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
Encoder x(M) ChannelP
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)
Encoder x(M) ChannelP
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
EnergyHarvester Energy
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
⌦
(M̂1, M̂2)InformationDecoder
EnergyHarvester Energy
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
⌦
(M̂1, M̂2)InformationDecoder
EnergyHarvester Energy
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
⌦
(M̂1, M̂2)InformationDecoder
EnergyHarvester Energy
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
⌦
(M̂1, M̂2)InformationDecoder
EnergyHarvester Energy
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
(M̂1, M̂2)InformationDecoder
Min energy rate b
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PLAYER 1
PLAYER 2
PLAYER 3
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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.
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⌘-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
B[energy units/ch.use]
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
B[energy units/ch.use]
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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