fundamental limits of simultaneous energy and …fundamental limits of simultaneous energy and...

Fundamental Limits of Simultaneous Energy and InformationTransmission

Selma Belhadj Amor and Samir M. Perlaza

Inria, Lyon, France

International Conference on Telecommunications (ICT)Thessaloniki, Greece

May 17, 2016

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Simultaneous Energy and Information Transmission: A Trade-Off??

When Tesla meets Shannon

Conflict =) Trade-off between information and energy transmission rates

Example (Noiseless Transmission of a 4-PAM Signal in {�2, �1, 1, 2})

If no constraint is imposed on received energy rate �! can transmit 2 bits/ch.useIf received energy rate is constrained to be

I at maximum possible value �! can transmit 1 bit/ch.useI larger than a given value �! in some cases, can transmit 0 bits/ch.use

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Outline

1 Point-to-point Information-Energy Trade-off

2 Multi-User Simultaneous Energy and Information Transmission

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Discrete Memoryless Point-to-Point Channel

Encoder x(M) ChannelP

Y |X Y DecoderM̂

(n)M 2 M

Transmission blocklength n

Finite input and output alphabets X and YTransition law P

Y |X (memoryless)

Transmitter sends message M 2 M , {1, 2, . . . , 2nR}Information rate R

Decoder forms estimate M̂(n)

Probability of Error

P(n)error(R) , Pr

�

M̂(n) 6= M

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Discrete Memoryless Channel with Energy Harvester


YS|X Y

S

Decoder

EnergyHarvester

M̂

(n)M 2 M

Additional output alphabet S; Transition law PYS|X

Energy function ! : S ! R+

Harvested energy from s = (s1, . . . , sn) is !(s) =P

n

t=1 !(st

)

Average energy rate (in energy-units per channel use) at the EH:

B (n) , 1n

n

X

t=1

!(St

).

Minimum energy rate b at EH (in energy units per channel use)Energy rate B, with b 6 B 6 Bmax (Bmax is the maximum feasible energy rate)Guarantee B (n) > B with high probability

Probability of Energy Outage

P(n)outage(B) , Pr

n

B (n) < B � ✏o

, ✏ > 0 arbitrarily small

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Simultaneous Energy and Information Transmission (SEIT)


YS|X Y

S

Decoder

EnergyHarvester

M̂

(n)M 2 M

Objective of SEITProvide blocklength-n coding schemes such that:(i) information transmission occurs at rate R with P

(n)error(R) ! 0; and

(ii) energy transmission occurs at rate B with P(n)outage(B) ! 0 and B � b.

Under these conditions, the information-energy rate-pair (R, B) is achievable.

) What is the fundamental limit on information rate for a given energy rate?

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Information Capacity Under Minimum Energy Rate b

For each blocklength n, define the function C (n)(b) as follows:

C (n)(b) , maxX

n :B(n)>b

I (X n; Y n).

Definition: Information-Energy Capacity Function [Varshney’08]

The information-energy capacity function for a minimum energy rate b is defined as

C(b) , lim supn!1

1nC (n)(b).

Theorem: Information Capacity Under Minimum Energy Rate [Varshney’08]

The supremum over all achievable information rates in the DMC under a minimumenergy rate b in energy-units per channel use is given by C(b) in bits/ch. use.

L. R. Varshney, “Transporting information and energy simultaneously,” in Proc. IEEE

International Symposium on Information Theory, Jul. 2008, pp. 1612–1616.

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Example: Noiseless binary channel

S = Y

P(1|1) = P(0|0) = 1 and P(1|0) = P(1|0) = 0Channel capacity: C = 1 bit/ch.use.Capacity-achieving dist: Ber( 1

2 )

1

1

0 0X Y = S

1 1

Symbol 1 �! 1 energy-unit & Symbol 0 �! 0 energy-unitMaximum energy: Bmax = 1 energy-units/ch.use (Symbol ’1’ always sent)

Information-Energy Capacity Function [Varshney’08]

CNC(b) =

⇢

1, if 0 6 b 6 12 ,

H2(b), if 12 6 b 6 1,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

b [energy-units/ch. use]

CN

C(b

)[bit

s/ch

.use

]

Information-Energy Capacity Function of Noiseless Channel

Bmax = 1

) The more stringent the energy rate constraint is, the more the transmitter needs toswitch over to using the most energetic symbol

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Gaussian Memoryless Channel with Energy Harvester

TransmitterM

h1

h2

x

t

Z

t

Receiver

Q

t

M̂

InformationDecoder

EnergyHarvester Energy

Y

t

S

t

⌦ �

�

Information Decoder: Yt

= h1Xt

+ Zt

,

Energy harvester: St

= h2Xt

+ Qt

Xt

, Qt

, Zt

2 R;Constant channel coefs h1, h2 > 0 satisfying :

khk2 6 1, with h , (h1, h2)T (energy conservation principle)

{Zt

}, {Qt

} i.i.d. ⇠ N (0, 1)

Input power constraints:1n

n

X

t=1

E⇥

X 2t

⇤

6 P.

Fully described by signal-to-noise ratios:

SNRi

, |hi

|2P, i 2 {1, 2}x .

Output energy function: !(s) , s28 / 28

Gaussian Memoryless Channel with Energy Harvester

TransmitterM

h1

h2

x

t

Z

t

Receiver

Q

t

M̂

InformationDecoder


Y

t

S

t

⌦ �

�

Capacity C(0, P) = 12 log2(1 + SNR1)

Maximum energy rate Bmax , 1 + SNR2

Optimal dist: N (0, P)

Information-Energy Capacity Function CGC

(b, P) [Belhadj Amor et al.’16]

For any 0 6 b 6 1 + SNR2, the information-energy capacity function is

CGC(b, P) = maxX :E[X2]P and E[S2]>b

I (X ; Y ) =12

log2 (1 + SNR1) = C(0, P).

=) For any feasible energy rate 0 6 b 6 1 + SNR2 the information-optimal strategy isunchanged.

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Outline

1 Point-to-point Information-Energy Trade-off

2 Multi-User Simultaneous Energy and Information Transmission

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Multi-Access Channel With Minimum Energy Rate Constraint b

Energy Harvester

Transmitter 1

M1

M2

Transmitter 2

Receiver

(M̂1, M̂2)InformationDecoder

Min energy rate b

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Gaussian Multiple Access Channel With Energy Harvester

Transmitter 1M1h11

h21

h22

h12

x1,t

x2,tTransmitter 2M2

Z

t

Receiver

Q

t

⌦



Y1,t

Y2,t

⌦

⌦

�

�

Information Decoder: Y1,t = h11X1,t + h12X2,t + Zt

Energy harvester: Y2,t = h21X1,t + h22X2,t + Qt

n: blocklengthX1,t , X2,t , Qt

, Zt

2 R;Constant channel coefs h11, h12, h21, h22 > 0 satisfying :

8j 2 {1, 2}, khj

k2 1, with hj

, (hj1, hj2)

T (energy conservation principle)

{Zt

}, {Qt

} i.i.d. ⇠ N (0, 1)

Input power constraints: Pi

, 1n

n

X

t=1

E⇥

X 2i,t

⇤

6 Pi,max, i 2 {1, 2}.

Fully described by signal-to-noise ratios: SNRji

, |hji

|2Pi,max, (i , j) 2 {1, 2}2.

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Information Transmission

Transmitter 1M1h11

h21

h22

h12

x1,t

x2,tTransmitter 2M2

Z

t

Receiver

Q

t

⌦



Y1,t

Y2,t

⌦

⌦

�

�

Transmitters 1 and 2 send M1 and M2 to the information decoder

Messages M1 and M2 independent ; Mi

⇠ U{1, . . . , 2nR

i }

R1 and R2 are information transmission rates

Common randomness ⌦ known to all terminals

Probability of Error

P(n)error(R1, R2) , Pr

�

(M̂(n)1 , M̂(n)

2 ) 6= (M1, M2)

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Energy Transmission

Transmitter 1M1h11

h21

h22

h12

x1,t

x2,tTransmitter 2M2

Z

t

Receiver

Q

t

⌦



Y1,t

Y2,t

⌦

⌦

�

�

Minimum energy rate b at EH (in energy units per channel use) such that

0 6 b 6 1 + SNR21 + SNR22 + 2p

SNR21SNR22

Average energy rate: B (n) , 1n

n

X

t=1

Y 22,t

B energy rate such that with b 6 B 6 1 + SNR21 + SNR22 + 2p

SNR21SNR22

Guarantee B (n) > B with high probability

Probability of Energy Outage

P(n)outage(B) , Pr

n

B (n) < B � ✏o

, ✏ > 0 arbitrarily small

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Simultaneous Energy and Information Transmission (SEIT)

Transmitter 1M1h11

h21

h22

h12

x1,t

x2,tTransmitter 2M2

Z

t

Receiver

Q

t

⌦



Y1,t

Y2,t

⌦

⌦

�

�

Objective of SEITProvide blocklength-n coding schemes such that:(i) information transmission occurs at rates R1 and R2 with P

(n)error(R1, R2) ! 0; and

(ii) energy transmission occurs at rate B with P(n)outage(B) ! 0 and B � b.

Under these conditions, the information-energy rate-triplet (R1, R2, B) is achievable inthe G-MAC with minimum energy rate b.

) What are the fundamental limits on achievable information-energy rate-triplets?14 / 28

Information-Energy Capacity Region Eb

(SNR11, SNR12, SNR21, SNR22)[Belhadj Amor et al.’15]

Theorem: Information-Energy Capacity Region Eb

(SNR11

, SNR12

, SNR21

, SNR22

)

Eb

(SNR11, SNR12, SNR21, SNR22) contains all (R1, R2, B) that satisfy

06 R1 612

log2 (1 + �1 SNR11) ,

06 R2 612

log2 (1 + �2 SNR12) ,

06R1 + R2612

log2�

1 + �1 SNR11 + �2 SNR12�

,

b6 B 61 + SNR21 + SNR22 + 2p

(1 � �1)SNR21(1 � �2)SNR22,

with (�1, �2) 2 [0, 1]2.

�i

: power-splitting coefficient at transmitter i

�i

Pi,max to transmit information-carrying (IC) component ([Cover’75] and

[Wyner’76])(1 � �

i

)Pi,max to transmit energy-carrying (EC) component (common randomness)

S. B., S. M. Perlaza, I. Krikidis and H. V. Poor. “Feedback enhances simultaneous wirelessinformation and energy transmission in multiple access channels”, Technical Report, INRIA,No. 8804, Lyon, France, Nov., 2015.

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3-D Representation of E0(SNR11, SNR12, SNR21, SNR22)SNR11 = SNR12 = SNR21 = SNR22 = 10

Q3

Q2

Q1

Q4 Q5

R1 [bits/ch.use] R2 [bits/ch.use]

B[energy units/ch.use]

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Centralized Versus Decentralized SEIT

Centralized:I A central controller determines a network operating pointI Tx/Rx configuration or each component is imposed by controllerI Central controller optimizes a network metric! All (R1,R2,B) 2 E

b

(SNR11, SNR12, SNR21, SNR22) are feasible operating points

Decentralized:I Each component is autonomousI Each component determines its own Tx/Rx configurationI Each component optimizes an individual metric! Only some (R1,R2,B) 2 E

b

(SNR11, SNR12, SNR21, SNR22) are stable

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Decentralized MAC with Minimum Energy Rate Constraint b

Multi-Access Channel With Minimum Energy Rate Constraint b

Energy Harvester

Transmitter 1

M1

M2

Transmitter 2

Receiver


Min energy rate b

1 / 27

PLAYER 1

PLAYER 2

PLAYER 3

18 / 28

Game Formulation

Consider the following game in normal form:

G(b) =�

K, {Ak

}k2K , {u

k

}k2K

�

b 2 [0, 1 + SNR21 + SNR22 + 2p

SNR21SNR22]

Set of players K = {1, 2}Sets of actions A1 and A2

Utility function ui

: A1 ⇥A2 ! R+ such that

ui

(s1, s2) =

⇢

R

i

(s1, s2), if P

(n)error(R1,R2) < ✏ and P

(n)outage(b) < �

�1, otherwise,

where ✏ > 0 and � > 0 are arbitrarily small.

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Game Formulation

A transmit configuration si

2 Ai

can be described in terms of:I Information rates R

i

I Block-length n

I Power-split �i

I Average input power P

i

I Common randomness ⌦

I Channel input alphabet Xi

I Encoding functions f

(1)i

, . . . , f(n)i

, etc

Receiver adopts a fixed decoding strategy

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⌘-Nash Equilibrium (⌘-NE)

Definition (⌘-Nash Equilibrium)

Let ⌘ > 0. In the game G(b) =�

K, {Ak

}k2K , {u

k

}k2K

�

, an action profile (s⇤1 , s⇤2 ) is an⌘-Nash equilibrium if for all i 2 K and for all s

i

2 Ai

, it holds that

ui

(si

, s⇤j

)6ui

(s⇤i

, s⇤j

) + ⌘.

If ⌘ = 0, we obtain the classical definition of Nash equilibrium.

At any ⌘-NE and for all i 2 K, player i cannot obtain a utility improvement biggerthan ⌘ by changing its own action s

i

(stability)

J. F. Nash, “Equilibrium points in n-person games,” Proc. of the National Academy of

Sciences, vol. 36, pp. 48–49, 1950.

21 / 28

⌘-Nash Equilibrium Region

Definition (⌘-Nash Equilibrium Region)

Let ⌘ > 0. An (R1, R2, B) 2 Eb

(SNR11, SNR12, SNR21, SNR22) is said to be in the ⌘-NE

region of the game G(b) =�

K, {Ak

}k2K , {u

k

}k2K

�

if there exists an action profile

(s⇤1 , s⇤2 ) 2 A1 ⇥A2 that is an ⌘-NE and the following holds:

u1(s⇤1 , s⇤2 ) = R1 and u2(s

⇤1 , s⇤2 ) = R2.

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⌘-Nash Equilibrium Region with Single User Decoding (SUD)[Belhadj Amor et al.’16]

Theorem: NSUD(b): ⌘-Nash Equilibrium Region of G(b) with SUD

The set NSUD(b) contains all (R1, R2, B) 2 Eb

(SNR11, SNR12, SNR21, SNR22) such that:

06R1=12

log2

✓

1 +�1SNR11

1 + �2SNR12

◆

,

06R2=12

log2

✓

1 +�2SNR12

1 + �1SNR11

◆

,

b6B 61 + SNR21 + SNR22 + 2p

(1 � �1)SNR21(1 � �2)SNR22,

where �1 = �2 = 1 when b 2 [0, 1 + SNR21 + SNR22] and (�1, �2) satisfy

b = 1 + SNR21 + SNR22 + 2p

(1 � �1)SNR21(1 � �2)SNR22

when b 2 (1 + SNR21 + SNR22, 1 + SNR21 + SNR22 + 2p

SNR21SNR22].

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⌘-Nash Equilibrium Region with Successive Interference Cancellation (SIC)[Belhadj Amor et al.’16]

SIC(i ! j): receiver uses SIC with decoding order: transmitter i before j .

Theorem: NSIC(i!j)(b): ⌘-Nash Equilibrium Region of G(b) with SIC(i ! j)

The set NSIC(i!j)(b) contains all (R1, R2, B) 2 Eb

(SNR11, SNR12, SNR21, SNR22) suchthat:

06Ri

=12

log2

✓

1 +�i

SNR1i

1 + �j

SNR1j

◆

,

06Rj

=12

log2 (1 + �j

SNR1j) ,

b6B 61 + SNR21 + SNR22 + 2p

(1 � �1)SNR21(1 � �2)SNR22,

where �1 = �2 = 1 when b 2 [0, 1 + SNR21 + SNR22] and (�1, �2) satisfy

b = 1 + SNR21 + SNR22 + 2p

(1 � �1)SNR21(1 � �2)SNR22

when b 2 (1 + SNR21 + SNR22, 1 + SNR21 + SNR22 + 2p

SNR21SNR22].

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⌘-Nash Equilibrium Region of G(b)[Belhadj Amor et al.’16]

Any time-sharing combination between SUD, SIC(1 ! 2), and SIC(2 ! 1)

Theorem: N (b) , ⌘-Nash Equilibrium Region of G(b)The set N (b) is defined as:

N (b) = Convex hull✓

NSUD(b) [NSIC(1!2)(b) [NSIC(2!1)(b)

◆

.

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⌘-Nash Equilibrium Region for b 6 1 + SNR21 + SNR22Projection over the plane R1-R2 for SNR11 = SNR12 = SNR21 = SNR22 = 10

Q3 Q2

Q1

Q4

Q5Q6


R1[bits/ch.use] R2[bits/ch.use]

R1[bits/ch.use]

R2[b

its/

ch.u

se]

SIC(1 ! 2)

SIC(2 ! 1)

SUD

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Square point: Projection of NSUD(b)

Round points: Projection of NSIC(i!j)(b)

Region inside solid lines: Projection of E0(10, 10, 10, 10)

Blue region: Projection of convex hull of NSUD(b) [NSIC(1!2)(b) [NSIC(2!1)(b)

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⌘-Nash Equilibrium Region for b = 0.7Bmax > 1 + SNR21 + SNR22.Projection over the plane R1-R2 for SNR11 = SNR12 = SNR21 = SNR22 = 10


R1[bits/ch.use] R2[bits/ch.use]

B = b

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

R1[bits/ch.use]

R2[b

its/

ch.u

se]

SIC(1 ! 2)

SIC(2 ! 1)

SUD

Dotted line: Projection of NSUD(b)

Dashed line: Projection of NSIC(i!j)(b)

Region inside solid lines: Projection of E0(10, 10, 10, 10)

Blue region: Projection of convex hull of NSUD(b) [NSIC(1!2)(b) [NSIC(2!1)(b)

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Summary

SEIT in point-to-point channelsI Fundamental limits on information rate for a minimum energy rate b characterized by

information-energy capacity functionI Information-energy trade-off is not always observed!

SEIT in multi-user channelsI Centralized G-MAC with minimum energy rate constraint:

FFundamental limits characterized by information-energy capacity region

I Decentralized G-MAC with minimum energy rate constraintF

Fundamental limits characterized by ⌘-NE information-energy region

FThere always exists an ⌘-NE

FThere always exists a Pareto-optimal ⌘-NE

Open problems:I Extension to K > 2-usersI Other Equilibria concepts (Stackelberg, Satisfaction, etc.)I SEIT in other multi-user channels (Broadcast channel, interference channel, etc)

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fundamental limits of simultaneous energy and …fundamental limits of simultaneous energy and...

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