Transmitter cooperation in wireless networks: potential and challenges

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Transmitter Cooperation in Wireless Networks: Potential and Challenges

Presented by David Gesbert and Paul de Kerret [email protected], [email protected]

May 4th, 2014

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Shameless (but useful) Self-references

Multi-cell MIMO cooperative networks : A new look at interference D. Gesbert, S. Hanly, H. Huang, S. Shamai, O. Simeone, and W. Yu, IEEE Journal on Selected Areas in Communications, December 2010.

CSI sharing strategies for transmitter cooperation in wireless networks, P. de Kerret and D. Gesbert, in IEEE Wireless Communications Magazine, Feb. 2013.

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The Essence of this Talk

Let’s solve conflict! our

Sorry, but I don’t have time to talk!

Okay, tell me your point of

view

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Outline

1 Fundamentals for Transmitter Coordination

2 Thinking Practical

3 Team Decisions Problems in Wireless Networks

4 Information Allocation in Wireless Networks

5 Team Decision for Multi-Antenna Precoding

6 Key Aspects and Open Problems

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Outline

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The Dimensions of Interference Management

REJECT AVOID

EXPLOIT

COORDINATE

CONTAIN

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Transmitter Cooperation Domains

Transmitter Cooperation

Domains

User selection

time

codes frequency

Space (MIMO)

power

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Example 1: Interference Coordination using Scheduling and 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

Distributed max-SINR scheduler exploits the variability (fading) of interference

Power control/beamforming couplesthe decisions at all cells

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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

Alignment can be carried out in space, frequency, time domains A optimal DoF of 1/2 can be achieved (everyone getshalf the cake)

[Maddah-Ali et al., 2008, TIT] [Cadambe and Jafar, 2008, TIT]

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Interference Alignment: Algorithm Design

Exploiting uplink downlink duality of alignment[Gomadam et al., 2011, TIT]

Notations:

LetUi be the RX filter at RXi LetWj be the TX precoder at TXj

LetIi be the total noise summed at RX i, with covarianceQi

Algorithm:

1 TakeU_i as minimum eigenvector ofQ_i, ∀i

2 UseU_i as TX precoder from RXi, on reciprocal channel

3 TakeW_i as RX vector at TXi, on reciprocal channel

4 FindW_i as minimum eigenvector of noise covariance matrix at base i

5 iterate

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Interference Alignment: Finite SNR

IA is optimal at infinite SNR case only

At finite SNR, key is to balance interference canceling with desired signal enhancement

Maximum SINR[Gomadam et al., 2011, TIT]

Minimum MSE[Peters and Heath, 2011, TVT]

Maximum sum rate[Santamar´ıa et al., 2010, GLOBECOM]

Game theoretic approach (Altruism vs. Egoism)[Ho and Gesbert, 2010, ICC]

...

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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 Ncells

RX 1 ^{RX Ncells}

N_t

Network MIMO Transmit antennas:

Nt x Ncells

ŝNcells Data: S1,S2,…,SNcells

N_r

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

Modify standard MU-MIMO schemes to reflect per base power constraint (ZF, MMSE, non-linear precoding: Dirty Paper Coding, vector

perturbation, ..)

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Performance Modeling of Joint Precoding

Transmission from K TXs toK RXs where theith TX and theith RX are equipped respectively withM_i andN_i antennas

Ntot,

K

X

i=1

Ni, Mtot,

K

X

i=1

Mi

Multi-user channelH^H∈C^N^tot^×M^tot

H^H_ik ∈C^Nⁱ^×M^k channel matrix from TXk to RXi with its elements i.i.d. as CN(0, ρ²_ik)

Perfect CSI at the RXs treating interference as noise Linear precoding and filtering

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Received Signal

Received signal





 y1

... yK





=H^Hx+η=





 H^H₁

... H^H_K







t1 . . . tK





 s1

... sK





+





 η1

... ηK







with ktik²=P,∀i,s_i i.i.d. NC(0,1) andη_i i.i.d. NC(0,I_N_i)

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Received Signal

Received signal





 y1

... yK





=

Channel matrix (Ntot×Mtot)

z}|{

H^H x+η=

Channel matrix (Ntot×Mtot)

z }| {





 H^H₁

... H^H_K







t1 . . . tK





 s1

... s_K





+





 η1

... η_K







with ktik² =P,∀i,s_i i.i.d. NC(0,1) and η_i i.i.d. NC(0,I_N_i)

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Received Signal

Received signal





 y1

... yK





=H^Hx +η=





 H^H₁

... H^H_K







Precoder (Mtot×K)

z }| {

t1 . . . tK





 s₁

... sK





+





 η₁

... ηK







with ktik²=P,∀i,si i.i.d. NC(0,1) andηi i.i.d. NC(0,INi)

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Received Signal

Received signal





 y1

... yK





=H^Hx+η=





 H^H₁

... H^H_K







t1 . . . tK

Users data symbols (K×1)

z }| {





 s1

... s_K





 +





 η1

... η_K







withktik² =P,∀i,s_i i.i.d. NC(0,1) and η_i i.i.d. NC(0,I_N_i)

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Received Signal

Received signal





 y1

... yK





=H^Hx+η=





 H^H₁

... H^H_K







t1 . . . tK





 s1

... s_K





+

Noise at the RXs (Ntot×1)

z }| {





 η1

... η_K







with ktik² =P,∀i,s_i i.i.d. NC(0,1) andη_i i.i.d. NC(0,I_N_i)

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Figures-of-Merit

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

R_i,E

"

log₂ 1 + |g_i^HH^H_i ti|² 1 +P

j6=i|g_i^HH^H_i tj|²

!#

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

defined as [Tse and Viswanath, 2005]

DoF_i, lim

P→∞

R_i log₂(P).

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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.

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Outline

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Thinking practical

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

Hardware impairments

Channel estimation and tracking Feedback limitation

Inter-transmitter signaling limitation

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Channel Estimation

Usually relying on a pilot training phase [Hassibi and Hochwald, 2003, TIT],[Caire et al., 2010, TIT]

TX transmitsαpilots per antenna (α≥1) Received signals

y =√

αPh+η

MSE channel estimate

hˆ=

√αP N₀+αPy This yields[Caire et al., 2010, TIT]

h= ˆh+ ˜h with ˜h∼ NC(0, ¹

1+^αP_N

0

) being independent of ˆh y Challenging in massive MIMO [Jose et al., 2011, TWC]

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Hardware Impairments

Transceiver impairments leads to a modified RX signal model[Studer et al., 2010, WSA],[Bjornson et al., 2012, WSA]

y =√

PH(x+η) +n with η∝tr(E[xx^H]) (radiated power)

DoF is shown to be 0

Need for robust signal processing taking the imperfections into accounts[Bjornson et al., 2012, GLOBECOM]

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Modeling Feedback Limitation in TX Cooperation

Quantized feedback Noisy analog feedback Delayed feedback

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Imperfect Centralized CSIT

Imperfect CSIT at the TX modeled as [Wagner et al., 2012, TIT]

{Hˆ}ik =σik

p1−2^−B^ik{H˜}ik +σik2^−B^ik{∆}ik, ∀i,k where{∆}ik ∼ CN(0,1)

CSIT allocationB defined as

{B}ik =B_ik, ∀i,k

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Quantized Feedback in MU-MIMO

Assume a total of M antennas across all cooperating BS, single antenna users

No feedback

Theorem ([Jafar and Goldsmith, 2005, TIT]) With no CSIT, then DoF →1

Quantized feedback

Theorem ([Jindal, 2006, TIT])

Assume Random Vector Quantization with B bits used to encode the channel of one user. Then B ≥α(M−1) log(SNR) is sufficient to achieve DoF of Mα.

Crucial assumption: The quantized feedback is ideally shared across all transmit antennas.

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CSIT in Interference Alignement

Current IA schemes are based on

CSIT feedback followed exchange across links Pilot based CSI estimation + TX-RX iterations

Direct broadcast of the estimates[Ayach and Heath, 2012, TWC]

Quantized feedback in Interference Alignement

Similar results as broadcast channels apply (feedback bits must grow withK(MN−1) logSNR, to achieve maximum DoF)[Thukral and Boelcskei, 2009, ISIT]

Possible to improve over this feedback rate by exploiting rotational invariance[Rezaee and Guillaud, 2012, ITW]

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Inter-transmitter Signaling Limitation

Virtually all cooperation methods require shared channel state information at the transmitters (CSIT)

Perfect sharing of CSIT is not scalablein large networks

In some networks inter-TX links are capacity limited (e.g. wireless backhaul, corgnitive radios)

Any exchange of CSIT is likely to induce latency and/or quantization

→ TX-dependent CSIT noise

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Transmitter Cooperation with Clustering

Cooperation Clusters

Approaches:

Network-centric clustering

User-centric clustering [Papadogiannis et al., 2008, ICC]

Limitations:

Cluster too big: feedback sharing overhead heavy[Lozano et al., 2013, TIT]

Cluster too small: edge-effects (inter-cluster interference) predominant

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Limited Signaling: Backhaul Quantization Model

Backhaul signaling introduces delays and possible quantization noise LTE compliant feedback: User feeds back to its home eNB only

 )2(

H j

H⁽²⁾=[h1(2)h2(2)h3(2)]

H⁽³⁾=[h1(3)h2(3)h3(3)]

h2(3)

h2(2)

h2(1)

h3

h2

h1

CSI Sharing Scenario

Delay/

Quantization

H⁽¹⁾=[h1(1)

h2(1)

h3(1)

]

Delay/

Quantization

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Over-the-air Signaling: Feedback Broadcast

CSIT can be shared directly over-the-air without backhaul links

 )2(

H j

H⁽³⁾=[h1(3)h2(3)h3(3)]

H⁽²⁾=[h1(2)h2(2)h3(2)]

H⁽¹⁾=[h1(1)h2(1)h3(1)]

h2

h3

h1

CSI Broadcast Scenario

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A Distributed Channel State Information Model

Imperfect CSIT at TX j modeled as {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

x1=w₁^H( ˆH⁽¹⁾)s

x2=w₂^H( ˆH⁽²⁾)s TX1

TX2

•

• Hˆ⁽¹⁾,s

Hˆ⁽²⁾,s

x1

x2

• 1

• 2

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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:

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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Three Coordination Problems

Team Decision

Information allocation

Signaling

Coordination with limited signaling

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Outline

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Wireless Coordination Problems

K nodes in a network seek tocooperatetowards the maximization of acommon utility

Each nodei must make bestdecisionbased on:

local measurement or feedback

finite rate signaling with neighbor nodes

TX1

TX2

TX𝑗

TX𝑖

TX𝐾

Message/interference Coordination domains:

• Power

• Time/freq/code

• Antenna/beam

• …

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Coordination Problems

The coordination problem is decomposed into three related optimizations:

Problem 1: Channel state information allocation

What should the local information ˆH⁽ⁱ⁾ be? (under a typical feedback constraint)

How uncorrelated can it be?

Problem 2: Signaling

What to communicate overR_ij bits? →Z_ij Very challenging as it involves second problem Problem 3: (Team) decision making

W_i( ˆH⁽ⁱ⁾,Q₁,Q₂, ..Q_K,Z_1i,Z_2i, ..Z_Ki)

Team decision optimization (challenging enough..)

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One-shot Coordination Graph

TX1

TX2

TX𝑗

TX𝑖

TX𝐾 𝑅₁₂

𝑅₂₁

𝑅_1𝑗 𝑅_𝑗1 𝑅_𝑗𝑖

𝑅_𝑖𝑗

𝑅_𝐾𝑖 𝑅_𝑖𝐾

𝐻 ⁽²⁾ {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ⁽¹⁾ {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝑗) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝑖) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝐾) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

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

A priori information: Coordination link rates:

From i to j: R _ij

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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,{H^(j⁾}^Kj=1

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 to K >2.)

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Typical Distributed Information Structures

Consider the K transmitter (N antennas each) K user (single antenna) channel. Leth^H_ij be the 1×N vector channel between the jth transmitter and the ith user.

Local CSIT with TDD reciprocity

( ˆH^(j))^H=







0 h_1j^H 0 ... ... ... 0 h^H_Kj 0







Local CSIT with LTE feedback mode

( ˆH^(j))^H=





0 0 0 h_j^H₁ . . . h_jK^H

0 0 0





Fully local CSIT

( ˆH^(j⁾)^H=





0 0 0 0 h_jj^H 0 0 0 0





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Problem 2: Signaling for Coordination

What is most relevantto communicate of the signaling link?

Many interesting heuristics (precoding decisions, measurements, etc.) Optimal signaling an open problem

Optimal signaling strategy coupled with optimum decision making W_i

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Signaling for Coordination

Heuristic strategies:

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

But poorly informed nodes make bad decisions !

2 Exchange quantized CSI ˆH⁽ⁱ⁾ overRij bits But this ignoresQi !

3 Exploit coordination graph for CSI improvement:

UseRij bit signaling to create optimal estimatesHˆˆ⁽ⁱ⁾

Optimal strategy (source coding with side-information): Create locally optimal codebooks, that are function of local CSI and neighbor CSI qualities [Li et al., 2014]

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Some Open Problems

Coordination over more than one shot

No constraint over number bits exchanged: Convergence? Speed?

Constraint on total number of bits exchanged

With more signaling slots, opportunity tolearn from pastactions With fewer signaling slots,less latencyeffects

Implicit coordination[Larrousse and Lasaulce, 2013, ISIT]

Sequentialcoordination over the graph

TX1

TX2

TX𝑗

TX𝑖

TX𝐾 𝑅₁₂

𝑅21

𝑅_1𝑗 𝑅𝑗1 𝑅_𝑗𝑖

𝑅_𝑖𝑗

𝑅𝐾𝑖 𝑅_𝑖𝐾

𝐻 ⁽²⁾ {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ⁽¹⁾ {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝑗) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝑖) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

𝐻 ^(𝐾) {𝑄𝑖, 𝑖 = 1, … , 𝐾}

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

A priori information: Coordination link rates:

From i to j: R _ij

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Problem 3: Team Decisional Coordination

Team Decision theoretic problems:

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

Classical ”robust” design does not work...

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

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Team Decision with Finite Signaling

TX1

TX2

TX𝑗

TX𝑖

TX𝐾 𝑅𝐾𝐾

𝐻�⁽²⁾ {𝑄𝐾,𝑖= 1, … ,𝐾}

{𝑍𝑛2,𝑛= 1, … ,𝐾,𝑛 ≠2}

𝐻�⁽¹⁾ {𝑄𝐾,𝑖= 1, … ,𝐾}

{𝑍𝑛1,𝑛= 1, … ,𝐾,𝑛 ≠1}

𝐻�^(𝑗) {𝑄𝐾,𝑖= 1, … ,𝐾}

{𝑍𝑛𝑗,𝑛= 1, … ,𝐾,𝑛 ≠ 𝑗}

𝐻�^(𝐾) {𝑄𝐾,𝑖= 1, … ,𝐾}

{𝑍𝑛𝐾,𝑛= 1, … ,𝐾,𝑛 ≠ 𝑖}

𝐻�^(𝐾) {𝑄𝐾,𝑖= 1, … ,𝐾}

{𝑍𝑛𝐾,𝑛= 1, … ,𝐾,𝑛 ≠ 𝐾}

𝐻�^(.): local CSI

𝑄𝐾: Error covariance, i=1..K

𝑍𝑚𝐾: Coordination signal from TX𝑚 to TX𝑖 Local information at TXi after signaling

Decision making: Example, precoder W 𝑖

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Team Decision Theory: Buying a Baguette or not?

In 1936, a french 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 ci,. We assume that the ci 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?

Optimal decision of

threshold form







 



th i i

th i i i

i c c

c c c

if home Go

if bread ) Buy

 (

) ( _i

i^ c



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The Distributed Rendez-vous Problem

Two visitors arrive independently in Firenze 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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Team Decisional Transmitter Cooperation

Let us consider the following K-transmitter framework:

1 Distributedinformation structure:

Each transmitter i has knowledge of ˆH⁽ⁱ⁾, which exhibits some arbitrary correlation with H

2 Zero rate coordination links (R_ij)

3 Decision spacefor each transmitteri

Example: W_i( ˆH⁽ⁱ⁾) wherew_i is a complex matrix. (e.g. W_i ∈C^N×K for K-user network MIMO)

4 A network utility u =

K

X

i=1

u_i(W₁( ˆH⁽¹⁾), ..,W_K( ˆH^(K)),H)

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Can Team Problems be Solved with Games?

Key idea: Let autonomous transmitting devices interact to solve their interference conflicts

Players → transmitters

Actions → transmit decision (power, frequency, beam, ..)

Strategy → Utility maximization (max rate, min power, min delay,..) Timing→ simultaneous, sequential,..

Equilibrium → Nash, Stackelberg, Nash Bargaining,..

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From Selfish Games to ”Team Playing”

Why interference coordination can be different from a typical

”game”’:

Team agents (network nodes) are not conflicting players (different from players in a cooperative game)

Agents seek maximization of the samenetwork utility

It is thelack of shared informationwhich hinders cooperation, not the selfish of their interests

Agents are not required to improve over the performance of the Nash equilibrium

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

Wi( ˆH⁽ⁱ⁾),i=1..K

E (

X

i

ui(W1( ˆH⁽¹⁾), . . . ,WK( ˆH^(K)),H) )

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Model-based Decision Making

The model-based approach:

Replacew_i( ˆH⁽ⁱ⁾) byf(a_i,Hˆ⁽ⁱ⁾) wheref(., .) is afunctional model and a_i a vector of deterministic parameters to be determined at

transmitteri.

Solve for (still hard ;-) )

ai,i=1..Kmax E (

X

i

ui(f(a1,Hˆ⁽¹⁾), ..,f(aK,Hˆ^(K)),H) )

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Solving the Problem

We now target the following problems:

What information isreally needed where?

Distributedcooperative precoding: conventional and robust solutions We use the following approaches:

The high SNR regime The large scale analysis

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Outline

Interference Alignment with Incomplete CSIT Sharing Joint Precoding with Distance Based CSIT Allocation

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Information Allocation

Study here theCSI feedback allocation problem

No signaling possible between TXs Intuitively: CSI allocation should be TX dependent

y Can we quantify this intuition?

 )2(

H j

H⁽¹⁾

H⁽²⁾ H⁽³⁾

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

How does the required information accuracy depend on the network geometry?

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Key Intuition with IA

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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Key Intuition with IA

H22

t1 t2 t3

H21 H23

RX 1 RX 2 RX 3

H12t2

H13t3

H21t1

H23t3 H32t2

H31t1

Interference free

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Exploiting Distributed Precoding

Can we reduce the CSIT requirements in general antenna configurations without additional antennas?

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Model for Incomplete CSIT

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

Hˆ^(j⁾=F^(j⁾H

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

K

X

j=1

kF^(j⁾k²F

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Example

(1,1)(2,3)(2,3) IC:

Conventional CSIT allocation F⁽¹⁾=F⁽²⁾=F⁽³⁾=1_5×7 IA possible with

F⁽¹⁾ =05×7,F⁽²⁾=15×7,F⁽³⁾=15×7

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Minimal CSIT Allocation

Fcomp completeCSIT allocation:

Size(Fcomp) =K(Size(1_N_tot_×M_tot)) Study only feasible settings: IA feasible withFcomp

Study only single-streams transmissions Optimization Problem

Find the most incomplete CSIT allocation where IA remains feasible

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Tightly-Feasible and Super-Feasible Settings

Definition

Tightly-feasible IC ⇔ feasible IC and removing any antenna makes IA unfeasible⇔ feasible and PK

i=1Ni +Mi =K(K + 1) Definition

Super-feasible IC ⇔ feasible IC and it is possible to remove at least one antenna without making IA unfeasible

Definition

A sub-IC is the (generalized) IC formed by any subset SRX of RXs and any subsetSTX of TXs

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Feasibility Condition

Theorem (reformulation of [Razaviyayn et al., 2012, TSP]) IA is feasible if and only if,

Nvar(SRX,STX)≥ Neq(SRX,STX), ∀STX,SRX⊆ {1, . . . ,K} where

Nvar(SRX,STX), X

i∈S_RX

Ni −1 + X

i∈S_TX

Mi −1 Neq(SRX,STX), X

k∈S_TX

X

j∈S_RX,j6=k

1

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Tightly Feasible Setting

Theorem

In a tightly-feasible [Q_K

k=1(N_k,M_k)]IC, if there exists atightly-feasible sub-ICformed by the set of TXs STX and the set of RXsSRX, i.e.,

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

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

F^(j⁾=FS_RX,S_TX, ∀j ∈ STX

F^(j⁾=FK,K=1_N_tot×Mtot, ∀j ∈ S/ TX.

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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

Remark: Tightly-feasible approach exploits heterogeneity of antenna configuration

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

TX 5

RX 5

TX 2 TX 3 TX 4

RX 2 RX 3 RX 4

TX 1

RX 1 ZF ZF

CSIT allocation for tightly-feasible and super-feasible settings in

[de Kerret and Gesbert, 2014b, TWC]

Remark: Tightly-feasible approach exploits heterogeneity of antenna configuration

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Super-feasible Settings

1..830 2 2.16 2.33 2.5 2.66

20 40 60 80 100 120

Average number of antenna per node

Average size of the CSIT allocation

Complete CSIT allocation Proposed approach Exhaustive search

Figure: Average feedback size for K = 3 users with the antennas allocated uniformly at random to the TXs and the RXs

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Key Aspects

Perfect IA possible without full CSIT sharing

Adapted CSI allocation provides strong gains with no DoF reduction In fact, we also designed a new IA algorithm: Joint CSI

allocation/precoding design

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CSIT Reduction in Centralized IA

Large literature on IA feasibility assume perfect centralized CSIT

[Razaviyayn et al., 2012, TSP],[Gonzalez et al., 2014, TIT]

Previous intuition developed in[Rao et al., 2013, TSP]

Investigate trade-off Antenna/CSI/DoF

Propose new precoding schemes and new IA feasibility conditions Exploit Grassmanian invariance to reduce feedback from the RXs

[Rezaee and Guillaud, 2012, ITW]

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Exploiting Distributed Precoding

CSI exchange is reduced by using iterative exchange between TXs in

[Cho et al., 2012, TSP]

Only for particular stream/antenna configurations Requires iterations between the TXs/RXs

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Outline

Interference Alignment with Incomplete CSIT Sharing Joint Precoding with Distance Based CSIT Allocation

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TX Cooperation in Large Networks

Joint precoding: user’s data symbols are shared

Consider large networks: TX cooperation even more challenging Channel estimation

CSI exchange

Amount of CSI increases quickly

Recent results suggest that even with perfect transmitter cooperation, performances are fundamentally limited as the network size increases

[Lozano et al., 2013, TIT]

Is it possible to manage interference via TX cooperation in asymptotically large networks?

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Goal

Find a CSIT allocation verifying TX coordination: Global interference management Scalability: Two users

asymptotically far away should not exchange any 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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Received Signal

Received signal





 y1

... yK





=H^Hx +η=





 h^H₁

... h^H_K







t1 . . . tK





 s₁

... s_K





+





 η₁

... η_K







with

ktik²=P,∀i s_i i.i.d. NC(0,1) η_i i.i.d. NC(0,1)

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Generalized DoF and Interference Level Matrix

Define thegeneralized DoF DoFi(Γ), lim

P→∞

Ri

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

Γ represents the network geometry:

Transmission at SNRP0with channel variancesσ²_i,j,0 DefineΓas

Γ_i,j ,−log(σ²_i,j,0)

log(P0) , ∀i,j

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Example (Generalized DoF)

2-user IC, single-antenna nodes, α² = 10⁻¹²,Hi,j ∼ NC(0,1)

H11 H22

P P

α H12

α H21

RX 1 RX 2

TX 1 TX 2

DoF analysis: DoFi = 0.5[Etkin et al., 2008, TIT]

Generalized DoF analysis: For P = 20dB, Γ=

0 6 6 0

and DoF_i(Γ) = 1

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Distributed CSIT

Imperfect CSIT at TX j modeled as {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 CSIT allocation characterized by {B^(j⁾}^Kj=1

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Distributed Zero Forcing

TX j computesT^(j⁾= [t₁^(j), . . . ,t_K^(j)]∈C^K×K where

t_i^(j),

Hˆ^(j)−1

ei

k Hˆ^(j⁾

−1

eik

√P, ∀i

TX j transmits onlyx_j =e_j^HT^(j⁾s such that

T^DCSI,





 e₁^HT⁽¹⁾ e₂^HT⁽²⁾

... e_K^HT^(K)







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