Scalable and distributed transmitter cooperation in wireless networks: Team decision problems in wireless networks

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Scalable and distributed transmitter cooperation in wireless networks

Team decision problems in wireless networks

A presentation to WiOpt 2014, Hammamet Tunisia David Gesbert [email protected]

With thanks to Paul de Kerret

May 13th, 2014

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The Dimensions of Interference Management

REJECT AVOID

EXPLOIT

COORDINATE

CONTAIN

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Outline

1 Fundamentals for Transmitter Coordination

2 Thinking Practical

3 Team Decisions Problems in Wireless Networks

4 Spatial Allocation of CSI in Multi-antenna Coordinated Networks

5 Team Decision for Multi-Antenna Precoding

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Fundamentals for Transmitter Coordination

Outline

6 Lessons learned and open problems

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Transmitter Cooperation Domains

Transmitter Cooperation

Domains

User selection

time

codes frequency

Space

power

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Example 1: Coordination using Scheduling, Power Control

RX k

TX 3 TX 2

TX 1

0 5 10 15 20 25 30 35 40 45 50

4 6 8 10 12 14 16

Rates [Bits/s/Hz]

K No interference max−SINR Scheduler

Picking the right user at any time/freq exploits the variability of interference

Can be combined with power control/beamforming (will couplethe decisions at all cells)

Simple max-SINR scheduler isdistributedand worksbeautifully

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Example 2: Coordination using Alignment

Interference Alignment Conditions with N_r=2:

H12

S₁ S₂

ŝ₃ S₃

w₁ w₂ w₃

2 32 1 31

3 23 1 21

3 13 2 12

w H w H



H32

RX 3 RX 2

RX 1

ŝ₂ ŝ₁

TX 3 TX 2

TX 1

H22

H11 H21

H31 H13

H23 H33

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Example 3: Cooperation with Joint MIMO Precoding

Joint MIMO Precoding[Hanly, 1993] [Shamai and Zaidel, 2001, VTC]

Backhaul Links (fibers,wireless,..) TX 2

RX 2 RX 3

RX 1 RX 5

RX 4 TX 4

TX 3

TX 1 TX 5

TX 1

RX 2

TX 2 TX N_cells

RX 1 ^{RX Ncells}

N_t

Network MIMO Transmit antennas:

Nt x Ncells

ŝNcells Data: S₁,S₂,…,SNcells

Nr

ŝ2 ŝ1

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How does Joint MIMO Precoding Work?

S1,S2

H12

FULL CSIT FEEDBACK

H H

TX 2 TX 1

Ideal Network MIMO

  



 







) (

) ) (

( )

( _H

2 H 1 2

1 w H

H H w

t H t T

F 1

1

||

e.g. _

 

H T H

RX 1 RX 2

ŝ1 ŝ2

)

H(

1 H

w w^H₂(H)

H22

H11

H21

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Figures-of-Merit

Average rate of user i given by [Cover and Thomas, 2006]

R_i,E

"

log₂ 1 + |g^H_i H^H_i t_i|² 1 +P

j6=i|g^H_i H^H_i t_j|²

!#

Number of Degrees-of-Freedom (DoF) at user i –or prelog factor–

defined as [Tse and Viswanath, 2005]

DoFi, lim

P→∞

Ri

log₂(P). Sum DoF over K TX-RX pairs:

DoF, X

i=1..K

DoFi

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Myth and Reality of Transmitter Cooperation

* A. Lozano et al, “Fundamental limits of cooperation”, IEEE Trans. On Information Theory, Sept. 2013.

Cooperation Clusters

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Thinking Practical

Outline

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Thinking Practical

Thinking practical

A number of issues arise in the implementation of cooperation mechanisms:

Hardware impairments

Channel estimation and tracking

Channel State information (CSI) Feedback limitation Inter-transmitterinformation sharing limitation

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Thinking Practical

Inter-transmitter Information Sharing Limitation

Perfect sharing of CSIT is not scalablein large networks CSIT sharing is:

Latency limited (often)

Capacity limited (sometimes) as in wireless mmw backhauling Privacy limited (cognitive radios)

→ TX-dependent CSIT noise

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Thinking Practical

A Distributed Channel State Information Model

CSIT imperfection at TX j modeled asLocal quantization noise

{Hˆ^(j⁾}i,k = q

1−2^−B

(j)

i,kσ_i,k{H}i,k+ q

2^−B

(j)

i,kσ_i_,k{∆}^(j)_i,k, ∀i,k

where{∆}^(j_i_,k⁾ ∼ CN(0,1)

CSIT allocationB^(j⁾ at TXj defined as{B^(j)}i,k =B_i^(j)_,k, ∀i,k

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Thinking Practical

Joint Precoding with Distributed CSIT

x1=w^H₁( ˆH⁽¹⁾)s

x2=w^H₂( ˆH⁽²⁾)s TX1

TX2

•

• Hˆ⁽¹⁾,s

Hˆ⁽²⁾,s

x1

x2

•

1

• 2

Key questions (assuming single-shot coordination):

1 What kind of CSI should over-the-air feedback convey?

2 What should be exchanged over the signaling links?

3 Assuming TX 1 finally hasH⁽¹⁾ and TX 2 hasH⁽²⁾, how should precoders w₁(H⁽¹⁾) andw₂(H⁽²⁾) be designed?

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Team Decisions Problems in Wireless Networks

Outline

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One-shot coordination problems

TX1

TX2

TX𝑗

TX𝑖

TX𝐾 𝑅₁₂

𝑅₂₁

𝑅1𝑗 𝑅𝑗1 𝑅𝑗𝑖

𝑅𝑖𝑗

𝑅_𝐾𝑖 𝑅_𝑖𝐾

𝐻 ⁽²⁾ {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ⁽¹⁾ {𝑄_𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝑗) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝑖) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝐾) {𝑄_𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(.): local CSI 𝑄𝑖: Error covariance

A priori information: Coordination link rates:

From i to j: R _ij

Problem 1: Channel state information allocation (what should ˆH⁽ⁱ⁾ be?

Problem 2: Signaling (what to send over R_ij bits?) Problem 3: Team decision making

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Problem 1: Channel State Information Allocation

The nodes to be coordinated are initially assigned someCSIT-related data. The spatial distribution of CSIT is called the information structure.

A CSI structure isperfect if ˆH⁽ⁱ⁾=H, ∀i.

A CSI structure iscentralizedif ˆH⁽ⁱ⁾= ˆH^(j),∀i,j.

A CSI structure isdistributed if there existi andj such that Hˆ⁽ⁱ⁾6= ˆH^(j).

maximize objective

{H^(j)}^Kj=1,H

subject tosize({B^(j)}^Kj=1)≤τ

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Some Distributed Information Structures

Incomplete CSIT: A CSI structure is incompleteif ˆH⁽ⁱ⁾ takes the form ∀i Hˆ⁽ⁱ⁾={H_kl,k ∈ Stx,l ∈ Srx}, where Stx (resp. Srx) are subsets of the transmitter set (resp. receiver set).

Hierarchical CSIT: A CSI structure is hierarchicalif there exists an order of transmitter indices i₁,i₂,i₃..such that

Hˆ⁽ⁱ¹⁾⊂Hˆ⁽ⁱ²⁾⊂Hˆ⁽ⁱ³⁾⊂...

Master Slave: Hierarchical where ˆH⁽ⁱ¹⁾= [], and ˆH⁽ⁱ²⁾=H(can be extended toK >2.)

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Problem 2: Signaling for Coordination

TX1

TX2 𝑅𝑅12

𝑅𝑅21 𝐻𝐻�⁽²⁾ {𝑄𝑄1,𝑄𝑄2} 𝐻𝐻�⁽¹⁾

{𝑄𝑄1,𝑄𝑄2}

Heuristic strategies:

1 Local decisionW_i based on ˆH⁽ⁱ⁾ andQ_i,i = 1, ..,K, exchange quantized decisions overRij bits

But poorly informed nodes make bad decisions !

2 Exchange quantized CSI ˆH⁽ⁱ⁾ overRij bits

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Problem 3: Team Decision

Several network agents wish to cooperate towards maximization of a common utility

Each agent has its ownlimited view over the system state All need to come up withconsistent actions

Introduced first in economics and control [Ho, 1980, IEEE], recently in wireless[Zakhour and Gesbert, 2010, ITA]

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Team Decision Theory: Buying a Baguette or not?

In 1936, a couple returns separately from work and wants baguette for dinner. Personal cost for stopping at the baker is ci . Each person knows its own cost c_i,. We assume that thec_i are uniformly distributed over [0,1].

Goal: maximize expectation of joint utility given by:

Person 2\Person 1 Buy bread Go home

Buy bread a-c₁-c₂ 1-c₁

Go home 1-c₂ 0

When should each person buy bread?





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The Distributed Rendez-vous Problem

Two visitors arrive independently in Hammamet and seek to meet as quickly as possible.

They have differentand impreciseinformation about their own and each other’s position.

Problem: Pick a direction to walk into

p1

p2 ) 2 (

p2 ) 1 (

p2 )

1 (

p1 ) 2 (

p1

) (i

pjEstimated position of person j available at person i Meet!

No meet 

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Games vs. Teams

Team agents (network nodes) are not conflicting players Agents seek maximization of the samenetwork utility

It is the lack of shared informationwhich hinders cooperation, not selfishness

Connections to Bayesian games (see work by 1994 Nobel Prize winner John Harsanyi[Harsanyi, 1967] )

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Team Decision Making

Distributed coordination = team decision making= A difficult problem in general! (functional optimization).

max

W_i( ˆH⁽ⁱ⁾),i=1..K

E (

X

i

u_i(W₁( ˆH⁽¹⁾), . . . ,W_K( ˆH^(K)),H) )

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Solving the Problem

Some pragmatic approaches:

Model based decision

Hierarchical (nested) CSI structure High SNR regime

Large scale analysis

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Spatial Allocation of CSI in Multi-antenna Coordinated Networks

Outline

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spatial Allocation of CSIT

 )2(

H j

H⁽¹⁾

H⁽²⁾ H⁽³⁾

Is it optimal for each TX to receive the same information?

How can we save on sharing overhead?

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CSIT vs. antennas in interference alignment

H22

t1 t2 t3

H21 H23

H12t2 H13t3

H21t1 H23t3

H32t2 H31t1

RX 1 RX 2 RX 3

Inte rferenc

e free

Interfere

nce free Interference free

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CSIT vs. antennas in interference alignment

H22

t1 t2 t3

H21 H23

RX 1 RX 2 RX 3

Ht

H21t1

H31t1

Interference free

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Modeling the CSIT sharing overhead

H_ik either known perfectly or not at all at TXj F^(j⁾∈ {0,1}^N^tot^×M^tot theCSIT index matrix

Size(F) the sizeof a CSIT allocationF ={F^(j⁾|j = 1, . . . ,K}

Size(F) =

K

X

j=1

kF^(j⁾k²F

One question: How can wereduce CSIT allocation size while preserving alignment?

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An efficient CSI allocation result

Theorem

[de Kerret and Gesbert, 2014b, TWC] In a tightly-feasible [QK

k=1(Nk,Mk)] IC, if there exists a tightly-feasible sub-IC formed by the set of TXs STX and the set of RXs SRX, i.e.,

Nvar(SRX,STX) =Neq(SRX,STX),

then the incomplete CSIT allocation F ={F^(j⁾|j ∈ K}preserves IA feasibility, if

F^(j⁾=F , ∀j ∈ S

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Example (IC (5,4),(2,2),(2,2),(2,2),(5,4))

TX 1

RX 1

TX 2 TX 3 TX 4 TX 5

RX 2 RX 3 RX 4 RX 5

Tightly-feasible IC

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Random Antenna Settings

1..830 2 2.16 2.33 2.5 2.66

20 40 60 80 100 120

Average size of the CSIT allocation

Complete CSIT allocation Proposed approach Exhaustive search

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A result for joint MIMO precoding

Goal is to find a frugalspatial CSIT allocation giving same DoF as perfect CSIT.

Intuition: Two far away nodes should exchange little (or no) CSI

TX

RX TX

RX

TX RX

TX

RX

TX RX

TX

RX TX

RX

TX

RX TX

RX

TX

RX

TX RX

TX

RX

TX RX TX

RX TX

RX

TX RX TX

RX

TX RX

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Generalized DoF and Interference Level Matrix

Define thegeneralized DoF DoF_i(Γ), lim

P→∞

Ri

log₂(P) , subject toσ_i,j² =P^−{Γ}^i,j, ∀i,j DefineΓ∈[0,∞]^K×K theinterference level matrix

Γ results from the network topology

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Conventional CSIT Allocation

Proposition

The following “conventional” CSIT allocation {B^conv,(j⁾}^Kj=1 such that {B^conv,(j)}k,i = [dlog₂(Pσ²_k,i)e]⁺, ∀k,i,j

=d[1−Γ_k,i]⁺log₂(P)e is DoF achieving, i.e., {B^conv,(j⁾}^Kj=1∈BDoF.

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Shortest Path: Example

Example

RX 1 RX 2

TX 1 TX 2

RX 3 TX 3

Γ31

Γ21 Γ32

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CSIT Allocation for regular networks

Theorem ([de Kerret and Gesbert, 2014a, TIT]) If

Symmetry: Γk,i = Γi,k,∀i,k

Triangular inequality: Γi→k = Γk,i,∀k,i (⇔Γk,i ≤Γk,`+ Γ`,i,∀k,i, `) then the distance based CSIT allocation simplifies to

{B^dist,(j⁾}k,i =d[1−Γ_k,i −min (Γ_i,j,Γ_k_,j)]⁺log₂(P)e, ∀k,i,j

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Team Decision for Multi-Antenna Precoding

Outline

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The Naive Zero Forcing in the distributed setting

TX j computesT^(j⁾= [t^(j₁⁾, . . . ,t^(j_K⁾]∈C^K×K where

t^(j_i ⁾, Hˆ^(j_i ⁾

−1

e_i k

Hˆ^(j_i ⁾−1

e_ik

√P, ∀i

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Applying the Naive ZF in distributed CSI setting

Define CSIT scaling coefficientsA^(j_i ⁾ at TX j such that

A^(j_i ⁾, lim

P→∞

PK k=1B_i,k^(j⁾ Klog(P)

Theorem

The DoF achieved with conventional ZF for user i is equal to [de Kerret and Gesbert, 2012, TIT]

DoF^ZF=K min

i,j∈{1,...,K}A^(j)_i .

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Conventional Robust Design

Classical robust designs target CSI imperfectionbut not CSI inconsistencies [Shenouda and Davidson, 2006, ICASSP]

t^rZF(j_i ⁾, rP

2

(R^(j_∆⁾+H^(j)HH^(j⁾)⁻¹H^(j)He_i

with R^(j_∆⁾ the covariance matrix of the multiuser channel estimation error at TXj

Theorem

Conventional robust ZF precoder achieves the same number of DoFs as conventional ZF.

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Hierarchical Feedback

IntroduceHierarchical/Layered Quantization[Ng et al., 2009, TIT]

CSI encoded such that each TX decodes up to a number of bits depending on the quality of the feedback link

Ifh^(j_i ¹⁾ more accurate thanh^(j_i²⁾, then TXj1 has also knowledge of h^(j_i²⁾

Remark: IfA^(j_i¹⁾=A^(j_i ²⁾, thenh^(j_i ¹⁾=h^(j_i²⁾

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Degrees of Freedom with Hierarchical Feedback

Theorem

The number of DoFs achieved by user i with Conventional ZF and hierarchical feedback is

DoF^cZF=

K

X

i=1

j∈{1,...,K}min A^(j)_i .

Strong improvement of the number of DoFs achieved

CSI scaling of user i impacts solely number of DoFs of useri y Hierachical quantifization enforces coordination between TXs

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Active-Passive Zero Forcing (two user case)

Assume w.l.o.g. thatA_¯⁽²⁾_i ≥A_¯⁽¹⁾_i , then

t^APZF_i , s

P 2 log₂(P)



 1

−^{^˜^h

(2)

¯i }1

{˜h_¯⁽²⁾_i }₂





Theorem

Active-Passive ZF achieves the number of DoFs at user i

DoF^APZF = max A^(j)₁ + max A^(j₂⁾

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Simulations

0 10 20 30 40 50 60

0 5 10 15 20 25 30

Average transmit SNR [dB]

Average sum rate [Bits/s/Hz]

ZF with perfect CSI AP ZF − 3 bits power control AP ZF − heuristic power control Beacon ZF

Conventional ZF

Figure: Sum rate in terms of the SNR with a statistical modeling of the error from RVQ using [A⁽¹⁾₁ ,A⁽²⁾₁ ] = [1,0.5] and [A⁽¹⁾₂ ,A⁽²⁾₂ ] = [0,0.7].

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Lessons learned and open problems

Outline

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Lessons learned

Coordination a powerful method to optimize wireless networks

Nodes may not (must not) share a common channel state information Team decision methods allow dealing with lack of consistency in CSI at various users

Some solutions for certain regimes (high nb users, SNR) but a general optimal strategy remains elusive.

Some interesting future work:

Sequential coordination

Connections with many-shot coordination (Convergence speed vs timeliness)

Implicit coordination[Larrousse and Lasaulce, 2013, ISIT]

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