Cooperation and Competition in Multi-user Wireless Networksewh.ieee.org/r7/montreal/vts/djeumou-ets-26-03-10.pdf · 3/26/2010 · Cognitiveradio Goal Dynamically exploit the available

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



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



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



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



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











Virtual MIMO
















Virtual MIMO
















Virtual MIMO








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



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



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



Background and goalThe discrete caseThe Gaussian case with only private messages

Outline of the talk




7 / 40




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













Our contributions





8 / 40













Our contributions





8 / 40













Our contributions





8 / 40




Outline of the talk




9 / 40




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.


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




Rate-splitting ([Carleial, 1978]) at each source node

(W10,W11) at S1 and (W20,W22) at S2 .



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




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

(


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








∑

i∈I1

Ri ≤ I

(


1

)

for all I1 ⊆ S1 = {10, 11, 20},

∑

i∈I2

Ri ≤ I

(


2

)

for all I2 ⊆ S2 = {20, 22, 10},


I (Yr ; Yr2|U2,Y2) ≤ I (U2;Y2),



11 / 40








∑

i∈I1

Ri ≤ I

(


1

)

for all I1 ⊆ S1 = {10, 11, 20},

∑

i∈I2

Ri ≤ I

(


2

)

for all I2 ⊆ S2 = {20, 22, 10},


I (Yr ; Yr2|U2,Y2) ≤ I (U2;Y2),



11 / 40




Theorem (EF-based strategy: Single-level compression)


∑

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 ),


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.

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Theorem (EF-based strategy: Single-level compression)


∑

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 ),


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




Outline of the talk




13 / 40




System model

power constraints: E|X1|2 ≤ P1, E|X2|

2 ≤ P2 and E|Xr |2 ≤ Pr .

14 / 40




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




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




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




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




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




Bi-level resolution vs single-level resolution

With the double resolution strategy, the cost of the additional interference bythe relay is significant.

20 / 40




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




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



BackgroundSystem modelEquilibrium analysis for some relaying protocols

Outline of the talk




23 / 40




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




Outline of the talk




25 / 40




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




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




Outline of the talk




27 / 40




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





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





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








, . . . , θ(Q)i


θi ∈ [0, 1]Q |

Q∑

q=1

θ(q)i

≤ 1

;


Q∑

q=1

R(q),DF

i(θ

(q)i

, θ(q)−i

).






∗i , θ

∗−i ) ≥ ui (θ

′i , θ

∗−i ).



h(q)∗ri




Ai =

θi ∈ [0, 1]Q

∣

∣

∣

∣

∣

∣

Q∑

q=1

θ(q)i

≤ 1


29 / 40








, . . . , θ(Q)i


θi ∈ [0, 1]Q |

Q∑

q=1

θ(q)i

≤ 1

;


Q∑

q=1

R(q),DF

i(θ

(q)i

, θ(q)−i

).






∗i , θ

∗−i ) ≥ ui (θ

′i , θ

∗−i ).



h(q)∗ri




Ai =

θi ∈ [0, 1]Q

∣

∣

∣

∣

∣

∣

Q∑

q=1

θ(q)i

≤ 1


29 / 40








, . . . , θ(Q)i


θi ∈ [0, 1]Q |

Q∑

q=1

θ(q)i

≤ 1

;


Q∑

q=1

R(q),DF

i(θ

(q)i

, θ(q)−i

).






∗i , θ

∗−i ) ≥ ui (θ

′i , θ

∗−i ).



h(q)∗ri




Ai =

θi ∈ [0, 1]Q

∣

∣

∣

∣

∣

∣

Q∑

q=1

θ(q)i

≤ 1


29 / 40





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





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




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


Ai =

θi ∈ [0, 1]Q

∣

∣

∣

∣

∣

∣

Q∑

q=1

θ(q)i

≤ 1


31 / 40




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


Ai =

θi ∈ [0, 1]Q

∣

∣

∣

∣

∣

∣

Q∑

q=1

θ(q)i

≤ 1


31 / 40




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




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




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 |


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






α(q)i = 1, Q = 2 and ∀q ∈ {1, 2}, a

(q)r = a

(q)r,1 = a

(q)r,2 = A




ui (θi , θj) =

∣

∣

∣

∣

∣

∣



, gij = h(2)ij


θj +dicii




2ρi ;





2ρj ;




2ρj+A(2)r |gri |


2(A(1)r |hri |

2+1).



∂θ(q)i

= 0), there



33 / 40






α(q)i = 1, Q = 2 and ∀q ∈ {1, 2}, a

(q)r = a

(q)r,1 = a

(q)r,2 = A




ui (θi , θj) =

∣

∣

∣

∣

∣

∣



, gij = h(2)ij


θj +dicii




2ρi ;





2ρj ;




2ρj+A(2)r |gri |


2(A(1)r |hri |

2+1).



∂θ(q)i

= 0), there



33 / 40




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




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




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




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




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




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



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



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



Questions?

40 / 40

Documents

Cooperation and Competition in Multi-user Wireless Networksewh.ieee.org/r7/montreal/vts/djeumou-ets-26-03-10.pdf · 3/26/2010 · Cognitiveradio Goal Dynamically exploit the available