound its performance. Extend this sub-optimal policy for general energy arrivals. Bernoulli is the worst energy arrival for the proposed algorithm. Obtain a near-optimal online power policy. rate general-ffp.72 B B E i general-optimal upper bound Bernoulli-optimal Bernoulli-FFP lower bound data queue Tx Rx B 27

119 Online Policy for the Single-User Channel Bernoulli energy arrivals: B B E i data queue Tx Rx P[E i = B]=1 P[E i = ]= p When an energy arrives, a renewal occurs. 28

120 Long-Term Average Throughput Using Renewal Theory Long-term average throughput, under Bernoulli energy arrivals: [ ] n lim E 1 1 L ] n n i=1 2 log(1+p i) = 1 E[L] E[ 1 i=1 2 log(1+p i) = p = k=1 i=1 k=i L is inter-energy arrival time, geometric withe[l]= 1 p. = p(1 p) k 1 k 1 i=1 2 log(1+p i) p 2 (1 p) k log(1+p i) p(1 p) i 1 1 i=1 2 log(1+p i) 29

121 Resulting Optimization Problem To characterize{p i } which achieves the maximum, we solve max {P i } s.t. i=1 p(1 p) i log(1+p i) P i B i=1 P i, i Solution: P i = p(1 p)i 1 λ 1, i=1,...,ñ Decreasing power for a finite duration Ñ that depends on B. 3

122 Online Policy for the Single-User Channel Bernoulli energy arrivals: Optimal power allocation with Ñ = 4: B B E i data queue Tx Rx Sub-optimal fractional power allocation, P i = Bp(1 p) i 1 : B B E i data queue Tx Rx 31

123 Upper bound from offline policy: Bounds on the Online Policies r 1 2 log(1+µ) Lower bound algebraically for Bernoulli arrivals: Sketch of the proof: r 1 2 log(1+µ).72 L ] r= 1 E[L] E[ 1 i=1 2 log( 1+Bp(1 p) i 1) L 1 E[L] E[ 1 i=1 2 log(1+bp)+ 1 2 log( (1 p) i 1) Bernoulli is the worst energy arrival for the fractional policy: ] 1 2 log(1+µ).72 T upper.72 T Bern T any T upper 32

124 Bernoulli energy arrivals: Online Policies for the Broadcast Channel B E i B data queue 1 Rx 1 data queue 2 Tx Rx 2 P[E i = B]=1 P[E i = ]= p When an energy arrives, a renewal occurs. 33

125 Long-Term Weighted Average Throughput Long-term weighted average throughput, under Bernoulli energy arrivals: [ ] n lim E 1 L ] n n (µ 1 r 1i + µ 2 r 2i ) = 1 i=1 E[L] E[ (µ 1 r 1i + µ 2 r 2i ) i=1 = p = = k=1 i=1 k=i p(1 p) k 1 k (µ 1 r 1i + µ 2 r 2i ) i=1 p 2 (1 p) k 1 (µ 1 r 1i + µ 2 r 2i ) p(1 p) i 1 (µ 1 r 1i + µ 2 r 2i ) i=1 34

126 Resulting Optimization Problem for the Broadcast Channel Problem becomes where max {r 1i,r 2i } s.t. i=1 p(1 p) i 1 (µ 1 r 1i + µ 2 r 2i ) g(r 1i,r 2i ) B i=1 r 1i,r 2i, i P i = σ 2 1 e2(r 1i+r 2i ) +(σ 2 2 σ2 1 )e2r 2i σ 2 2 g(r 1i,r 2i ) Modified offline problem: One energy arrival. Generalized fading due to p(1 p) i 1 35

127 Structure of the Optimal Online Policy P i user 1 user 2 P 1 P 2 P c P 3 P M Ñ P 5 slots (i) User 1 is served for a time no shorter than user 2. Both users powers are decreasing. Cut-off level P c : P c =( µ1 σ 2 2 µ 2σ 2 1 µ 2 µ 1 ) + 36

128 Proposed Sub-optimal Policy for Bernoulli Energy Arrivals P i = pb i user 1 user 2 P 1 P c P 2 P 3 P P 5 slots (i) Sub-optimal fractional total power policy: Total power per slot: P i = P 1i + P 2i = pb i = Bp(1 p) i 1 Optimally divided power according to cut-off: P 1i = min{p c,bp(1 p) i 1 } P 2i = Bp(1 p) i 1 P 1i 37

129 Proposed Sub-optimal Policy for General Energy Arrivals P i = qb i user 1 user 2 P 1 P2 P c P 3 P 5 P 6 P slots (i) Defining q=µ/b. Total power per slot: P i = qb i Optimally divided power according to cut-off: P 1i = min{p c,qb i } P 2i = qb i P 1i 38

130 Bounds on the Online Policies Bernoulli energy arrivals gives a lower bound for general energy arrivals. Lower bound: r 1 1 ( 2 log 1+ αµ ) σ r log ( 1+ (1 α)µ αµ+σ 2 2 ).99 Upper bound: r 1 1 ( 2 log 1+ αµ ) σ 2 1 r 2 1 ( 2 log 1+ (1 α)µ ) αµ+σ 2 2 for some α [,1], where µ=e[e i ] is the average recharge rate. 39

131 Illustration of Bounds r 2 upper bound optimal policy a = b fractional policy (FPCC) lower bound r 1 Distance between any two points with the same α on the upper and lower bounds is equal to: =

132 Conclusions for the Online Broadcasting Scenario Energy harvesting transmitter with finite capacity battery Maximize the departure region. Obtain the structure of the solution, such as: the monotonicity of the transmit power the cut-off power property Near-optimal policy. 41

133 Multiple Access Channel with Common Source Bernoulli energy arrivals: E i B 1 B common energy source data queue Tx 1 E i B 2 Rx data queue Tx 2 P[E i = B]=1 P[E i = ]= p, where B max{b 1,B 2 }. Average admitted energies at the two users are not the same. When an energy arrives, a renewal occurs. 42

134 Long-Term Weighted Average Throughput Long-term weighted average throughput, under Bernoulli energy arrivals: [ ] n lim E 1 L ] n n (µ 1 r 1i + µ 2 r 2i ) = 1 i=1 E[L] E[ (µ 1 r 1i + µ 2 r 2i ) i=1 = p = = k=1 i=1 k=i p(1 p) k 1 k (µ 1 r 1i + µ 2 r 2i ) i=1 p 2 (1 p) k 1 (µ 1 r 1i + µ 2 r 2i ) p(1 p) i 1 (µ 1 r 1i + µ 2 r 2i ) i=1 43

135 Online Policies for the Multiple Access Channel with Common Source For Bernoulli energy arrivals: max {P 1i,P 2i p(1 p) i 1 (µ 1 r 1i + µ 2 r 2i ) } i=1 s.t. (r 1i,r 2i ) C(P 1i,P 2i ) P 1i B 1, i=1 P 2i B 2 i=1 where C(P 1i,P 2i ) of this channel in slot i is: r 1i 1 ( 2 log 1+ P ) 1i σ 2 r 2i 1 ( 2 log 1+ P ) 2i σ 2 r 1i + r 2i 1 ( 2 log 1+ P ) 1i+ P 2i σ 2 Modified offline problem: One energy arrival. Generalized fading due to p(1 p) i 1 44

136 Online Policies for the Multiple Access Channel with Common Source Achievable rate region r 2 a b c d e f r 1 Each feasible policy achieves a pentagon Rate region is the union of all such pentagons Points a and f are single-user rates 45

137 Online Policies for the Multiple Access Channel with Common Source Achievable rate region r 2 a b c d e f r 1 Each feasible policy achieves a pentagon Rate region is the union of all such pentagons Points a and f are single-user rates 46

138 Point b User 2 power is fixed to: P 2i = p(1 p)i 1 λ 2 σ 2, i=1,...,ñ 2 Optimization problem becomes: max {P 1i } p(1 p) i 1 r 1i i=1 s.t. r 1i C(P 1i,P 2i ), i=1p 1i B 1 The optimal power: P 1i = p(1 p)i 1 λ 1 ν 1i σ 2 P 2i At point b, user 1 transmits for a duration no shorter than user 2. Power of both users are monotonically decreasing. 47

139 Sum-Rate r 2 a b c d e f r 1 48

140 Sum-Rate µ 1 = µ 2 = 1 The optimization problem becomes: max {P 1i,P 2i } s.t. 1 2 i=1 ( p(1 p) i 1 log 1+ P ) 1i+ P 2i σ 2 P 1i B 1, i=1 P 2i B 2 i=1 A relaxed problem: Equivalent problems. max {P 1i,P 2i } s.t. 1 2 i=1 ( p(1 p) i 1 log 1+ P ) 1i+ P 2i σ 2 P 1i + P 2i B 1 + B 2 i=1 Use P 1i =(P 1i + P 2i ) B 1 B 1 +B 2 Hence, solve a single-user problem for(p 1i + P 2i ). 49

141 Sum-Rate (P 1i + P 2i ) is positive for a duration Ñ s max{ñ 1,Ñ 2 } It is sufficient to show that: (P 1i + P 2i ) P2i Implies that the single-user power allocation is feasible r 2 a b c d e f r 1 5

142 Online Policies for the Multiple Access Channel with Common Source Optimal capacity region with Bernoulli arrivals is a single pentagon r 2 a b,c d,e f r 1 Distributed sub-optimal policy, let q k P k B k : For Bernoulli energy arrivals: For general energy arrivals: P 1i = B 1 p(1 p) i 1 P 2i = B 2 p(1 p) i 1 P 1i = q 1 b 1i P 2i = q 2 b 2i 51

143 Bounds for the Multiple Access Channel with Common Source Bernoulli energy arrivals gives a lower bound for general energy arrivals. Lower bound: r 1 1 ( 2 log 1+ P ) 1 σ 2.72 r log ( 1+ P 2 σ 2 ).72 r 1 + r log ( 1+ P 1 + P 2 σ 2 ).72 Upper bound for any energy arrival: r 1 1 ( 2 log 1+ P ) 1 σ 2 ) r log ( 1+ P 2 σ 2 r 1 + r log ( 1+ P 1 + P 2 σ 2 ) 52

144 Multiple Access Channel with General (Arbitrarily Correlated) Arrivals Bounds are the same for any arbitrary energy arrivals. E 1i data queue Tx 1 E 2i Rx data queue Tx 2 Using q k = P k B k, the lower bound is: r log ( 1+ P 1 σ 2 r log ( 1+ P 2 σ 2 ).72 ).72 r 1 + r log ( 1+ P 1 + P 2 σ 2 53 ).72

145 Illustration of Bounds for the Multiple Access Channel r 2 upper bound optimal policy lower bound fractional policy (DFP).72 r 1 54

146 Multiple Access Channel with Arbitrary Number of Users Bounds are the same for any arbitrary number of users. E 1i data queue Tx 1 E 2i data queue Tx 2 Rx E Ki data queue Tx K Using q k = P k B k, the lower bound is: Upper bound: r i 1 ( i S 2 log 1+ i S P i σ 2 ).72, S {1,...,K} r i 1 ( i S 2 log 1+ i S P ) i σ 2, S {1,...,K} 55

147 Multiple Access Channel with Large Number of Users Sum-rate approaches the capacity for very large number of users. E 1i data queue Tx 1 E 2i data queue Tx 2 Rx E Ki data queue Tx K Using q k = P k B k, the lower bound is: Upper bound: K r i 1 ( i=1 2 log 1+ K i=1 P i σ 2 ).72 K r i 1 ( i=1 2 log 1+ K i=1 P ) i σ 2 56

148 Conclusions for the Online Multiple Access Scenario Energy harvesting transmitters sending messages to a single access point. The problem: maximization of the departure region. Obtain the structure of the solution, such as: Monotonicity of the power. Synchronous multiple access capacity region is a pentagon. Near-optimal policy. 57

149 Wireless Energy Transfer Newly emerging technologies have enabled us to perform wireless energy transfer efficiently. Inductive coupling can be used to wirelessly transfer energy. 58

150 Energy Cooperation in Multi-user Energy Harvesting Communications Wireless energy transfer is a new cooperation paradigm. Energy cooperation: Nodes share their energy as well as their information. 59

151 Gaussian Two-Hop Relay Channel with Energy Cooperation E i δ i Ē i energy queue data queue S AWGN Channel data queue R AWGN Channel D Energy harvesting source and relay with deterministic energy arrivals E i, Ē i. Wireless energy transfer unit that allows the source to transfer some of its energy to the relay (with α 1efficiency). Unlimited data and energy buffers at the source and the relay. New energy arrivals at every slot i, 1 i T. The source transfers δ i energy to the relay at slot i. Relay receives αδ i of this transferred energy at the next slot. 6

152 Two Hop Relay Channel without Energy Cooperation E i Ē i energy queue data queue S AWGN Channel data queue R AWGN Channel D Optimal source/relay profile is a separable policy. Source performs single-user throughput maximization with respect to its own energy arrivals. Relay forwards as many of the received bits as possible, satisfying data causality and energy causality. 61

153 Two Hop Relay Channel without Energy Cooperation E i Ē i log(1+7) = 3 energy queue data queue S AWGN Channel data queue R AWGN Channel D Separable policy, source maximizes its own throughput. Relay tries to send as much as it can. 1 bit sent to destination, 2 bits remaining at the relay. End-to-end throughput is 1 bit. 62

154 Two Hop Relay Channel without Energy Cooperation E i Ē i energy queue log(1+1) = 1 data queue S AWGN Channel data queue R AWGN Channel D Separable policy, source maximizes its own throughput. Relay tries to send as much as it can. 1 bit sent to destination, 2 bits remaining at the relay. End-to-end throughput is 1 bit. 63

155 Two Hop Relay Channel without Energy Cooperation E i Ē i energy queue data queue S AWGN Channel data queue R AWGN Channel D Separable policy, source maximizes its own throughput. Relay tries to send as much as it can. 1 bit sent to destination, 2 bits remaining at the relay. End-to-end throughput is 1 bit. 64

156 Two Hop Relay Channel with Energy Cooperation E i α =.5 Ē i energy queue 4.5 = 2 log(1+3) = 2 data queue S AWGN Channel data queue R AWGN Channel D Source sends less data, but some energy to assist the relay. Relay uses this extra energy to forward more data. 2 bits sent to destination, bits remaining at the relay. End-to-end throughput is 2 bits. 65

157 Two Hop Relay Channel with Energy Cooperation E i Ē i energy queue log(1+3) = 2 data queue S AWGN Channel data queue R AWGN Channel D Source sends less data, but some energy to assist the relay. Relay uses this extra energy to forward more data. 2 bits sent to destination, bits remaining at the relay. End-to-end throughput is 2 bits. 66

158 Two Hop Relay Channel with Energy Cooperation E i Ē i energy queue data queue S AWGN Channel data queue R AWGN Channel D Source sends less data, but some energy to assist the relay. Relay uses this extra energy to forward more data. 2 bits sent to destination, bits remaining at the relay. End-to-end throughput is 2 bits. 67

159 End-to-end Throughput Maximization Maximize end-to-end throughput subject to: max s.t. T 1 i=1 2 log(1+ P i ) k P i i=1 k i=1 P i Data causality at the relay node Energy causality at both nodes k i=1 k i=1 (Possibly) non-zero energy transfers (E i δ i ), k (Ē i + αδ i ), k k k 1 i=1 2 log(1+ 1 P i ) i=1 2 log(1+p i), k 68

160 Gaussian Two Way Channel with Energy Cooperation Energy harvesting users with deterministic energy arrivals E i, Ē i One-way wireless energy transfer with efficiency <α<1. E i δ i Ē i energy queue energy queue data queue User 1 User 2 data queue Physical layer is a Gaussian two-way channel: Y 1 = X 1 + X 2 + N 1 Y 2 = X 1 + X 2 + N 2 N 1,N 2 are Gaussian noises with zero mean and unit power. 69

161 Capacity Region R 1 1 θr 2 3 R 2 Convex region, boundary is characterized by solving max P i,p i,δ i s.t. T 1 θ 1 i=1 2 log(1+p 1 i)+θ 2 2 log(1+ P i ) (δ,p, P) F Point 1 is achieved by δ = : no energy transfer. Point 3 is achieved by δ = E: full energy transfer. 7

162 Water-filling Approach Generalized two-dimensional directional water-filling algorithm. Transfer energy from one user to another while maintaining optimal allocation in time. Spread the energy as much as possible in time and user dimensions. Now we give a numerical example for θ 1 = θ 2 and α=1. 71

163 Numerical Example E=[,12,] mj Ē=[6,6,] mj OFF OFF 12 User 1 1 OFF 2 3 OFF OFF OFF OFF User

164 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 12 User 1 1 OFF 2 3 OFF OFF ON ON User

165 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 6 User OFF 2 3 OFF OFF ON 2 ON 4 User

166 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 6 User OFF 2 3 OFF ON ON 2 ON 4 User

167 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 6.9 User OFF 2 3 OFF 1.8 ON ON 1.4 ON 2.8 User

168 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 7.2 User OFF 2 3 OFF 2.4 ON ON 1.2 ON 2.4 User

169 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 7.2 User ON 2 3 OFF 2.4 ON ON 1.2 ON 2.4 User

170 Numerical Example E=[,12,] mj Ē=[6,6,] mj ON ON 7.2 User ON 2 3 ON 2.4 ON ON 1.2 ON 2.4 User

171 Conclusions for Offline Energy Cooperation Scenarios Energy harvesting users with infinite capacity batteries. Energy transfer capability in an orthogonal channel in one way. Energy transfer provides a new degree of freedom to smooth out the energy profiles. Optimal policies identified for Gaussian two-hop relay and two-way channels. End-to-end throughput maximization for the two-hop relay channel. Capacity regions for two-way channels. 8

172 Acknowledgements NSF CNS , CCF , CCF , and CNS Slides: Abdulrahman Baknina, Ahmed Arafa, Berk Gurakan, and Omur Ozel 81

173 References [1] J. Yang and S. Ulukus, Optimal Packet Scheduling in an Energy Harvesting Communication System, IEEE Transactions on Communications, January 212. [2] K. Tutuncuoglu and A. Yener, Optimum Transmission Policies for Battery Limited Energy Harvesting Nodes, IEEE Transactions on Wireless Communications, February 212. [3] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus and A. Yener, Transmission with Energy Harvesting Nodes in Fading Wireless Channels: Optimal Policies, IEEE Journal on Selected Areas in Communications, September 211. [4] J. Yang, O. Ozel and S. Ulukus, Broadcasting with an Energy Harvesting Rechargeable Transmitter, IEEE Transactions on Wireless Communications, February 212. [5] O. Ozel, J. Yang and S. Ulukus, Optimal Broadcast Scheduling for an Energy Harvesting Rechargeable Transmitter with a Finite Capacity Battery, IEEE Transactions on Wireless Communications, June 212. [6] J. Yang and S. Ulukus, Optimal Packet Scheduling in a Multiple Access Channel with Energy Harvesting Transmitters, JCN Special Issue on Energy Harvesting in Wireless Networks, April 212. [7] D. Shaviv and A. Ozgur, Universally Near Optimal Online Power Control for Energy Harvesting Nodes, 215. Available at arxiv: v1. 82

174 References [8] A. Baknina and S. Ulukus, Online Scheduling for Energy Harvesting Broadcast Channels with Finite Battery, IEEE ISIT, July 216. [9] A. Baknina and S. Ulukus, Online Policies for Multiple Access Channel with Common Energy Harvesting Source, IEEE ISIT, July 216. [1] H. A. Inan and A. Ozgur, Online Power Control for the Energy Harvesting Multiple Access Channel, WiOpt, May 216. [11] C. Huang, R. Zhang and S. Cui, Throughput maximization for the Gaussian Relay Channel with Energy Harvesting Constraints, IEEE Journal on Selected Areas in Communications, August 213. [12] D. Gunduz and B. Devillers, Two-Hop Communication with Energy Harvesting, IEEE CAMSAP, San Juan, PR, December 211. [13] B. Gurakan, O. Ozel, J. Yang and S. Ulukus, Energy Cooperation in Energy Harvesting Communications, IEEE Transactions on Communications, December 213. [14] K. Tutuncuoglu and A. Yener, Energy Harvesting Networks with Energy Cooperation: Procrastinating Policies, IEEE Transactions on Communications, August

175 IEEE ISIT 216 Tutorial July 1, 216 Energy Harvesting and Remotely Powered Wireless Networks- Part II Aylin Yener

176 Outline Aylin- Part II Information theory of energy harvesting transmitters Energy harvesting AWGN channel with infinite battery Energy harvesting AWGN channel with no battery Binary noiseless energy harvesting channel State amplification and state masking 7/1/216 IEEE ISIT 216, Barcelona, Spain 66

177 Information Theory of EH Transmitters So far, we have assumed E i sufficiently long time slots and utilized the known rate expressions. Tx Rate : r(p) Rx What if energy harvesting is E i at the symbol level, i.e., each input symbol is individually limited by EH constraints? ENC DEC X i Y i 7/1/216 IEEE ISIT 216, Barcelona, Spain 67

178 Energy Harvesting (EH) Channel [Tutuncuoglu-Ozel-Ulukus-Yener 13] The channel input is restricted by an external energy harvesting process. E i State: available energy Has memory (due to energy storage) Depends on channel input W ENCODER X i Causally known to Tx (causal CSIT) 7/1/216 IEEE ISIT 216, Barcelona, Spain 68

179 Energy Harvesting (EH) Channel E i E max X i S i (Ch. input constrained by state) S i1 min S i X i E i, E max (State has memory) (State evolves based on ch. input) S i W ENCODER X i CHANNEL Yi DECODER Wˆ P Y X 7/1/216 IEEE ISIT 216, Barcelona, Spain 69

180 Energy Harvesting AWGN Channel E i E max N i W ENCODER X i Yi DECODER Wˆ [Ozel-Ulukus 12] Battery capacity E max is infinite. P E[E i ] Average recharge rate: Capacity without energy harvesting: C 1 2 log1 P 7/1/216 IEEE ISIT 216, Barcelona, Spain 7

181 Energy Harvesting AWGN Channel Code symbols are constrained by the energy in the battery at each channel use, i.e., k i1 X 2 i E i1 1,2,..., n. Conversely, the average power constraint for a non-eh AWGN channel would be a single constraint: 1 C log1 P 2 1 n n i1 is an upper bound on the capacity of the energy harvesting AWGN channel. X k 2 i i, 1 n n i1 k E i P. 7/1/216 IEEE ISIT 216, Barcelona, Spain 71

182 Achievability This upper bound is achievable. Two sources of error: 1. Decoding error, 2. Energy shortage. Idea: Design the codebook as if the channel is non-eh and show that energy shortages are insignificant. Two achievable schemes: 1. Save-and-Transmit, 2. Best-Effort-Transmit. 7/1/216 IEEE ISIT 216, Barcelona, Spain 72

183 Save-and-Transmit hnon hn n. Suppose, i.e., n Save energy for the first channel uses, do not transmit. h Transmit i.i.d. Gaussian signals in the remaining channel uses. n h n n The energy saved during the first channel uses is sufficient to guarantee no energy shortages occur in the remaining channel uses. n h n h 7/1/216 IEEE ISIT 216, Barcelona, Spain 73

184 Save-and-Transmit np harvested energy n h hnp n P X hn1 hn2 n n, hn h 1 log1 P 2 Since there is no loss in rate. Rates are achievable. expended energy X X n 7/1/216 IEEE ISIT 216, Barcelona, Spain 74 n

185 Best-Effort-Transmit Codewords are i.i.d. Gaussian with variance P. S i : the energy in the battery, i.e., the battery state in the th channel use. i S If i X i, i.e., there is enough energy in the battery, send. Otherwise, send nothing. X i 2 The battery state updates according to S S E X 1 2 S X. 2 i1 i i i i i 7/1/216 IEEE ISIT 216, Barcelona, Spain 75

186 Best-Effort-Transmit X P 2 E i EE i P With and, it is shown by SLLN that, finitely many energy shortages occur. Finitely many symbols are infeasible, i.e., the transmitter puts to the channel instead of the desired code symbol finitely many times. Finitely many mismatches are insignificant for joint typical decoding. 1 log1 P 2 Rates are achievable. 7/1/216 IEEE ISIT 216, Barcelona, Spain 76

187 EH AWGN Channel with No Battery E i N i W ENCODER X i Yi DECODER Wˆ [Ozel-Ulukus 11] There is no battery at the transmitter, i.e., E max. The code symbols are amplitude constrained: X 2 i E i, i 1,2,..., n. IEEE ISIT 216, Barcelona, Spain 7/1/216 77

188 EH AWGN Channel with No Battery The transmitter has causal information of energy arrivals. The receiver does not know the energy arrivals. The harvested energy amount is one of finitely many possibilities. For simplicity, assume binary E 1 E., 2 Background: 1. Static amplitude constrained AWGN channel [Smith 71] 2. State dependent channel with causal state information at the transmitter [Shannon 58] IEEE ISIT 216, Barcelona, Spain 7/1/216 78

189 Static Amplitude Constrained AWGN Channel [Smith 71] At each channel use, the code symbol is amplitude constrained by A. The channel capacity under this constraint is C Sm A max IX ; Y X A which is a convex program. The capacity achieving distribution was shown to have finitely many mass points. IEEE ISIT 216, Barcelona, Spain 7/1/216 79

190 State Dependent Channel with Causal State Information at the Tx [Shannon 58] Channel model: s S State is causally available at the transmitter only. The channel capacity is p y x, s C Sh max p T t IT ; Y. T [ T, T2,, T ] is an extended channel input satisfying 1 S p Y T S y 1 2 i e i1 2 y t Ps s t i 2. IEEE ISIT 216, Barcelona, Spain 7/1/216 8

191 Capacity of the EH AWGN Channel with No Battery 7/1/216 IEEE ISIT 216, Barcelona, Spain Suppose the harvested energy is Apply Shannon s result with and The capacity achieving distribution is observed to have finitely many mass points. ], [ 2 T 1 T T. e 2 e 2, E t t y E t t y p p t t y p. 1 w.p., w.p., p p E p E 81

192 Mass Points Symmetric about the origin Constrained to the blue line binary quaternary quinary t 2 E 2 E 2 E 2 2 E 1 E 1 E 1 E 1 t 1 IEEE ISIT 216, Barcelona, Spain 7/1/216 82

193 Numerical Results 7/1/216 IEEE ISIT 216, Barcelona, Spain 83

194 Binary Noiseless EH Channel S i,1 E i E max 1 W ENCODER X i,1 Y i X i DECODER Wˆ [Tutuncuoglu-Ozel-Ulukus-Yener 13] Transmitting requires units of energy Unit battery, X i,1 X i E max 1 Binary noiseless channel, Y i X i 7/1/216 IEEE ISIT 216, Barcelona, Spain 84

195 Energy Model In channel use i, the transmitter first puts input symbol X i to the channel, and then harvests energy E i : One channel use S i S i1 Channel input Energy harvest At the beginning of channel use i, battery state is S i. State evolution: S i1 mins i X i E i,1 (next state depends on input) Energy harvest: E i are i.i.d. Bernoulli with Pr[E i 1] q h 7/1/216 IEEE ISIT 216, Barcelona, Spain 85

196 Timing Channel A representation that simplifies the problem. X i E i Z i,1,... : # of channel uses spent waiting for energy, ~ Geometric (q h ), i.i.d. V i 1,2,... : # of channel uses the energy is kept in storage 1,2,... : # of channel uses between 1s at the receiver side T i Z 6 V Z 4 V 8 2 E i =1 X i =1 2 T T... T i V i Z i Memoryless! 7/1/216 IEEE ISIT 216, Barcelona, Spain 86

197 Timing Channel m m m n n n The two sets of variables, ( V, Z, T ) and ( X, E, Y ), are alternative representations of the same sequences. n m X,,,1,,,1,,1,,,,,,1, V 1,2,2,2 n Y,,,1,,,1,,1,,,,,,1, m T 4,3,2,6 n m E,,1,,1,,1,,,,,,1,,1, Z 3,1,,4 Lemma: The timing channel capacity with additive causally known state C T and the originally formulated binary EH channel capacity C are equal, i.e., C = C T. 7/1/216 IEEE ISIT 216, Barcelona, Spain 87

198 Capacity of the Timing Channel [Shannon 1958] Capacity of a memoryless channel with causal CSIT: C CSIT max p(u),v(u,z) I(U;T) [Anantharam-Verdu 1996] I(X;T) Capacity of the timing channel: C T max p(x) [T] Capacity of the timing channel with causal CSIT I( U; T) CT max CBEHC p( u), v( u, z) [ T] Main challenge: selection of auxiliary variable U Z, V U, v :(U, Z) V 7/1/216 IEEE ISIT 216, Barcelona, Spain 88

199 Modulo Encoding U,1,..., N 1, U ~ p U (u), V U Z mod N1 Binary encoding interpretation: The encoder indexes channel uses in mod N, and sends U i by transmitting a 1 at the earliest feasible channel use with index U i. Achievable rate: R A (N ) max p(u) I(U;T ) [V Z] max p(u) H(U) [V ][Z] Example: N 5 U 2,1,3,..., Z 2,3,1,... i i U1 2 U 2 1 U /1/216 IEEE ISIT 216, Barcelona, Spain 89

200 Extended Modulo Encoding Choose V U Z 1 U Z (U Z mod N)1 U Z U {,1, } Decoder: T ' T 1 mod N U mod N (U mod N decoded without error) Achievable Rate: I(U;T ) R ext A max max N p(u) [V Z] U 2 1 U 2 6 or U 2 1 N T 1mod N 7/1/216 IEEE ISIT 216, Barcelona, Spain 9

201 Genie Upper Bound Provide channel state as side information at the decoder. I(V;T Z) C genie UB max p(v) [V Z] max H(V ) p(v) [V ][Z] 1 max [Z] max H(V ) [V ] The entropy maximizing distribution on with [V ] Z i is Geometric(1/μ). V 1,2,... C genie UB max q u [,1] q h H(q u ) q h q u (1 q h ) 7/1/216 IEEE ISIT 216, Barcelona, Spain 91

202 Asymptotic Optimality Modulo Encoding: Genie Upper bound: R A mod max q u,n H(U) [V ][Z] C genie UB max q u [,1] q h H(q u ) q h q u (1 q h ) Choose N 1 *, U ~ Unif ({,1,, N 1}) q u genie C lim UB q h mod R A 1 Modulo encoding is asymptotically optimal for low harvesting rates 7/1/216 IEEE ISIT 216, Barcelona, Spain 92

203 Leakage Upper Bound [Tutuncuoglu-Ozel-Yener-Ulukus 14] Timing Channel Capacity: C T max I( U; T) [ ] p( u), v( u, z) T I( U; T) H( T) I( Z; T U) (Mutual dependence of Z and T given U) I( Z; T U) t1 u p( t, u)[ H( Z) H( Z T t, U u)] (Entropy of Z upon observing T=t and decoding U=u) Lemma: H Z T t, U u) H( Z ) ( t where p Zt (z) q h (1 q h ) z 1 (1 q h ) t, if z t, otherwise (Truncated geometric) 7/1/216 IEEE ISIT 216, Barcelona, Spain 93

204 Leakage Upper Bound C T max p(u),v(u,z) max p(u),v(u,z) I(U;T ) [T ] H(T ) I(Z;T U) [T] C T max p(t ) H(T) p(t)[h(z) H(Z t )] t1 [T] Leakage Upper Bound C T Easier to evaluate than since the maximization is p(t) over instead of p( u), v( u, z) 7/1/216 IEEE ISIT 216, Barcelona, Spain 94

205 Computing the Leakage Upper Bound H(T) p(t)[h(z) H(Z t )] t1 C T max p(t) E[T] 1 max max H(T) t p(t) p(t),e[t ] t1 (Inner problem is convex) KKT optimality conditions give t t 1 n1 n A t t1 t n1 n p(t) Aexp t t Calculate UB by exhaustive search over for each 7/1/216 IEEE ISIT 216, Barcelona, Spain 95

206 Interpretation of the UB t t CT H( T) p( t)[ H( Z) H( Z 1 max p( t) [ T] )] T i T 8 T T... Revealed: Z 8 Z 4 Z We inadvertently waste part of the potential rate of the channel 7/1/216 IEEE ISIT 216, Barcelona, Spain 96

207 Numerical Results 1.9 Upper bounds Asymptotic optimality of timing-based encoding Capacity / rate achieved (bits/ch. use) Capacity with infinite storage (C IS ) genie Genie upper bound (C UB ) Leakage upper bound (C ) UB Extended encoding rate (R A ext ) Modulo encoding rate (R A mod ) Capacity with zero storage (C ZS ) Energy harvest probability (q ) h Achievable rates 7/1/216 IEEE ISIT 216, Barcelona, Spain 97

208 Numerical Results Rate (bits/ch. use) leakage Leakage upper bound (C UB ) Extended encoding rate (R A ext ) Modulo encoding rate (R A mod ) 2nd order Markov Shannon strategy rate (R M2 ) 1st order Markov Shannon strategy rate (R M1 ) Optimal i.i.d. Shannon strategy rate (R OIID ).63 Capacity within.3 bits/ch.use Energy harvest probability (q h ) 7/1/216 IEEE ISIT 216, Barcelona, Spain 98

209 Binary Symmetric EH Channel E i E max S i W ENCODER X i CHANNEL p Y X Y i DECODER Wˆ Binary symmetric channel: Pr[ Yi X i ] 2, 1 pe, X i, Y i,1 E i The energy arrivals are i.i.d. Bernoulli, ~ Bernoulli( q). Two sources of errors: 1. Energy shortage: Without energy, the encoder must send a zero. 2. Channel errors: Any bit sent can be flipped by the channel. 7/1/216 IEEE ISIT 216, Barcelona, Spain 99

210 Binary Symmetric EH Channel n Observing Y, decoder also obtains information about n E Rate of this information flow can be quantified by n n n 1 n H( E ) H( E Y ) I( E ; Y ). 1 n n n Randomness of energy arrival process Randomness remaining after channel output is observed The encoder may wish to [Tutuncuoglu-Ozel-Yener-Ulukus 14ITW]: Maximize entropy reduction rate : State Amplification (Cooperative scenario) [Kim et al. 8] Minimize entropy reduction rate : State Masking (Privacy or stealth scenario) [Merhav-Shamai 7] 7/1/216 IEEE ISIT 216, Barcelona, Spain 1

211 No Battery Case [Tutuncuoglu-Ozel-Yener-Ulukus 14ITW] The encoder can send X 1 only when 1. For i.i.d. arrivals, this is a memoryless channel with CSIT. Capacity achieved using Shannon strategies: i E i U {,1} n, U i Bern ( p), X i 1 E i 1, U else i 1 Shorthand for U (,) and U (,1). 7/1/216 IEEE ISIT 216, Barcelona, Spain 11

212 No Battery Case State Amplification State Masking where ) ( ) ( ) ( ) ( ) (1 ) ( ) ( e e e e e p H p pq H R q H p H p p q ph p pq H R ) ( ) ( ) ( ) (1 ) ( ) ( e e e e e p ph p q ph p H p p q ph p pq H R q p q p q p ) (1 ) 1 ( 7/1/216 IEEE ISIT 216, Barcelona, Spain 12

213 Infinite Battery Case [Tutuncuoglu-Ozel-Yener-Ulukus 14ITW] Capacity achieved via extending the save-and-transmit scheme [Ozel-Ulukus 12]. Channel input constrained as EX q C C BSC H ( q pe) H ( p 1 H ( pe) e ) q q /1/216 IEEE ISIT 216, Barcelona, Spain 13

214 as State Amplification ) log( ) ( ) log( ) ( ) ( ) ; ( ) ;, ( ) ; ( nr H nr n nr H W H Y E I Y W E I Y X I n n n n n n Lemma: The region is given by ) (, q H C R BSC ), ( R Achievability: Compress part of and send as a part of the message, i.e., decoder obtains Converse: Using the Markov Chain n E ), ( k E W W n n n Y X E W ), ( 7/1/216 IEEE ISIT 216, Barcelona, Spain 14

215 State Masking E max For, perfect state masking is possible, i.e., ( R, ) ( CBSC,) is achievable In the save-and-transmit scheme, channel input is independent of harvested energy R C BSC Any rate is also achievable. Due to converse proof, no better rate can be achieved E n X n 7/1/216 IEEE ISIT 216, Barcelona, Spain 15

216 Unit-sized Battery Case Capacity of this channel is open as we have just seen. Some achievable rates proposed: [Tutuncuoglu-Ozel-Y.- Ulukus 13][Mao-Hassibi 13]. [Mao-Hassibi 13]: Two strategies, U i,1 Channel input: R IID S i lim n 1 n X i I ( U 1 n ; Y n ) S 1, U else 1 If was memoryless, this would be capacity achieving. i i 7/1/216 IEEE ISIT 216, Barcelona, Spain 16

217 Numerical Results State Amplification Perfect state amplification with zero message rate Noiseless channel p e 1 q 2 Better state amplification with instantaneous Shannon strategies 7/1/216 IEEE ISIT 216, Barcelona, Spain 17

218 Numerical Results State Masking Noiseless channel p e 1 q 2 Better state masking with timing encoding Perfect masking for infinitesized battery 7/1/216 IEEE ISIT 216, Barcelona, Spain 18

219 Conclusion New wireless communications paradigm: energy harvesting nodes New design insights arising from new energy constraints energy storage limitations and inefficiencies interaction of multiple EH transmitters energy cooperation New problems in the information theory domain Lots of open problems related to all layers of the network design: e.g. Signal processing/phy design; MAC protocol design; channel capacity 7/1/216 IEEE ISIT 216, Barcelona, Spain 19

220 References-Part II [Ozel-Ulukus 12] O. Ozel and S. Ulukus, Achieving AWGN Capacity Under Stochastic Energy Harvesting. IEEE Trans. on Information Theory, 58(1): , October 212. [Ozel-Ulukus 11] O. Ozel and S. Ulukus, AWGN Channel under Time-Varying Amplitude Constraints with Causal Information at the Transmitter, Proceedings of the 45 th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, November 211. [Tutuncuoglu-Ozel-Yener-Ulukus 13]: Kaya Tutuncuoglu, Omur Ozel, Aylin Yener and Sennur Ulukus, Binary Energy Harvesting Channel with Finite Energy Storage, Proceedings of the IEEE International Symposium on Information Theory, ISIT'13, Istanbul, Turkey, July 213. [Tutuncuoglu-Ozel-Yener-Ulukus 14]: Kaya Tutuncuoglu, Omur Ozel, Aylin Yener, and Sennur Ulukus, Improved Capacity Bounds for the Binary Energy Harvesting Channel, Proceedings of the IEEE International Symposium on Information Theory, ISIT'14, Honolulu, HI, July 214. [Tutuncuoglu-Ozel-Yener-Ulukus 14ITW]: Kaya Tutuncuoglu, Omur Ozel, Aylin Yener, and Sennur Ulukus, State Amplification and State Masking for the Binary Energy Harvesting Channel, Proceedings of Information Theory Workshop, ITW'14, Hobart, Australia, November 214. [Mao-Hassibi 13]: W. Mao and B. Hassibi. On the capacity of a communication system with energy harvesting and a limited battery. Proceedings of the IEEE International Symposium on Information Theory, ISIT, July 213. [Shannon 58] C. E. Shannon. Channels with side information at the transmitter. IBM Journal of Research and Development, 2(4): , [Anantharam-Verdu 96]: V. Anantharam and S. Verdu. Bits through queues. IEEE Transactions on Information Theory, 42(1):4 18, January /1/216 IEEE ISIT 216, Barcelona, Spain 111

221 Information-Theoretic Capacity of Energy Harvesting and Remotely Powered Systems Ayfer Özgür Tutorial on Energy Harvesing and Remotely Powered Communication ISIT 216, Barcelona, Spain Ayfer Özgür Information-Theoretic Capacity July 16 1 / 31

222 Model E t E max Battery N t N (, 1) Transmitter X t Y t Receiver X t 2 B t B t+1 = min ( B t X t 2 + E t+1, E max ). E t : i.i.d. energy harvesting process known causally at the transmitter and not at the receiver. State-dependent channel: State process has memory and is input dependent. State is known causally at the transmitter but not at the receiver. Ayfer Özgür Information-Theoretic Capacity July 16 2 / 31

223 E max = : Capacity equal to that of a classical AWGN channel with P = E [E t ]: C = log (1 + E[E t ]) First-order questions: How does the capacity of the energy harvesting AWGN channel depend on system parameters such as E max and E t? Ayfer Özgür Information-Theoretic Capacity July 16 3 / 31

224 E max = : Capacity equal to that of a classical AWGN channel with P = E [E t ]: C = log (1 + E[E t ]) First-order questions: How does the capacity of the energy harvesting AWGN channel depend on system parameters such as E max and E t? What are the properties of E t most relevant to capacity? What are more favorable and less favorable energy profiles? Ayfer Özgür Information-Theoretic Capacity July 16 3 / 31

225 E max = : Capacity equal to that of a classical AWGN channel with P = E [E t ]: C = log (1 + E[E t ]) First-order questions: How does the capacity of the energy harvesting AWGN channel depend on system parameters such as E max and E t? What are the properties of E t most relevant to capacity? What are more favorable and less favorable energy profiles? Are there different operating regimes where the dependence to E max and E t is qualitatively different? Ayfer Özgür Information-Theoretic Capacity July 16 3 / 31

226 E max = : Capacity equal to that of a classical AWGN channel with P = E [E t ]: C = log (1 + E[E t ]) First-order questions: How does the capacity of the energy harvesting AWGN channel depend on system parameters such as E max and E t? What are the properties of E t most relevant to capacity? What are more favorable and less favorable energy profiles? Are there different operating regimes where the dependence to E max and E t is qualitatively different? For a given E t, how can we optimally choose E max? Ayfer Özgür Information-Theoretic Capacity July 16 3 / 31

227 E max = : Capacity equal to that of a classical AWGN channel with P = E [E t ]: C = log (1 + E[E t ]) First-order questions: How does the capacity of the energy harvesting AWGN channel depend on system parameters such as E max and E t? What are the properties of E t most relevant to capacity? What are more favorable and less favorable energy profiles? Are there different operating regimes where the dependence to E max and E t is qualitatively different? For a given E t, how can we optimally choose E max? Ayfer Özgür Information-Theoretic Capacity July 16 3 / 31

228 A channel with random battery recharges E t E max Battery N t N (, 1) Transmitter X t Y t Receiver Ayfer Özgür Information-Theoretic Capacity July 16 4 / 31

229 A channel with random battery recharges E t E max Battery N t N (, 1) Transmitter X t Y t Receiver X t 2 B t B t+1 = min ( B t + E t+1 X t 2, E max ) Ayfer Özgür Information-Theoretic Capacity July 16 4 / 31

230 A channel with random battery recharges E t E max Battery N t N (, 1) Transmitter X t Y t Receiver X t 2 B t B t+1 = min ( B t + E t+1 X t 2, E max ) We focus on i.i.d. Bernoulli energy arrival process: { Emax w.p. p E t = w.p. 1 p, Ayfer Özgür Information-Theoretic Capacity July 16 4 / 31

231 A channel with random battery recharges E t E max Battery N t N (, 1) Transmitter X t Y t Receiver X t 2 B t B t+1 = min ( B t + E t+1 X t 2, E max ) We focus on i.i.d. Bernoulli energy arrival process: { Emax w.p. p E t = w.p. 1 p, The energy arrival process {E t } is causally known both at the transmitter and the receiver. Ayfer Özgür Information-Theoretic Capacity July 16 4 / 31

232 Results for this model n-letter expression for capacity: C = lim max N p(x N ): X N 2 E max N p 2 (1 p) k 1 I (X k ; X k + Z k ) k=1 Connection to online power control: Bounded gap to AWGN capacity: T 1.5 C T. log(1 + pe max ) 1.77 C log(1 + pe max ). Ayfer Özgür Information-Theoretic Capacity July 16 5 / 31

233 Clipping Channel Y j [t] = { X j [t] + Z j [t], j L[t], j > L[t]... where L[t] are i.i.d. Geometric(p) and X [t] 2 E max. Theorem C EH = p C clp Ayfer Özgür Information-Theoretic Capacity July 16 6 / 31

234 Capacity of the Clipping Channel C clp = lim max I (X N ; Y L L) N p(x N ): X N 2 E max = lim max N p(x N ): X N 2 E max N p(1 p) k 1 I (X k ; X k + Z k ) k=1 Ayfer Özgür Information-Theoretic Capacity July 16 7 / 31

235 Bounding C EH C EH = lim max N p(x N ): X N 2 E max N p 2 (1 p) k 1 I (X k ; X k + Z k ) k=1 Ayfer Özgür Information-Theoretic Capacity July 16 8 / 31

236 Bounding C EH C EH = lim max N p(x N ): X N 2 E max lim max N p(x N ): E X N 2 E max N p 2 (1 p) k 1 I (X k ; X k + Z k ) k=1 N p 2 (1 p) k 1 k=1 k I (X i ; X i + Z i ) i=1 Ayfer Özgür Information-Theoretic Capacity July 16 8 / 31

237 Bounding C EH C EH = lim max N p(x N ): X N 2 E max lim max N p(x N ): E X N 2 E max = lim max N p(x N ): E X N 2 E max N p 2 (1 p) k 1 I (X k ; X k + Z k ) k=1 N p 2 (1 p) k 1 k=1 k I (X i ; X i + Z i ) i=1 N p(1 p) i 1 I (X i ; X i + Z i ) i=1 Ayfer Özgür Information-Theoretic Capacity July 16 8 / 31

238 Bounding C EH C EH = lim max N p(x N ): X N 2 E max lim max N p(x N ): E X N 2 E max = lim max N p(x N ): E X N 2 E max = lim max N {E i } N i=1 : E i i N i=1 E i E max N p 2 (1 p) k 1 I (X k ; X k + Z k ) k=1 N p 2 (1 p) k 1 k=1 k I (X i ; X i + Z i ) i=1 N p(1 p) i 1 I (X i ; X i + Z i ) i=1 N p(1 p) i log(1 + E i) i=1 Ayfer Özgür Information-Theoretic Capacity July 16 8 / 31

239 Bounding C EH C EH lim N max {E i } N i=1 : E i i N i=1 E i E max C EH lim max N {E i } N i=1 : E i i N i=1 E i E max N p(1 p) i log(1 + E i) i=1 N i=1 p(1 p) i 1 max X i E i I (X i ; X i + Z i ) Ayfer Özgür Information-Theoretic Capacity July 16 9 / 31

240 Bounding C EH C EH lim N max {E i } N i=1 : E i i N i=1 E i E max C EH lim max N {E i } N i=1 : E i i N i=1 E i E max N p(1 p) i log(1 + E i) i=1 N i=1 p(1 p) i 1 max X i E i I (X i ; X i + Z i ) Ayfer Özgür Information-Theoretic Capacity July 16 9 / 31

241 Bounding C EH C EH lim N max {E i } N i=1 : E i i N i=1 E i E max C EH lim max N {E i } N i=1 : E i i N i=1 E i E max We can show N p(1 p) i log(1 + E i) i=1 N i=1 p(1 p) i 1 max X i E i I (X i ; X i + Z i ) max X i I (X i ; X i + Z i ) 1 log(1 + E) 1.5. E 2 Ayfer Özgür Information-Theoretic Capacity July 16 9 / 31

242 Bounding C EH C EH lim N max {E i } N i=1 : E i i N i=1 E i E max C EH lim max N {E i } N i=1 : E i i N i=1 E i E max We can show C EH N p(1 p) i log(1 + E i) i=1 N i=1 p(1 p) i 1 max X i E i I (X i ; X i + Z i ) max X i I (X i ; X i + Z i ) 1 log(1 + E) 1.5. E 2 lim N max {E i } N i=1 : E i i N i=1 E i E max N p(1 p) i log(1 + E i) 1.5. i=1 Ayfer Özgür Information-Theoretic Capacity July 16 9 / 31

243 Bounding C EH C EH lim N max {E i } N i=1 : E i i N i=1 E i E max C EH lim max N {E i } N i=1 : E i i N i=1 E i E max We can show C EH N p(1 p) i log(1 + E i) i=1 N i=1 p(1 p) i 1 max X i E i I (X i ; X i + Z i ) max X i I (X i ; X i + Z i ) 1 log(1 + E) 1.5. E 2 lim N max {E i } N i=1 : E i i N i=1 E i E max N p(1 p) i log(1 + E i) 1.5. i=1 Ayfer Özgür Information-Theoretic Capacity July 16 9 / 31

244 Connection to the online throughput: T 1.5 C T. Bounded gap to AWGN capacity: log(1 + pe max ) 1.77 C TxRx log(1 + pe max ). Ayfer Özgür Information-Theoretic Capacity July 16 1 / 31

245 Bounding C EH Bounds for p = log(1+pe max) C Smith capacity lower bound C 1.5 Rate [bits] Battery Size (E max ) Ayfer Özgür Information-Theoretic Capacity July / 31

246 Capacity Improvement due to CSIR Proposition In a general channel (not necessarily stationary memoryless), capacity improvement due to receiver side information is bounded by the entropy rate of the side information itself. For the Bernoulli case, capacity improvement is bounded by H(p) 1. Capacity with no receiver energy arrival information: log(1 + pe max ) 2.77 C Tx log(1 + pe max ). Ayfer Özgür Information-Theoretic Capacity July / 31

247 Capacity Theorem The capacity of the energy harvesting channel with i.i.d. energy arrivals is given by C causal Tx C causal TxRx C noncausal TxRx 1 = lim sup I (U n ; Y n ), (1) n n P U n P n(b) = lim 1 sup I (U n ; Y n E n ), (2) n n P U n P n(b) 1 = lim sup I (X n ; Y n E n ), (3) n n P X n E n F n(b) where { F n (b) = P X n E n s.t. en E n, a.s. for t = 1,..., n : } Xt 2 B t, B = b, B t = min{b t 1 X t e t, E max }. Ayfer Özgür Information-Theoretic Capacity July / 31

248 Capacity Theorem The capacity of the energy harvesting channel with i.i.d. energy arrivals is given by C causal Tx C causal TxRx C noncausal TxRx 1 = lim sup I (U n ; Y n ), (1) n n P U n P n(b) = lim 1 sup I (U n ; Y n E n ), (2) n n P U n P n(b) 1 = lim sup I (X n ; Y n E n ), (3) n n P X n E n F n(b) where { P n (b) = P U n s.t. U t : e t X for t = 1,..., n and e n E n a.s. : } U t (e t ) 2 B t, B = b, B t = min{b t 1 U t 1 (e t 1 ) 2 + e t, E max }. Ayfer Özgür Information-Theoretic Capacity July / 31

249 Connection to the Energy Allocation Problem Theorem The capacities of the energy harvesting channel with various levels of energy arrival information can be bounded by T online 1.5 H(g t (E t )) C causal Tx T online, T online 1.5 C causal TxRx T online, T offline 1.5 C noncausal TxRx T offline where H(g t (E t )) is the entropy rate of the power control process. Ayfer Özgür Information-Theoretic Capacity July / 31

250 Connection to the Energy Allocation Problem Theorem The capacities of the energy harvesting channel with various levels of energy arrival information can be bounded by T online 1.5 H(g t (E t )) C causal Tx T online, T online 1.5 C causal TxRx T online, T offline 1.5 C noncausal TxRx T offline where H(g t (E t )) is the entropy rate of the power control process. Also for η.7473, η T online H(g t (E t )) C causal Tx T online, η T online C causal TxRx T online, η T offline C noncausal TxRx T offline. Ayfer Özgür Information-Theoretic Capacity July / 31

251 Approximate Capacity for General i.i.d. Processes Theorem: The capacity of the energy harvesting channel can be approximated as 1 causal log(1 + µ) 3.85 CTx 1 log(1 + µ), causal log(1 + µ) 1.77 CTxRx 2 1 log(1 + µ), 2 1 noncausal log(1 + µ) 1.77 CTxRx 1 log(1 + µ). 2 2 Proof: For the case where the the receiver does not have side information devise a new online power control policy which is universally near-optimal and at the same time has low entropy rate: g t = q(1 q) j E max, where j = t max{t t : B t = E max } and q = µ/e max. Ayfer Özgür Information-Theoretic Capacity July / 31

252 Insights C 1 2 log(1 + E[min{E t, E max }]) f E (x) Ē E max x E max > Ē C 1 2 log(1 + E[E t]) Ayfer Özgür Information-Theoretic Capacity July / 31

253 Insights C 1 2 log(1 + E[min{E t, E max }]) f E (x) fẽ (x) Ē E max x E max x E max > Ē E max < Ē C 1 2 log(1 + E[E t]) C 1 2 log(1 + E[Ẽ t ]) Ayfer Özgür Information-Theoretic Capacity July / 31

254 Home IoT Ayfer Özgür Information-Theoretic Capacity July / 31

255 Two topologies for home IoT Current practice: Transfer energy at a constant rate. Periodically charge transmitter s battery. Ayfer Özgür Information-Theoretic Capacity July / 31

256 Exploit Side Information Charger observes the output of the channel. Charger observes the input to the channel. Ayfer Özgür Information-Theoretic Capacity July / 31

257 Binary Example Charger E t M Transmitter X t Receiver ˆM Y t = X t X t {, 1} E t {, 1} Charger has no side information: Et = 1, t: C = 1 bits/channel use, Γ = 1 unit/channel use. Ayfer Özgür Information-Theoretic Capacity July 16 2 / 31

258 Binary Example Charger E t M Transmitter X t Receiver ˆM Y t = X t X t {, 1} E t {, 1} Charger has no side information: Et = 1, t: C = 1 bits/channel use, Γ = 1 unit/channel use. Charger knows the message: Charge when the transmitter intends to send a 1: C M = 1 bits/channel use, Γ = 1/2 units/channel use. Ayfer Özgür Information-Theoretic Capacity July 16 2 / 31

259 Binary Example Charger X t 1 E t M Transmitter X t Receiver ˆM Y t = X t X t {, 1} E t {, 1} Charger has no side information: Et = 1, t: C = 1 bits/channel use, Γ = 1 unit/channel use. Charger knows the message: Charge when the transmitter intends to send a 1: C M = 1 bits/channel use, Γ = 1/2 units/channel use. Charger can observe the transmitted signal X t 1 : Charge when battery is empty: C X = 1 bits/channel use, Γ = 1/2 units/channel use. Ayfer Özgür Information-Theoretic Capacity July 16 2 / 31

260 Charger Side Information Charger E t X t 1 Y t 1 E max M Transmitter P Y X Receiver X t Y t ˆM C : Generic Charger; ft C : E C M : Charger and Tx connected through a backhaul link; ft C : M E C X : Charger observes the transmitted signal; ft C : X t 1 E. C Y : Receiver charges the transmitter; ft C : Y t 1 E Ayfer Özgür Information-Theoretic Capacity July / 31

261 Remotely Powered Communications Charger E t X t 1 Y t 1 E max M Transmitter P Y X Receiver X t Y t ˆM Charger: Dynamically decide how much energy to transfer to the receiver based on its side information regarding the transmission (subject to an average power constraint Γ). Transmitter: Dynamically adapt its transmission scheme based on its instantaneous battery level. Exploiting side information at the charger can enable performance close to the centralized case. Ayfer Özgür Information-Theoretic Capacity July / 31

262 Receiver powering transmitter E t (Y t 1 ) E max Battery X t Y t Transmitter Channel Receiver Receiver can convey both feedback information and energy with its charging actions. Ayfer Özgür Information-Theoretic Capacity July / 31

263 Simultaneous Information and Energy Transfer M Transmitter P Y X Receiver ˆM X t Y t Maximize information rate under a minimum received power constraint. For a BSC(α), C(P) = max I (X ; Y ). p(x ):E[b(Y )] P C(P) = { 1 h2 (α), P 1/2 h 2 (P) h 2 (p), 1/2 P 1 α. Ayfer Özgür Information-Theoretic Capacity July / 31

264 Can feedback increase capacity? E t E max = 1 Battery Transmitter X t BEC Channel Y t Receiver Y t 1 X = {, 1} Y = {, 1, e} 1 α α α e 1 1 α 1 B t = E t = { 1, t odd, t even { 1, t odd 1 X t 1, t even Ayfer Özgür Information-Theoretic Capacity July / 31

265 A claim by Shannon Ayfer Özgür Information-Theoretic Capacity July / 31

266 A claim by Shannon Ayfer Özgür Information-Theoretic Capacity July / 31

267 Feedback increases capacity.8.7 Capacity of the EH-BEC with and without feedback Cfb C α Ayfer Özgür Information-Theoretic Capacity July / 31

268 Open Questions and Directions Extension to more realistic energy harvesting and battery models. Ayfer Özgür Information-Theoretic Capacity July / 31

269 Open Questions and Directions Extension to more realistic energy harvesting and battery models. Coding and modulation techniques. Ayfer Özgür Information-Theoretic Capacity July / 31

270 Open Questions and Directions Extension to more realistic energy harvesting and battery models. Coding and modulation techniques. Networking and multi-user systems. Ayfer Özgür Information-Theoretic Capacity July / 31

271 Acknowledgement CCF , Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant agreement CCF , and Stanford SystemX. Slides: Dor Shaviv. Ayfer Özgür Information-Theoretic Capacity July / 31