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Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Cooperation and Competition in Multi-user
Wireless Networks
Brice Djeumou
Joint work with E.V. Belmega and S. LasaulceLaboratoire des signaux et systemes (LSS)
Gif-sur-Yvette, France
ETS Talk
March 26, 2010
1 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
multiuser wireless networks: Motivations
Licence-free frequency band
Deal with the inter-user interference.
Low transmit power.
How can we increase the transmission rate?Cognitive radio: dynamically exploit the available radio frequency spectrum inorder to efficiently avoiding interference.
Cooperation between transmitter/receiver nodes or with additional relay nodes.
2 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
multiuser wireless networks: Motivations
Licence-free frequency band
Deal with the inter-user interference.
Low transmit power.
How can we increase the transmission rate?Cognitive radio: dynamically exploit the available radio frequency spectrum inorder to efficiently avoiding interference.
Cooperation between transmitter/receiver nodes or with additional relay nodes.
2 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Cognitive radio
Goal
Dynamically exploit the available radio frequency spectrum in orderto efficiently avoiding interference.
Remarks
Resource allocation game between users.
Making use of tools from game theory.
3 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Cooperative channels
Major information-theoretic works: [Cover and El Gamal, 1979]
Useful to assess the benefits of cooperation in terms of communication rate(standard relay channel).
They introduced two major relaying strategies: decode-and-forward andestimate-and-forward.
The relaying strategy is a key point for the cooperation between users.
The Return of Cooperative ChannelsMIMO channels
[Telatar, 1995 and 1999], [Foschini, 1996 and 1998]: Information-theoretic analysis of
MIMO systems (diversity gain, multiplexing gain) → Increase communication rate.
Virtual MIMO
[Sendonaris et al., 1998 and 2003]: The benefits (rate, diversity) of MIMO systems
can be obtained in a distributive manner in wireless networks.
Objectives of user cooperation
Increase the range of wireless communications or the the communication rate.
Increase the reliability of communications in fading environnements.4 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Cooperative channels
Major information-theoretic works: [Cover and El Gamal, 1979]
Useful to assess the benefits of cooperation in terms of communication rate(standard relay channel).
They introduced two major relaying strategies: decode-and-forward andestimate-and-forward.
The relaying strategy is a key point for the cooperation between users.
The Return of Cooperative ChannelsMIMO channels
[Telatar, 1995 and 1999], [Foschini, 1996 and 1998]: Information-theoretic analysis of
MIMO systems (diversity gain, multiplexing gain) → Increase communication rate.
Virtual MIMO
[Sendonaris et al., 1998 and 2003]: The benefits (rate, diversity) of MIMO systems
can be obtained in a distributive manner in wireless networks.
Objectives of user cooperation
Increase the range of wireless communications or the the communication rate.
Increase the reliability of communications in fading environnements.4 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Cooperative channels
Major information-theoretic works: [Cover and El Gamal, 1979]
Useful to assess the benefits of cooperation in terms of communication rate(standard relay channel).
They introduced two major relaying strategies: decode-and-forward andestimate-and-forward.
The relaying strategy is a key point for the cooperation between users.
The Return of Cooperative ChannelsMIMO channels
[Telatar, 1995 and 1999], [Foschini, 1996 and 1998]: Information-theoretic analysis of
MIMO systems (diversity gain, multiplexing gain) → Increase communication rate.
Virtual MIMO
[Sendonaris et al., 1998 and 2003]: The benefits (rate, diversity) of MIMO systems
can be obtained in a distributive manner in wireless networks.
Objectives of user cooperation
Increase the range of wireless communications or the the communication rate.
Increase the reliability of communications in fading environnements.4 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Cooperative channels
Major information-theoretic works: [Cover and El Gamal, 1979]
Useful to assess the benefits of cooperation in terms of communication rate(standard relay channel).
They introduced two major relaying strategies: decode-and-forward andestimate-and-forward.
The relaying strategy is a key point for the cooperation between users.
The Return of Cooperative ChannelsMIMO channels
[Telatar, 1995 and 1999], [Foschini, 1996 and 1998]: Information-theoretic analysis of
MIMO systems (diversity gain, multiplexing gain) → Increase communication rate.
Virtual MIMO
[Sendonaris et al., 1998 and 2003]: The benefits (rate, diversity) of MIMO systems
can be obtained in a distributive manner in wireless networks.
Objectives of user cooperation
Increase the range of wireless communications or the the communication rate.
Increase the reliability of communications in fading environnements.4 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Objectives:
Cooperative multi-user system + MultiMultiband radio
Generalization of Cover and El Gamal’s results.
Introduction of a theoretic approach in a decentralized context.
Multiband radio with Q non-overlapping frequency bands.
Selfish users maximizing their individual transmission rate.
Power Allocation Game.
5 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Objectives:
Cooperative multi-user system + MultiMultiband radio
Generalization of Cover and El Gamal’s results.
Introduction of a theoretic approach in a decentralized context.
Multiband radio with Q non-overlapping frequency bands.
Selfish users maximizing their individual transmission rate.
Power Allocation Game.
5 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
6 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
7 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Cooperation for multiuser channels
Multiple access relay channel (MARC): [Sankaranarayanan et al., 2003]
Broadcast relay channel (BRC): [Liang et al., 2007], [Kramer, 2005]
Interference channel
Capacity region known for the special case of strong interference: [Carleial,1975], [Sato, 1978]
Best inner bound by [Han and Kobayashi, 1981]: rate-splitting + time-sharing.
Interference relay channel (IRC)
Introduced by [Sahin and Erkip, 2007]: rate region for the Gaussian case with aDF-based strategy.
Results of [Maric & al, 2008]: DF and interference forwarding.
Our contributions
Treat both discrete and Gaussian cases.
Coding theorems based on several strategies (DF and EF).
Two approaches for the EF-based strategy: Bi-level and single-levelcompressions.
A simple AF-based strategy for the Gaussian case.
8 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Cooperation for multiuser channels
Multiple access relay channel (MARC): [Sankaranarayanan et al., 2003]
Broadcast relay channel (BRC): [Liang et al., 2007], [Kramer, 2005]
Interference channel
Capacity region known for the special case of strong interference: [Carleial,1975], [Sato, 1978]
Best inner bound by [Han and Kobayashi, 1981]: rate-splitting + time-sharing.
Interference relay channel (IRC)
Introduced by [Sahin and Erkip, 2007]: rate region for the Gaussian case with aDF-based strategy.
Results of [Maric & al, 2008]: DF and interference forwarding.
Our contributions
Treat both discrete and Gaussian cases.
Coding theorems based on several strategies (DF and EF).
Two approaches for the EF-based strategy: Bi-level and single-levelcompressions.
A simple AF-based strategy for the Gaussian case.
8 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Cooperation for multiuser channels
Multiple access relay channel (MARC): [Sankaranarayanan et al., 2003]
Broadcast relay channel (BRC): [Liang et al., 2007], [Kramer, 2005]
Interference channel
Capacity region known for the special case of strong interference: [Carleial,1975], [Sato, 1978]
Best inner bound by [Han and Kobayashi, 1981]: rate-splitting + time-sharing.
Interference relay channel (IRC)
Introduced by [Sahin and Erkip, 2007]: rate region for the Gaussian case with aDF-based strategy.
Results of [Maric & al, 2008]: DF and interference forwarding.
Our contributions
Treat both discrete and Gaussian cases.
Coding theorems based on several strategies (DF and EF).
Two approaches for the EF-based strategy: Bi-level and single-levelcompressions.
A simple AF-based strategy for the Gaussian case.
8 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Cooperation for multiuser channels
Multiple access relay channel (MARC): [Sankaranarayanan et al., 2003]
Broadcast relay channel (BRC): [Liang et al., 2007], [Kramer, 2005]
Interference channel
Capacity region known for the special case of strong interference: [Carleial,1975], [Sato, 1978]
Best inner bound by [Han and Kobayashi, 1981]: rate-splitting + time-sharing.
Interference relay channel (IRC)
Introduced by [Sahin and Erkip, 2007]: rate region for the Gaussian case with aDF-based strategy.
Results of [Maric & al, 2008]: DF and interference forwarding.
Our contributions
Treat both discrete and Gaussian cases.
Coding theorems based on several strategies (DF and EF).
Two approaches for the EF-based strategy: Bi-level and single-levelcompressions.
A simple AF-based strategy for the Gaussian case.
8 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
9 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Rate-splitting ([Carleial, 1978]) at each source node
(W10,W11) at S1 and (W20,W22) at S2 .
D1 decodes the triplet (W10,W11,W20) and R1 = R10 + R11.
D2 decodes the triplet (W10,W20,W22) and R2 = R20 + R22.
Theorem (DF-based strategy)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, any rate quadruplet (R10, R11, R20, R22) satisfying
∑
i∈I
Ri ≤ I(
VI ; Yr | US , Xr ,VIC
)
for all I ⊆ S = {10, 11, 20, 22},
∑
i∈I1
Ri ≤ I
(
UI1, VI1
; Y1 | UIC1, V
IC1
)
for all I1 ⊆ S1 = {10, 11, 20},
∑
i∈I2
Ri ≤ I
(
UI2, VI2
; Y2 | UIC2, V
IC2
)
for all I2 ⊆ S2 = {20, 22, 10},
for some joint distribution p(u10)p(v10|u10)p(u11)p(v11|u11)p(x1|v10, v11)p(u20)p(v20|u20)p(u22)p(v22|u22)×
p(x2|v20, v22)p(xr |u10, u11, u20, u22), is achievable, where IC , IC1 and IC
2 and the complements of I, I1 andI2 respectively in S, S1 and S2 . We have VI = {Vj , j ∈ I}.
10 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Rate-splitting ([Carleial, 1978]) at each source node
(W10,W11) at S1 and (W20,W22) at S2 .
D1 decodes the triplet (W10,W11,W20) and R1 = R10 + R11.
D2 decodes the triplet (W10,W20,W22) and R2 = R20 + R22.
Theorem (DF-based strategy)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, any rate quadruplet (R10, R11, R20, R22) satisfying
∑
i∈I
Ri ≤ I(
VI ; Yr | US , Xr ,VIC
)
for all I ⊆ S = {10, 11, 20, 22},
∑
i∈I1
Ri ≤ I
(
UI1, VI1
; Y1 | UIC1, V
IC1
)
for all I1 ⊆ S1 = {10, 11, 20},
∑
i∈I2
Ri ≤ I
(
UI2, VI2
; Y2 | UIC2, V
IC2
)
for all I2 ⊆ S2 = {20, 22, 10},
for some joint distribution p(u10)p(v10|u10)p(u11)p(v11|u11)p(x1|v10, v11)p(u20)p(v20|u20)p(u22)p(v22|u22)×
p(x2|v20, v22)p(xr |u10, u11, u20, u22), is achievable, where IC , IC1 and IC
2 and the complements of I, I1 andI2 respectively in S, S1 and S2 . We have VI = {Vj , j ∈ I}.
10 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Bi-level compression feature
The relay increases interference at each receiver node.
Theorem (EF-based strategy: Bi-level resolution compression)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, the rate quadruplet (R10,R11,R20,R22) is achievable, where
∑
i∈I1
Ri ≤ I
(
VI1;Y1, Yr1 | U1,VIC
1
)
for all I1 ⊆ S1 = {10, 11, 20},
∑
i∈I2
Ri ≤ I
(
VI2;Y2, Yr2 | U2,VIC
2
)
for all I2 ⊆ S2 = {20, 22, 10},
under the constraintsI (Yr ; Yr1|U1,Y1) ≤ I (U1;Y1),
I (Yr ; Yr2|U2,Y2) ≤ I (U2;Y2),
for some joint distribution
p (v10, v11, v20, v22, x1, x2, u1, u2, xr , y1, y2, yr , yr1, yr2, yr1, yr2) =p(v10)p(v11)p(x1|v10, v11)p(v20)p(v22)p(x2|v20, v22)p (u1) p (u2) p (xr |u1, u2)×p (y1, y2, yr |x1, x2, xr ) p (yr1|yr , u1) p (yr2|yr , u2) .
11 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Bi-level compression feature
The relay increases interference at each receiver node.
Theorem (EF-based strategy: Bi-level resolution compression)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, the rate quadruplet (R10,R11,R20,R22) is achievable, where
∑
i∈I1
Ri ≤ I
(
VI1;Y1, Yr1 | U1,VIC
1
)
for all I1 ⊆ S1 = {10, 11, 20},
∑
i∈I2
Ri ≤ I
(
VI2;Y2, Yr2 | U2,VIC
2
)
for all I2 ⊆ S2 = {20, 22, 10},
under the constraintsI (Yr ; Yr1|U1,Y1) ≤ I (U1;Y1),
I (Yr ; Yr2|U2,Y2) ≤ I (U2;Y2),
for some joint distribution
p (v10, v11, v20, v22, x1, x2, u1, u2, xr , y1, y2, yr , yr1, yr2, yr1, yr2) =p(v10)p(v11)p(x1|v10, v11)p(v20)p(v22)p(x2|v20, v22)p (u1) p (u2) p (xr |u1, u2)×p (y1, y2, yr |x1, x2, xr ) p (yr1|yr , u1) p (yr2|yr , u2) .
11 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Bi-level compression feature
The relay increases interference at each receiver node.
Theorem (EF-based strategy: Bi-level resolution compression)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, the rate quadruplet (R10,R11,R20,R22) is achievable, where
∑
i∈I1
Ri ≤ I
(
VI1;Y1, Yr1 | U1,VIC
1
)
for all I1 ⊆ S1 = {10, 11, 20},
∑
i∈I2
Ri ≤ I
(
VI2;Y2, Yr2 | U2,VIC
2
)
for all I2 ⊆ S2 = {20, 22, 10},
under the constraintsI (Yr ; Yr1|U1,Y1) ≤ I (U1;Y1),
I (Yr ; Yr2|U2,Y2) ≤ I (U2;Y2),
for some joint distribution
p (v10, v11, v20, v22, x1, x2, u1, u2, xr , y1, y2, yr , yr1, yr2, yr1, yr2) =p(v10)p(v11)p(x1|v10, v11)p(v20)p(v22)p(x2|v20, v22)p (u1) p (u2) p (xr |u1, u2)×p (y1, y2, yr |x1, x2, xr ) p (yr1|yr , u1) p (yr2|yr , u2) .
11 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Theorem (EF-based strategy: Single-level compression)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, the rate quadruplet (R10,R11,R20,R22) is achievable, where
∑
i∈Ik
Ri ≤ I
(
VIk; Yk , Yr | Xr ,VIC
k
)
for all Ik ⊆ Sk , k ∈ {1, 2}
under the constraint
maxk
I (Yr ; Yr |Xr , Yk ) ≤ mink
I (Xr ;Yk ),
for some joint distribution
p (v10, v11, v20, v22, x1, x2, xr , y1, y2, yr , yr1, yr2, yr ) =p(v10)p(v11)p(x1|v10, v11)p(v20)p(v22)p(x2|v20, v22)p(xr )p (y1, y2, yr |x1, x2, xr ) p (yr |yr , xr ) .
Single-level compression feature
The estimation noise level is lower bounded by the worse receiver node.
12 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Theorem (EF-based strategy: Single-level compression)
For the DMIRC (X1 × X2 × Xr , p (y1, y2, yr |x1, x2, xr ) ,Y1 × Y2 × Yr ) with both private and commonmessages, the rate quadruplet (R10,R11,R20,R22) is achievable, where
∑
i∈Ik
Ri ≤ I
(
VIk; Yk , Yr | Xr ,VIC
k
)
for all Ik ⊆ Sk , k ∈ {1, 2}
under the constraint
maxk
I (Yr ; Yr |Xr , Yk ) ≤ mink
I (Xr ;Yk ),
for some joint distribution
p (v10, v11, v20, v22, x1, x2, xr , y1, y2, yr , yr1, yr2, yr ) =p(v10)p(v11)p(x1|v10, v11)p(v20)p(v22)p(x2|v20, v22)p(xr )p (y1, y2, yr |x1, x2, xr ) p (yr |yr , xr ) .
Single-level compression feature
The estimation noise level is lower bounded by the worse receiver node.
12 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
13 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
System model
power constraints: E|X1|2 ≤ P1, E|X2|
2 ≤ P2 and E|Xr |2 ≤ Pr .
14 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Corollary (DF-based strategy – Sahin & Erkip, 2008)When DF is assumed, the following region is achievable:
R1 ≤ min
{
C
(
|h1r |2(1 − τ1)P1
Nr
)
,C
(
|h11|2P1 + |hr1|
2ν1Pr + 2Re(h11h∗r1)√
τ1P1ν1Pr
|h21|2P2 + |hr1|2ν2Pr + 2Re(h21h∗r1)√
τ2P2ν2Pr + N1
)}
R2 ≤ min
{
C
(
|h2r |2(1 − τ2)P2
Nr
)
,C
(
|h22|2P2 + |hr2|
2ν2Pr + 2Re(h22h∗r2)√
τ2P2ν2Pr
|h12|2P1 + |hr2|2ν1Pr + 2Re(h12h∗r2)√
τ1P1ν1Pr + N2
)}
R1 + R2 ≤ C
(
|h1r |2(1 − τ1)P1 + |h2r |
2(1 − τ2)P2
Nr
)
,
where (ν1, ν2) ∈ [0, 1]2 s.t. ν1 + ν2 ≤ 1 and (τ1, τ2) ∈ [0, 1]2 .
15 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Corollary (EF strategy: Bi-level resolution compression with only private messages)
For the Gaussian IRC with only private messages and with the bi-level resolution estimate-and-forward strategy, therate pair (R11, R22) is achievable, where
1 if C
(
|hr1|2ν2Pr
|h11|2P1 + |h21|2P2 + |hr1|2ν1Pr + N1
)
≥ C
(
|hr2|2ν2Pr
|h22|2P2 + |h12|2P1 + |hr2|2ν1Pr + N2
)
,
we have
R11 ≤ C
|h11|2P1
N1 +|h21|
2P2
(
Nr+N(1)wz
)
|h2r |2P2+Nr+N
(1)wz
+|h1r |
2P1
Nr + N(1)wz +
|h2r |2P2N1
|h21|2P2+N1
,
R22 ≤ C
|h22|2P2
N2 + |hr2|2ν1Pr +|h12|
2P1
(
Nr+N(2)wz
)
|h1r |2P1+Nr+N
(2)wz
+|h2r |
2P2
Nr + N(2)wz +
|h1r |2P1
(
|hr2|2ν1Pr+N2
)
|h12|2P1+|hr2|
2ν1Pr+N2
,
subject to the constraints
N(1)wz ≥
(
|h11|2P1 + |h21|
2P2 + N1
)
A − A21
|hr1|2ν1Pr, N
(2)wz ≥
(
|h22|2P2 + |h12|
2P1 + |hr2|2ν1Pr + N2
)
A − A22
|hr2|2ν2Pr,
with (ν1, ν2) ∈ [0, 1]2, ν1 + ν2 ≤ 1, A = |h1r |2P1 + |h2r |
2P2 + Nr , A1 = 2Re(h11h∗1r )P1 + 2Re(h21h
∗2r )P2
and A2 = 2Re(h12h∗1r )P1 + 2Re(h22h
∗2r )P2. ...
The channel R-(D1,D2) is a Gaussian BC for which the capacity region is known.16 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Corollary (EF strategy: single-level resolution compression with only privatemessages)
For the Gaussian IRC with only private messages and with the bi-level resolution estimate-and-forward strategy, therate pair (R11, R22) is achievable, where
R11 ≤ C
|h11|2P1
N1 +|h21|
2P2(Nr+Nwz )
|h2r |2P2+Nr+Nwz
+|h1r |
2P1
Nr + Nwz +|h2r |
2P2N2|h21|
2P2+N1
,
R22 ≤ C
|h22|2P2
N2 +|h12|
2P1(Nr+Nwz )
|h1r |2P1+Nr+Nwz
+|h2r |
2P2
Nr + Nwz +|h1r |
2P1N2|h12|
2P1+N2
,
subject to the constraints Nwz ≥max
{
σ21, σ
22
}
22R0 − 1with
R0 = min
{
C
(
|hr1|2Pr
|h11|2P1 + |h21|2P2 + N1
)
, C
(
|hr2|2Pr
|h22|2P2 + |h12|2P1 + N2
)}
,
σ21 = |h1r |
2P1 + |h2r |
2P2 + Nr −
(
2Re(h11h∗1r )P1 + 2Re(h21h
∗2r )P2
)2
|h11|2P1 + |h21|2P1 + N1
σ22 = |h1r |
2P1 + |h2r |
2P2 + Nr −
(
2Re(h22h∗2r )P2 + 2Re(h12h
∗1r )P1
)2
|h22|2P2 + |h12|2P1 + N2
17 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Single-level compression vs Bi-level compression
Single-level compression Bi-level compression
Maximize the system sum-rate ×with low receiver SNRs asymmetry
Maximize the rate at the ”best” receiver ×with high received SNRs asymmetry
Maximize the rate at each receiver node ×with low receiver SNRs asymmetry
Maximize the system sum-rate ×with high receiver SNRs asymmetry
18 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Zero-delay scalar amplify-and-forward
Theorem (Transmission rate region for the IRC with ZDSAF)
Let Ri , i ∈ {1, 2}, be the transmission rate for the source node Si . When ZDSAF is assumed the following regionis achievable:
∀i ∈ {1, 2}, Ri ≤ C
|ar hir hri + hii |2 ρi
∣
∣ar hjr hri + hji∣
∣
2 ρj
Nj
Ni+ a2r |hri |
2 NrNi+ 1
where ρi =PiNi
and j = −i .
Observation
The achievable individual rates are not always concave.
Time-Sharing Techniques [El Gamal, Mohseni and Zahedi, 2006]
RTSi ≤ αi (1 − αj )C
|aTSr,i hir hri+hii |
2ρi
αi [(aTSr,i
)2|hri |2 NrNi
+1]
+ αiαjC
∣
∣
∣aTSr hir hri+hii
∣
∣
∣
2αjρi
αi
[
∣
∣
∣aTSr hjr hri+hji
∣
∣
∣
2ρj
NjNi
+αj [(aTSr )2|hri |
2 NrNi
+1]
]
∀i ∈ {1, 2}, where (α1, α2) ∈ (0, 1)2, aTSr,i =
√
Pr/µ
|hir |2Pi/αi+Nr
, aTSr =
√
Pr/µ
|h1r |2P1/α1+|h2r |
2P2/α2+Nr
and µ = max{α1, α2} are the relay amplification gains.
19 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Bi-level resolution vs single-level resolution
With the double resolution strategy, the cost of the additional interference bythe relay is significant.
20 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Bi-level resolution vs single-level resolution
Message
The bi-level resolution compression is better for the ”best” receiver if there is ahigh asymmetry in received SNRs between both receiver nodes.
With low asymmetry in received SNRs, the single-level resolution compression ispreferable to maximize the system sum-rate.
21 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Background and goalThe discrete caseThe Gaussian case with only private messages
Achievable system sum-rate versus xr (abscissa for the relay position) with AF, DF
and bi-level EF.
−6 −5 −4 −3 −2 −1 0 1 2 3 40.7
0.8
0.9
1
1.1
1.2
1.3
1.4
xr / d
0
Sys
tem
su
m−
rate
[b
pcu
]
P1 = 10, P
2 = 10 and P
r = 10
ZDSAF
Bi−level compression EF
DF
Message
Similar behavior as for the basic relay channel.
22 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
23 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Related works
[Xi & Yeh, 2008]: Traffic game in parallel relay networks withpower policy to minimize a certain cost function.
[Xi & Yeh, 2008]: Quite similar analysis for multi-hopnetworks.
[Shi & al., 2008]: Special case of IRCs with the DF protocolwithout direct links between the sources and destinations.
24 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
25 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Q non-overlapping frequency bands,
Signal transmitted by Si in band (q): X(q)i
with
Q∑
q=1
E|X(q)i
|2≤ Pi ., ∀i ∈ {1, 2},
θ(q)i
: fraction of power for Si in band (q) (
E|X(q)i
|2 = θ(q)i
Pi ),
the channel gains are considered to be static (large scalepropagation effects),
coherent communications assumption for eachtransmitter-receiver pair,
single user decoding.
Features and goals
Each transmitter optimize its transmission rate in a selfish manner,
a suitable model for this interaction: non-cooperative game,
Question: do some predictable outcomes exist to this conflict situation?
→ a solution concept to non-cooperative game: Nash Equilibrium [Nash, 1950].
26 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Q non-overlapping frequency bands,
Signal transmitted by Si in band (q): X(q)i
with
Q∑
q=1
E|X(q)i
|2≤ Pi ., ∀i ∈ {1, 2},
θ(q)i
: fraction of power for Si in band (q) (
E|X(q)i
|2 = θ(q)i
Pi ),
the channel gains are considered to be static (large scalepropagation effects),
coherent communications assumption for eachtransmitter-receiver pair,
single user decoding.
Features and goals
Each transmitter optimize its transmission rate in a selfish manner,
a suitable model for this interaction: non-cooperative game,
Question: do some predictable outcomes exist to this conflict situation?
→ a solution concept to non-cooperative game: Nash Equilibrium [Nash, 1950].
26 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Outline of the talk
1 Shannon theory for the interference relay channelBackground and goalThe discrete caseThe Gaussian case with only private messages
2 Power allocation Games in multiband IRCsBackgroundSystem modelEquilibrium analysis for some relaying protocols
3 Conclusion and perspectives
27 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Features
Signal transmitted by Si on band (q): X(q)i
= X(q)i,0
+
√
√
√
√
τ(q)i
ν(q)i
θ(q)i
Pi
P(q)r
X(q)r,i
;
Signal transmitted by Ri on band (q): X(q)r = X
(q)r,1 + X
(q)r,2 ;
Power allocation policies: ∀i ∈ {1, 2}, θi =(
θ(1)i
, ..., θ(Q)i
)
.
transmission rates
the source-destination pair (Si ,Di ) achieves the transmission rate∑Q
q=1 R(q),DF
iwhere
R(q),DF
1 = min{
R(q),DF
1,1 , R(q),DF
1,2
}
R(q),DF
2 = min{
R(q),DF
2,1 , R(q),DF
2,2
}
with
R(q),DF
1,1 = C
∣
∣
∣
∣
h(q)1r
∣
∣
∣
∣
2(
1−τ(q)1
)
θ(q)1
P1∣
∣
∣
∣
h(q)2r
∣
∣
∣
∣
2(
1−τ(q)2
)
θ(q)2
P2+N(q)r
R(q),DF
1,2 = C
∣
∣
∣
∣
h(q)11
∣
∣
∣
∣
2θ(q)1
P1+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +2Re
(
h(q)11
h(q),∗r1
)
√
τ(q)1
θ(q)1
P1ν(q)P
(q)r
∣
∣
∣
∣
h(q)21
∣
∣
∣
∣
2θ(q)2
P2+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +2Re
(
h(q)21
h(q),∗r1
)
√
τ(q)2
θ(q)2
P2ν(q)P
(q)r +N
(q)1
and (ν(q), τ(q)1 , τ
(q)2 ) is a given triple of parameters in [0, 1]3, τ
(q)1 + τ
(q)2 ≤ 1.
28 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Features
Signal transmitted by Si on band (q): X(q)i
= X(q)i,0
+
√
√
√
√
τ(q)i
ν(q)i
θ(q)i
Pi
P(q)r
X(q)r,i
;
Signal transmitted by Ri on band (q): X(q)r = X
(q)r,1 + X
(q)r,2 ;
Power allocation policies: ∀i ∈ {1, 2}, θi =(
θ(1)i
, ..., θ(Q)i
)
.
transmission rates
the source-destination pair (Si ,Di ) achieves the transmission rate∑Q
q=1 R(q),DF
iwhere
R(q),DF
1 = min{
R(q),DF
1,1 , R(q),DF
1,2
}
R(q),DF
2 = min{
R(q),DF
2,1 , R(q),DF
2,2
}
with
R(q),DF
1,1 = C
∣
∣
∣
∣
h(q)1r
∣
∣
∣
∣
2(
1−τ(q)1
)
θ(q)1
P1∣
∣
∣
∣
h(q)2r
∣
∣
∣
∣
2(
1−τ(q)2
)
θ(q)2
P2+N(q)r
R(q),DF
1,2 = C
∣
∣
∣
∣
h(q)11
∣
∣
∣
∣
2θ(q)1
P1+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +2Re
(
h(q)11
h(q),∗r1
)
√
τ(q)1
θ(q)1
P1ν(q)P
(q)r
∣
∣
∣
∣
h(q)21
∣
∣
∣
∣
2θ(q)2
P2+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +2Re
(
h(q)21
h(q),∗r1
)
√
τ(q)2
θ(q)2
P2ν(q)P
(q)r +N
(q)1
and (ν(q), τ(q)1 , τ
(q)2 ) is a given triple of parameters in [0, 1]3, τ
(q)1 + τ
(q)2 ≤ 1.
28 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Definition of the game: Non-cooperative strategic form game (SFG)
1 Players: S1 and S2;
2 Strategy of Si : θi = (θ(1)i
, . . . , θ(Q)i
) in its strategy set Ai =
θi ∈ [0, 1]Q |
Q∑
q=1
θ(q)i
≤ 1
;
3 Utility function (or payoff) of Si : uDFi (θi , θ−i ) =
Q∑
q=1
R(q),DF
i(θ
(q)i
, θ(q)−i
).
Assumption for the game
The game is played once (static game) and is with complete information i.e. every player knows the triplet
GDF = (K, (Ai )i∈K, (uDFi )i∈K), where K = {1, 2}
Definition [Nash Equilibrium]
The state (θ∗i , θ∗−i ) is a pure NE of the SFG G if ∀i ∈ K,∀θ′i ∈ Ai , ui (θ
∗i , θ
∗−i ) ≥ ui (θ
′i , θ
∗−i ).
Theorem [Existence of an NE for the DF protocol]
If the channel gains satisfy the condition Re(h(q)ii
h(q)∗ri
) ≥ 0, for all i ∈ {1, 2} and q ∈ {1, . . . ,Q} the game
defined by GDF = (K, (Ai )i∈K, (uDF
i (θi , θ−i ))i∈K) with K = {1, 2} and
Ai =
θi ∈ [0, 1]Q
∣
∣
∣
∣
∣
∣
Q∑
q=1
θ(q)i
≤ 1
, has always at least one pure NE.
29 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Definition of the game: Non-cooperative strategic form game (SFG)
1 Players: S1 and S2;
2 Strategy of Si : θi = (θ(1)i
, . . . , θ(Q)i
) in its strategy set Ai =
θi ∈ [0, 1]Q |
Q∑
q=1
θ(q)i
≤ 1
;
3 Utility function (or payoff) of Si : uDFi (θi , θ−i ) =
Q∑
q=1
R(q),DF
i(θ
(q)i
, θ(q)−i
).
Assumption for the game
The game is played once (static game) and is with complete information i.e. every player knows the triplet
GDF = (K, (Ai )i∈K, (uDFi )i∈K), where K = {1, 2}
Definition [Nash Equilibrium]
The state (θ∗i , θ∗−i ) is a pure NE of the SFG G if ∀i ∈ K,∀θ′i ∈ Ai , ui (θ
∗i , θ
∗−i ) ≥ ui (θ
′i , θ
∗−i ).
Theorem [Existence of an NE for the DF protocol]
If the channel gains satisfy the condition Re(h(q)ii
h(q)∗ri
) ≥ 0, for all i ∈ {1, 2} and q ∈ {1, . . . ,Q} the game
defined by GDF = (K, (Ai )i∈K, (uDF
i (θi , θ−i ))i∈K) with K = {1, 2} and
Ai =
θi ∈ [0, 1]Q
∣
∣
∣
∣
∣
∣
Q∑
q=1
θ(q)i
≤ 1
, has always at least one pure NE.
29 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Definition of the game: Non-cooperative strategic form game (SFG)
1 Players: S1 and S2;
2 Strategy of Si : θi = (θ(1)i
, . . . , θ(Q)i
) in its strategy set Ai =
θi ∈ [0, 1]Q |
Q∑
q=1
θ(q)i
≤ 1
;
3 Utility function (or payoff) of Si : uDFi (θi , θ−i ) =
Q∑
q=1
R(q),DF
i(θ
(q)i
, θ(q)−i
).
Assumption for the game
The game is played once (static game) and is with complete information i.e. every player knows the triplet
GDF = (K, (Ai )i∈K, (uDFi )i∈K), where K = {1, 2}
Definition [Nash Equilibrium]
The state (θ∗i , θ∗−i ) is a pure NE of the SFG G if ∀i ∈ K,∀θ′i ∈ Ai , ui (θ
∗i , θ
∗−i ) ≥ ui (θ
′i , θ
∗−i ).
Theorem [Existence of an NE for the DF protocol]
If the channel gains satisfy the condition Re(h(q)ii
h(q)∗ri
) ≥ 0, for all i ∈ {1, 2} and q ∈ {1, . . . ,Q} the game
defined by GDF = (K, (Ai )i∈K, (uDF
i (θi , θ−i ))i∈K) with K = {1, 2} and
Ai =
θi ∈ [0, 1]Q
∣
∣
∣
∣
∣
∣
Q∑
q=1
θ(q)i
≤ 1
, has always at least one pure NE.
29 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Definition of the game: Non-cooperative strategic form game (SFG)
1 Players: S1 and S2;
2 Strategy of Si : θi = (θ(1)i
, . . . , θ(Q)i
) in its strategy set Ai =
θi ∈ [0, 1]Q |
Q∑
q=1
θ(q)i
≤ 1
;
3 Utility function (or payoff) of Si : uDFi (θi , θ−i ) =
Q∑
q=1
R(q),DF
i(θ
(q)i
, θ(q)−i
).
Assumption for the game
The game is played once (static game) and is with complete information i.e. every player knows the triplet
GDF = (K, (Ai )i∈K, (uDFi )i∈K), where K = {1, 2}
Definition [Nash Equilibrium]
The state (θ∗i , θ∗−i ) is a pure NE of the SFG G if ∀i ∈ K,∀θ′i ∈ Ai , ui (θ
∗i , θ
∗−i ) ≥ ui (θ
′i , θ
∗−i ).
Theorem [Existence of an NE for the DF protocol]
If the channel gains satisfy the condition Re(h(q)ii
h(q)∗ri
) ≥ 0, for all i ∈ {1, 2} and q ∈ {1, . . . ,Q} the game
defined by GDF = (K, (Ai )i∈K, (uDF
i (θi , θ−i ))i∈K) with K = {1, 2} and
Ai =
θi ∈ [0, 1]Q
∣
∣
∣
∣
∣
∣
Q∑
q=1
θ(q)i
≤ 1
, has always at least one pure NE.
29 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Proof
The proof is based on Theorem 1 of [rosen, 1965]. It states that in game with a finitenumber of players, if for every player
1 the strategy set is convex and compact,
2 its utility is continuous in the vector of strategies and
3 concave in its own strategy,
then the existence of at least one NE is guaranteed.
Comments
Whatever the values of the channel gains, there exists an NE. Therefore
The transmitters are able to adapt their PA policies if the number of relay ismodified,
The transmitters are able to adapt their PA policies if the relay location ismodified.
30 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The decode-and-forward case
Proof
The proof is based on Theorem 1 of [rosen, 1965]. It states that in game with a finitenumber of players, if for every player
1 the strategy set is convex and compact,
2 its utility is continuous in the vector of strategies and
3 concave in its own strategy,
then the existence of at least one NE is guaranteed.
Comments
Whatever the values of the channel gains, there exists an NE. Therefore
The transmitters are able to adapt their PA policies if the number of relay ismodified,
The transmitters are able to adapt their PA policies if the relay location ismodified.
30 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The bi-level estimate-and-forward case
Decoding assumption and utility functions
Each receiver implements single-user decoding (SUD).
The utility function for Si is given by: uEF
i (θi , θ−i ) =∑Q
q=1 R(q),EF
i where, for
example,
R(q),EF
1 = C
(
∣
∣
∣
∣
h(q)2r
∣
∣
∣
∣
2θ(q)2
P2+N(q)r +N
(q)wz,1
)
∣
∣
∣
∣
h(q)11
∣
∣
∣
∣
2θ(q)1
P1+
(
∣
∣
∣
∣
h(q)21
∣
∣
∣
∣
2θ(q)2
P2+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +N
(q)1
)
∣
∣
∣
∣
h(q)1r
∣
∣
∣
∣
2θ(q)1
P1(
N(q)r +N
(q)wz,1
)(
|h(q)21
|2θ(q)2
P2+|h(q)r1
|2ν(q)P(q)r +N
(q)1
)
+|h(q)2r
|2θ(q)2
P2
(
|h(q)r1
|2ν(q)P(q)r +N
(q)1
)
N(q)wz,1 =
(
∣
∣
∣
∣
h(q)11
∣
∣
∣
∣
2θ(q)1
P1+
∣
∣
∣
∣
h(q)21
∣
∣
∣
∣
2θ(q)2
P2+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +N
(q)1
)
A(q)−
∣
∣
∣
∣
A(q)1
∣
∣
∣
∣
2
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r
,
ν(q) ∈ [0, 1], A(q) = |h(q)1r |2θ
(q)1 P1 + |h
(q)2r |2θ
(q)2 P2 + N
(q)r and A
(q)1 = h
(q)11 h
(q),∗1r θ
(q)1 P1 + h
(q)21 h
(q),∗2r θ
(q)2 P2.
Theorem [Existence of an NE for the bi-level EF protocol]
The game defined by GEF = (K, (Ai )i∈K, (uEF
i (θi , θ−i ))i∈K) with K = {1, 2} and
Ai =
θi ∈ [0, 1]Q
∣
∣
∣
∣
∣
∣
Q∑
q=1
θ(q)i
≤ 1
, has always at least one pure NE.
31 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The bi-level estimate-and-forward case
Decoding assumption and utility functions
Each receiver implements single-user decoding (SUD).
The utility function for Si is given by: uEF
i (θi , θ−i ) =∑Q
q=1 R(q),EF
i where, for
example,
R(q),EF
1 = C
(
∣
∣
∣
∣
h(q)2r
∣
∣
∣
∣
2θ(q)2
P2+N(q)r +N
(q)wz,1
)
∣
∣
∣
∣
h(q)11
∣
∣
∣
∣
2θ(q)1
P1+
(
∣
∣
∣
∣
h(q)21
∣
∣
∣
∣
2θ(q)2
P2+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +N
(q)1
)
∣
∣
∣
∣
h(q)1r
∣
∣
∣
∣
2θ(q)1
P1(
N(q)r +N
(q)wz,1
)(
|h(q)21
|2θ(q)2
P2+|h(q)r1
|2ν(q)P(q)r +N
(q)1
)
+|h(q)2r
|2θ(q)2
P2
(
|h(q)r1
|2ν(q)P(q)r +N
(q)1
)
N(q)wz,1 =
(
∣
∣
∣
∣
h(q)11
∣
∣
∣
∣
2θ(q)1
P1+
∣
∣
∣
∣
h(q)21
∣
∣
∣
∣
2θ(q)2
P2+
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r +N
(q)1
)
A(q)−
∣
∣
∣
∣
A(q)1
∣
∣
∣
∣
2
∣
∣
∣
∣
h(q)r1
∣
∣
∣
∣
2ν(q)P
(q)r
,
ν(q) ∈ [0, 1], A(q) = |h(q)1r |2θ
(q)1 P1 + |h
(q)2r |2θ
(q)2 P2 + N
(q)r and A
(q)1 = h
(q)11 h
(q),∗1r θ
(q)1 P1 + h
(q)21 h
(q),∗2r θ
(q)2 P2.
Theorem [Existence of an NE for the bi-level EF protocol]
The game defined by GEF = (K, (Ai )i∈K, (uEF
i (θi , θ−i ))i∈K) with K = {1, 2} and
Ai =
θi ∈ [0, 1]Q
∣
∣
∣
∣
∣
∣
Q∑
q=1
θ(q)i
≤ 1
, has always at least one pure NE.
31 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The zero-delay scalar amplify-and-forward case
transmission assumption and utility functions
Each transmitter use Time-Sharing techniques.
The utility function for Si is given by: uAF
i (θi , θ−i ) =∑Q
q=1 R(q),AF
i (θ(q)i , θ
(q)−i ) where
∀i ∈ {1, 2}, R(q),AF
i= α
(q)i
(1 − α(q)j
)C
|a(q)r,i
h(q)ir
h(q)ri
+h(q)ii
|2ρi θ(q)i
α(q)i
(
a(q)r,i
)2∣
∣
∣
∣
h(q)ri
∣
∣
∣
∣
2 N(q)r
N(q)i
+1
+α(q)i
α(q)j
C
∣
∣
∣
∣
a(q)r h
(q)ir
h(q)ri
+h(q)ii
∣
∣
∣
∣
2α(q)j
ρi
α(q)i
∣
∣
∣
∣
a(q)r hjr hri+hji
∣
∣
∣
∣
2ρjθ
(q)j
N(q)j
N(q)i
+α(q)j
(
a(q)r
)2∣∣
∣
∣
h(q)ri
∣
∣
∣
∣
2 N(q)r
N(q)i
+1
with ∀i ∈ {1, 2}, j = −i and ρ(q)i
=Pi
N(q)i
, (α(q)i
, α(q)j
) ∈ (0, 1)2, a(q)r,i
= a(q)r,i
(θ(q)i
) ,
√
√
√
√
Pr/µ(q)
∣
∣
∣
∣
h(q)ir
∣
∣
∣
∣
2Pi/αi+Nr
,
a(q)r = a
(q)r (θ
(q)1 , θ
(q)2 ) ,
√
√
√
√
Pr/µ(q)
∣
∣
∣
∣
h(q)1r
∣
∣
∣
∣
2P1/α
(q)1
(+|h2r |2P2/α
(q)2
+Nr
and µ(q) = max{α(q)1 , α
(q)2 }.
Theorem [Existence of an NE for ZDSAF when a(q)r = a
(q)r (θ
(q)1 , θ
(q)2 )]
There exists at least one pure NE in the PA game GAF.
32 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The zero-delay scalar amplify-and-forward case
transmission assumption and utility functions
Each transmitter use Time-Sharing techniques.
The utility function for Si is given by: uAF
i (θi , θ−i ) =∑Q
q=1 R(q),AF
i (θ(q)i , θ
(q)−i ) where
∀i ∈ {1, 2}, R(q),AF
i= α
(q)i
(1 − α(q)j
)C
|a(q)r,i
h(q)ir
h(q)ri
+h(q)ii
|2ρi θ(q)i
α(q)i
(
a(q)r,i
)2∣
∣
∣
∣
h(q)ri
∣
∣
∣
∣
2 N(q)r
N(q)i
+1
+α(q)i
α(q)j
C
∣
∣
∣
∣
a(q)r h
(q)ir
h(q)ri
+h(q)ii
∣
∣
∣
∣
2α(q)j
ρi
α(q)i
∣
∣
∣
∣
a(q)r hjr hri+hji
∣
∣
∣
∣
2ρjθ
(q)j
N(q)j
N(q)i
+α(q)j
(
a(q)r
)2∣∣
∣
∣
h(q)ri
∣
∣
∣
∣
2 N(q)r
N(q)i
+1
with ∀i ∈ {1, 2}, j = −i and ρ(q)i
=Pi
N(q)i
, (α(q)i
, α(q)j
) ∈ (0, 1)2, a(q)r,i
= a(q)r,i
(θ(q)i
) ,
√
√
√
√
Pr/µ(q)
∣
∣
∣
∣
h(q)ir
∣
∣
∣
∣
2Pi/αi+Nr
,
a(q)r = a
(q)r (θ
(q)1 , θ
(q)2 ) ,
√
√
√
√
Pr/µ(q)
∣
∣
∣
∣
h(q)1r
∣
∣
∣
∣
2P1/α
(q)1
(+|h2r |2P2/α
(q)2
+Nr
and µ(q) = max{α(q)1 , α
(q)2 }.
Theorem [Existence of an NE for ZDSAF when a(q)r = a
(q)r (θ
(q)1 , θ
(q)2 )]
There exists at least one pure NE in the PA game GAF.
32 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The zero-delay scalar amplify-and-forward case: a special case
The special case: parameters
α(q)i = 1, Q = 2 and ∀q ∈ {1, 2}, a
(q)r = a
(q)r,1 = a
(q)r,2 = A
(q)r ∈ [0, ar (1, 1)] are constant.
Best Response (BR) functions
BRi (θj ) = argmaxθi
ui (θi , θj) =
∣
∣
∣
∣
∣
∣
Fi (θj ) if 0 < Fi (θj ) < 11 if Fi (θj ) ≥ 10 otherwise
where j = −i , hij = h(1)ij
, gij = h(2)ij
, Fi (θj ) , −cijcii
θj +dicii
is an affine function of θj ; for
(i, j) ∈ {(1, 2), (2, 1)}, cii = 2|A(1)r hri hir + hii |
2|A(2)r gri gir + gii |
2ρi ;
cij = |A(1)r hri hir + hii |
2|A(2)r gri gjr + gji |
2ρj + |A(1)r hri hjr + hji |
2|A(2)r gri gir + gii |
2ρj ;
di = |A(1)r hri hir+hii |
2[|A(2)r gri gir+gii |
2ρi+|A(2)r gri gjr+gji |
2ρj+A(2)r |gri |
2+1]−|A(2)r gri gir+gii |
2(A(1)r |hri |
2+1).
Theorem [Number of Nash equilibria for ZDSAF]
For the game GAF with fixed amplification gains at the relays, (i.e., ∂ar
∂θ(q)i
= 0), there
can be a unique NE, two NE, three NE or an infinite number of NE, depending on the
channel parameters (i.e., hij , gij , ρi , A(q)r , (i , j) ∈ {1, 2, r}2, q ∈ {1, 2}).
33 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The zero-delay scalar amplify-and-forward case: a special case
The special case: parameters
α(q)i = 1, Q = 2 and ∀q ∈ {1, 2}, a
(q)r = a
(q)r,1 = a
(q)r,2 = A
(q)r ∈ [0, ar (1, 1)] are constant.
Best Response (BR) functions
BRi (θj ) = argmaxθi
ui (θi , θj) =
∣
∣
∣
∣
∣
∣
Fi (θj ) if 0 < Fi (θj ) < 11 if Fi (θj ) ≥ 10 otherwise
where j = −i , hij = h(1)ij
, gij = h(2)ij
, Fi (θj ) , −cijcii
θj +dicii
is an affine function of θj ; for
(i, j) ∈ {(1, 2), (2, 1)}, cii = 2|A(1)r hri hir + hii |
2|A(2)r gri gir + gii |
2ρi ;
cij = |A(1)r hri hir + hii |
2|A(2)r gri gjr + gji |
2ρj + |A(1)r hri hjr + hji |
2|A(2)r gri gir + gii |
2ρj ;
di = |A(1)r hri hir+hii |
2[|A(2)r gri gir+gii |
2ρi+|A(2)r gri gjr+gji |
2ρj+A(2)r |gri |
2+1]−|A(2)r gri gir+gii |
2(A(1)r |hri |
2+1).
Theorem [Number of Nash equilibria for ZDSAF]
For the game GAF with fixed amplification gains at the relays, (i.e., ∂ar
∂θ(q)i
= 0), there
can be a unique NE, two NE, three NE or an infinite number of NE, depending on the
channel parameters (i.e., hij , gij , ρi , A(q)r , (i , j) ∈ {1, 2, r}2, q ∈ {1, 2}).
33 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
The zero-delay scalar amplify-and-forward case: a special case
The special case: parameters
α(q)i = 1, Q = 2 and ∀q ∈ {1, 2}, a
(q)r = a
(q)r,1 = a
(q)r,2 = A
(q)r ∈ [0, ar (1, 1)] are constant.
Best Response (BR) functions
BRi (θj ) = argmaxθi
ui (θi , θj) =
∣
∣
∣
∣
∣
∣
Fi (θj ) if 0 < Fi (θj ) < 11 if Fi (θj ) ≥ 10 otherwise
where j = −i , hij = h(1)ij
, gij = h(2)ij
, Fi (θj ) , −cijcii
θj +dicii
is an affine function of θj ; for
(i, j) ∈ {(1, 2), (2, 1)}, cii = 2|A(1)r hri hir + hii |
2|A(2)r gri gir + gii |
2ρi ;
cij = |A(1)r hri hir + hii |
2|A(2)r gri gjr + gji |
2ρj + |A(1)r hri hjr + hji |
2|A(2)r gri gir + gii |
2ρj ;
di = |A(1)r hri hir+hii |
2[|A(2)r gri gir+gii |
2ρi+|A(2)r gri gjr+gji |
2ρj+A(2)r |gri |
2+1]−|A(2)r gri gir+gii |
2(A(1)r |hri |
2+1).
Theorem [Number of Nash equilibria for ZDSAF]
For the game GAF with fixed amplification gains at the relays, (i.e., ∂ar
∂θ(q)i
= 0), there
can be a unique NE, two NE, three NE or an infinite number of NE, depending on the
channel parameters (i.e., hij , gij , ρi , A(q)r , (i , j) ∈ {1, 2, r}2, q ∈ {1, 2}).
33 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Simulation: Number of Nash equilibria for the ZDSAF protocol
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ1
θ 2
Best Response Functions (ρ1=1, ρ
2=3, ρ
r=2)
BR2(θ
1)
BR1(θ
2)
(θ1NE,1, θ
2NE,1)=(0,1)
(θ1NE,2, θ
2NE,2)=(0.84,0.66)
(θ1NE,3, θ
2NE,3)=(1,0.37)
Message
Three Nash equilibria, in general.
The NE point can be predicted from the sole knowledge of the starting point ofthe game when making use of the Cournot tatnnement.
34 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Example of application: Optimal relay location
Stackelberg formulation
Introduction of a leader in the game (the network provider for example).
A bi-level game
At a first stage: The leader chooses its strategy.
At a second stage: The remaining players react according to the decision of theleader.
Strategy of the leader
2D propagation scenario.
Strategy: The pair of coordinates (xR; yR) corresponding to the relay location.
Utility function:
The social welfare u(xR, yR) = u1 [θ∗(xR, yR)] + u2 [θ
∗(xR, yR)];The utility function of one of the users.
35 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Example of application: Optimal relay location
Stackelberg formulation
Introduction of a leader in the game (the network provider for example).
A bi-level game
At a first stage: The leader chooses its strategy.
At a second stage: The remaining players react according to the decision of theleader.
Strategy of the leader
2D propagation scenario.
Strategy: The pair of coordinates (xR; yR) corresponding to the relay location.
Utility function:
The social welfare u(xR, yR) = u1 [θ∗(xR, yR)] + u2 [θ
∗(xR, yR)];The utility function of one of the users.
35 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Optimal relay location for the ZDSAF protocol with full power regime
−10
−8
−6
−4
−2
0
2
4
6
8
10
−10 −8 −6 −4 −2 0 2 4 6 8 10
R1(θ
1NE,θ
2NE)+R
2(θ
1NE,θ
2NE) [bpcu]
xR
yR
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42L=10m,ε=1m,P
1=20dBm,P
2=17dBm,P
r=22dBm,
N1=10dBm,N
2=9dBm,N
r=7dBm, γ(1)=2.5,γ(2)=2
(xR* ,y
R* )=(−1.2,1.7) m
Rsum* =0.42 bpcu
S1
S2
D1
D2
Message
The optimal relay location for the individual rates is one of the segmentsbetween Si and Di .
36 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Optimal relay location for the ZDSAF protocol with full power regime
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
xR
(θ1NE,θ
2NE)
y R
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1L=10m,ε=1m,P
1=20dBm,P
2=17dBm,P
r=22dBm,
N1=10dBm,N
2=9dBm,N
r=7dBm, γ(1)=2.5,γ(2)=2
D1
D2
S1
S2
user 1
user 2
Message
The selfish behavior of the transmitters leads to self-regulating the interferencein the network.
37 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
BackgroundSystem modelEquilibrium analysis for some relaying protocols
Optimal power allocation at the relay for DF and EF
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ν
R1(θ
1NE,θ
2NE)+
R2(θ
1NE,θ
2NE)
[bpc
u]
Achievable sum−rate
EFDF
νunif =1/2R
sumDF (νunif )=0.82 bpcu
νunif =1/2R
sumEF (νunif )=0.44 bpcu
ν*=1R
sumDF (ν*)=1.45 bpcu
L=10m, ε=1 m, P1=22dBm, P
2=17dBm,
Pr=23dBm, N
1=7dBm, N
2=9dBm, N
r=0dBm,
γ(1)=2,γ(2)=2.5
ν*=1R
sumDF (ν*)=1.09 bpcu
Message
The relay allocates all its available power to the better receiver.
38 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Conclusion
Multiband interference relay channels.
Shannon theory for the IRC.
Power allocation game for the decentralized multiband IRCs.
Perspectives
Improve the characterization of NE: analyze the uniqueness issue, for example.
Consider a more general game.
Distributed iterative algorithms that converge to NE.
39 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Conclusion
Multiband interference relay channels.
Shannon theory for the IRC.
Power allocation game for the decentralized multiband IRCs.
Perspectives
Improve the characterization of NE: analyze the uniqueness issue, for example.
Consider a more general game.
Distributed iterative algorithms that converge to NE.
39 / 40
Shannon theory for the interference relay channelPower allocation Games in multiband IRCs
Conclusion and perspectives
Questions?
40 / 40