Fundamental Limits of Simultaneous Energy and Information
Transmission
Selma Belhadj Amor and Samir M. Perlaza
Inria, Lyon, France
International Conference on Telecommunications (ICT)
Thessaloniki, Greece
May 17, 2016
1 / 28
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.use
If received energy rate is constrained to be
I
I
at maximum possible value ! can transmit 1 bit/ch.use
larger than a given value ! in some cases, can transmit 0 bits/ch.use
2 / 28
Outline
1
Point-to-point Information-Energy Trade-off
2
Multi-User Simultaneous Energy and Information Transmission
2 / 28
Discrete Memoryless Point-to-Point Channel
M2M
Encoder
x(M)
Channel
PY |X
Y
Decoder
M̂ (n)
Transmission blocklength n
Finite input and output alphabets X and Y
Transition law PY |X (memoryless)
Transmitter sends message M 2 M , {1, 2, . . . , 2nR }
Information rate R
Decoder forms estimate M̂ (n)
Probability of Error
(n)
Perror (R) , Pr M̂ (n) 6= M
3 / 28
Discrete Memoryless Channel with Energy Harvester
M2M
Encoder
Channel
PY S|X
x(M)
Y
Decoder M̂ (n)
S
Energy
Harvester
Additional output alphabet S; Transition law PYS|X
Energy function ! : S ! R+
Harvested energy from s = (s1 , . . . , sn ) is
!(s) =
Pn
t=1
!(st )
Average energy rate (in energy-units per channel use) at the EH:
B
(n)
n
1X
,
!(St ).
n t=1
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
n
(n)
Poutage (B) , Pr B (n) < B
o
✏ , ✏ > 0 arbitrarily small
4 / 28
Simultaneous Energy and Information Transmission (SEIT)
M2M
Encoder
x(M)
Channel
PY S|X
Y
Decoder M̂ (n)
S
Energy
Harvester
Objective of SEIT
Provide blocklength-n coding schemes such that:
(n)
(i) information transmission occurs at rate R with Perror (R) ! 0; and
(n)
(ii) energy transmission occurs at rate B with Poutage (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?
5 / 28
Information Capacity Under Minimum Energy Rate b
For each blocklength n, define the function C (n) (b) as follows:
C (n) (b) ,
max
X 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 sup
n!1
1 (n)
C (b).
n
Theorem: Information Capacity Under Minimum Energy Rate [Varshney’08]
The supremum over all achievable information rates in the DMC under a minimum
energy 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.
6 / 28
Example: Noiseless binary channel
S =Y
1
0
P(1|1) = P(0|0) = 1 and P(1|0) = P(1|0) = 0
0
Y =S
1
X
Channel capacity: C = 1 bit/ch.use.
1
1
Capacity-achieving dist: Ber( 12 )
Symbol 1 ! 1 energy-unit & Symbol 0 ! 0 energy-unit
Maximum energy: Bmax = 1 energy-units/ch.use
(Symbol ’1’ always sent)
Information-Energy Capacity Function of Noiseless Channel
1.1
1
CNC (b) =
⇢
1,
H2 (b),
if
if
0 6 b 6 12 ,
1
6 b 6 1,
2
CNC (b)[bits/ch. use]
Information-Energy Capacity Function [Varshney’08]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Bmax = 1
0
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]
) The more stringent the energy rate constraint is, the more the transmitter needs to
switch over to using the most energetic symbol
7 / 28
Gaussian Memoryless Channel with Energy Harvester
M
xt
Transmitter
Receiver
Zt
h1
⌦
Yt
h2
St
Information M̂
Decoder
Energy
Harvester
Energy
Qt
Information Decoder: Yt = h1 Xt + Zt ,
Energy harvester:
S t = h 2 X t + 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)
n
1 X ⇥ 2⇤
Input power constraints:
E Xt 6 P.
n t=1
Fully described by signal-to-noise ratios:
SNRi , |hi |2 P,
Output energy function: !(s) , s 2
i 2 {1, 2}x.
8 / 28
Gaussian Memoryless Channel with Energy Harvester
M
Transmitter
xt
Receiver
Zt
⌦
h1
Yt
h2
St
Information M̂
Decoder
Energy
Harvester
Energy
Qt
Capacity C(0, P) =
1
2
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) =
max
X :E[X 2 ]P and E[S 2 ]>b
I (X ; Y ) =
1
log2 (1 + SNR1 ) = C(0, P).
2
=) For any feasible energy rate 0 6 b 6 1 + SNR2 the information-optimal strategy is
unchanged.
9 / 28
Outline
1
Point-to-point Information-Energy Trade-off
2
Multi-User Simultaneous Energy and Information Transmission
9 / 28
Multi-Access Channel With Minimum Energy Rate Constraint b
Transmitter 1
M1
Receiver
Information (M̂1 , M̂2 )
Decoder
Min energy rate b
M2
Energy Harvester
Transmitter 2
10 / 28
Gaussian Multiple Access Channel With Energy Harvester
⌦
M1 Transmitter 1
x1,t
⌦
x2,t
⌦ h
22
Information (M̂1 , M̂2 )
Decoder
Y1,t
h21
h12
M2 Transmitter 2
Receiver
Zt
h11
Y2,t
Energy
Harvester
Energy
Qt
Information Decoder: Y1,t = h11 X1,t + h12 X2,t + Zt
Energy harvester:
Y2,t = h21 X1,t + h22 X2,t + Qt
n: blocklength
X1,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)
n
1X ⇥ 2⇤
Input power constraints: Pi ,
E Xi,t 6 Pi,max ,
n t=1
i 2 {1, 2}.
Fully described by signal-to-noise ratios: SNRji , |hji |2 Pi,max ,
(i, j) 2 {1, 2}2 .
11 / 28
Information Transmission
⌦
M1 Transmitter 1
x1,t
⌦
Y1,t
h21
h12
M2 Transmitter 2
x2,t
Receiver
Zt
h11
⌦ h
22
Y2,t
Information (M̂1 , M̂2 )
Decoder
Energy
Harvester
Energy
Qt
Transmitters 1 and 2 send M1 and M2 to the information decoder
Messages M1 and M2 independent ; Mi ⇠ U {1, . . . , 2nRi }
R1 and R2 are information transmission rates
Common randomness ⌦ known to all terminals
Probability of Error
(n)
(n)
(n)
Perror (R1 , R2 ) , Pr (M̂1 , M̂2 ) 6= (M1 , M2 )
12 / 28
Energy Transmission
⌦
x1,t
M1 Transmitter 1
Receiver
Zt
⌦
h11
h21
h12
M2 Transmitter 2
x2,t
Information (M̂1 , M̂2 )
Decoder
Y1,t
⌦ h
22
Y2,t
Energy
Harvester
Energy
Qt
Minimum energy rate b at EH (in energy units per channel use) such that
p
0 6 b 6 1 + SNR21 + SNR22 + 2 SNR21 SNR22
Average energy rate: B
(n)
n
1X 2
,
Y2,t
n t=1
p
B energy rate such that with b 6 B 6 1 + SNR21 + SNR22 + 2 SNR21 SNR22
Guarantee B (n) > B with high probability
Probability of Energy Outage
n
(n)
Poutage (B) , Pr B (n) < B
o
✏ , ✏ > 0 arbitrarily small
13 / 28
Simultaneous Energy and Information Transmission (SEIT)
⌦
M1 Transmitter 1
x1,t
⌦
Y1,t
h21
h12
M2 Transmitter 2
x2,t
Receiver
Zt
h11
⌦ h
22
Y2,t
Information (M̂1 , M̂2 )
Decoder
Energy
Harvester
Energy
Qt
Objective of SEIT
Provide blocklength-n coding schemes such that:
(n)
(i) information transmission occurs at rates R1 and R2 with Perror (R1 , R2 ) ! 0; and
(n)
(ii) energy transmission occurs at rate B with Poutage (B) ! 0 and B
b.
Under these conditions, the information-energy rate-triplet (R1 , R2 , B) is achievable in
the 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
1
6 log2 (1 + 1 SNR11 ) ,
2
1
06 R2 6 log2 (1 + 2 SNR12 ) ,
2
1
06R1 + R2 6 log2 1 + 1 SNR11 + 2 SNR12 ,
2
p
b6 B 61 + SNR21 + SNR22 + 2 (1
1 )SNR21 (1
06
with (
1,
2)
R1
2 )SNR22 ,
2 [0, 1]2 .
: 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)
i
S. B., S. M. Perlaza, I. Krikidis and H. V. Poor. “Feedback enhances simultaneous wireless
information and energy transmission in multiple access channels”, Technical Report, INRIA,
No. 8804, Lyon, France, Nov., 2015.
15 / 28
3-D Representation of E0 (SNR11 , SNR12 , SNR21 , SNR22 )
SNR11 = SNR12 = SNR21 = SNR22 = 10
B[energy units/ch.use]
Q1
Q4
Q5
Q2
Q3
R1 [bits/ch.use]
R2 [bits/ch.use]
16 / 28
Centralized Versus Decentralized SEIT
Centralized:
I
A central controller determines a network operating point
I
Tx/Rx configuration or each component is imposed by controller
I
Central controller optimizes a network metric
! All (R1 , R2 , B) 2 Eb (SNR11 , SNR12 , SNR21 , SNR22 ) are feasible operating points
Decentralized:
I
Each component is autonomous
I
Each component determines its own Tx/Rx configuration
I
Each component optimizes an individual metric
! Only some (R1 , R2 , B) 2 Eb (SNR11 , SNR12 , SNR21 , SNR22 ) are stable
17 / 28
Decentralized MAC with Minimum Energy Rate Constraint b
Multi-Access Channel With Minimum Energy Rate Constraint b
PLAYER 1
Transmitter 1
M1
PLAYER 3
Receiver
Information (M̂1 , M̂2 )
Decoder
Min energy rate b
M2
Energy Harvester
Transmitter 2
PLAYER 2
1 / 27
18 / 28
Game Formulation
Consider the following game in normal form:
G(b) = K, {Ak }k2K , {uk }k2K
p
b 2 [0, 1 + SNR21 + SNR22 + 2 SNR21 SNR22 ]
Set of players K = {1, 2}
Sets of actions A1 and A2
Utility function ui : A1 ⇥ A2 ! R+ such that
⇢
(n)
(n)
Ri (s1 , s2 ), if
Perror (R1 , R2 ) < ✏ and Poutage (b) <
ui (s1 , s2 ) =
1,
where ✏ > 0 and
otherwise,
> 0 are arbitrarily small.
19 / 28
Game Formulation
A transmit configuration si 2 Ai can be described in terms of:
I
Information rates Ri
I
Block-length n
I
Power-split
I
Average input power Pi
I
Common randomness ⌦
I
Channel input alphabet Xi
I
i
Encoding functions fi
(1)
, . . . , fi
(n)
, etc
Receiver adopts a fixed decoding strategy
20 / 28
⌘-Nash Equilibrium (⌘-NE)
Definition (⌘-Nash Equilibrium)
Let ⌘ > 0. In the game G(b) = K, {Ak }k2K , {uk }k2K , an action profile (s1⇤ , s2⇤ ) is an
⌘-Nash equilibrium if for all i 2 K and for all si 2 Ai , it holds that
ui (si , sj⇤ )6ui (si⇤ , sj⇤ ) + ⌘.
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 bigger
than ⌘ by changing its own action si (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 , {uk }k2K if there exists an action profile
(s1⇤ , s2⇤ ) 2 A1 ⇥ A2 that is an ⌘-NE and the following holds:
u1 (s1⇤ , s2⇤ ) = R1 and u2 (s1⇤ , s2⇤ ) = R2 .
22 / 28
⌘-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:
✓
◆
1
1 SNR11
06R1 = log2 1 +
,
2
1 + 2 SNR12
✓
◆
1
2 SNR12
06R2 = log2 1 +
,
2
1 + 1 SNR11
p
b6B 61 + SNR21 + SNR22 + 2 (1
1 )SNR21 (1
2 )SNR22 ,
where
= 1 when b 2 [0, 1 + SNR21 + SNR22 ] and ( 1 , 2 ) satisfy
p
b = 1 + SNR21 + SNR22 + 2 (1
1 )SNR21 (1
2 )SNR22
p
when b 2 (1 + SNR21 + SNR22 , 1 + SNR21 + SNR22 + 2 SNR21 SNR22 ].
1
=
2
23 / 28
⌘-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 ) such
that:
✓
◆
1
i SNR1i
06Ri = log2 1 +
,
2
1 + j SNR1j
1
06Rj = log2 (1 + j SNR1j ) ,
2
p
b6B 61 + SNR21 + SNR22 + 2 (1
1 )SNR21 (1
2 )SNR22 ,
where
= 1 when b 2 [0, 1 + SNR21 + SNR22 ] and ( 1 , 2 ) satisfy
p
b = 1 + SNR21 + SNR22 + 2 (1
1 )SNR21 (1
2 )SNR22
p
when b 2 (1 + SNR21 + SNR22 , 1 + SNR21 + SNR22 + 2 SNR21 SNR22 ].
1
=
2
24 / 28
⌘-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) .
25 / 28
⌘-Nash Equilibrium Region for b 6 1 + SNR21 + SNR22
Projection over the plane R1 -R2 for SNR11 = SNR12 = SNR21 = SNR22 = 10
2
B[energy units/ch.use]
SIC(1 ! 2)
1.8
R2 [bits/ch.use]
1.6
1.4
1.2
Q4
1
Q1
0.8
Q2
Q6
Q5
SUD
0.4
R1 [bits/ch.use]
0.2
0
Q3
SIC(2 ! 1)
0.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R2 [bits/ch.use]
2
R1 [bits/ch.use]
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)
26 / 28
⌘-Nash Equilibrium Region for b = 0.7Bmax > 1 + SNR21 + SNR22 .
Projection over the plane R1 -R2 for SNR11 = SNR12 = SNR21 = SNR22 = 10
2
B[energy units/ch.use]
1.8
R2 [bits/ch.use]
1.6
1.4
B=b
SIC(2 ! 1)
1.2
1
0.8
0.6
SUD
0.4
SIC(1 ! 2)
0.2
0
R1 [bits/ch.use]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R2 [bits/ch.use]
2
R1 [bits/ch.use]
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)
27 / 28
Summary
SEIT in point-to-point channels
I
I
Fundamental limits on information rate for a minimum energy rate b characterized by
information-energy capacity function
Information-energy trade-off is not always observed!
SEIT in multi-user channels
I
Centralized G-MAC with minimum energy rate constraint:
F
I
Fundamental limits characterized by information-energy capacity region
Decentralized G-MAC with minimum energy rate constraint
F
F
F
Fundamental limits characterized by ⌘-NE information-energy region
There always exists an ⌘-NE
There always exists a Pareto-optimal ⌘-NE
Open problems:
I
I
I
Extension to K > 2-users
Other Equilibria concepts (Stackelberg, Satisfaction, etc.)
SEIT in other multi-user channels (Broadcast channel, interference channel, etc)
28 / 28