Cooperation in Wireless Networks - Wireless Systems...

Cooperation in Wireless Networks

Ivana Maric and Ron Dabora

Stanford

15 September 2008

Ivana Maric and Ron Dabora Cooperation in Wireless Networks 1

Objectives

I The case for cooperation

I Types of cooperation

I Performance measures

I Cooperation schemesI Performance, limitationsI Building blocks

I Small networks, large scale networks

I Fundamental tradeoffs

I Introduce recent results


Outline

I Introduction

I Relaying strategies

I Conferencing and feedback

I Cooperation in networks with multiple communicating pairs

I Cooperation in fading channels

I Cooperation in large-scale networks

I Summary


Introduction


Introduction

I Motivation

I Basic measuresI CapacityI Scaling laws

I Channel modelsI Static channelsI Time-varying channels

I Diversity


Challenges

I Higher data rates and better coverageI USIIA 2007, RIAA 2006

I Dynamic nature: time-varying channel, users’ mobility,stochastically varying traffic

I Efficient spectrum allocation and coexistence of users

I Security and privacy

I Energy efficiency

I Operating large ad hoc networks

I Guaranteed rate (Quality-of-Service)


Traditional Approach: A Network is a Collection of

Point-to-Point Links

I Current wireless networks (cellular networks, Wi-Fi) areviewed as a collection of point-to-point links

I To increase data rates the point-to-point rate is increased

I What happens when this approach is exhausted (tooexpensive, approaching the fundamental limits)?

⇒ Need to find methods to significantly increase data rate forthe same PtP link performance


Current View: Interference is Harmful

I Wireless networks are inherently broadcastI Any transmission is overheard by neighbouring nodes

T3

R1

R3

R2T2

T1

Interference is extremely harmful for existing wireless networkdesigns


Addressing the Challenges via Cooperation

I Nodes which are not the source or destination of a givenmessage help communicating the message

I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback


Addressing the Challenges via Cooperation

I Nodes which are not the source or destination of a givenmessage help communicating the message

I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback

I Future applicationsI Ad-hoc networksI Sensor networks

I Cooperation takes advantage of the broadcast nature of thewireless channel


Not Just Theoretical

I Dlink High Speed 2.4GHz (802.11g) Wireless RangeExtender

I Under development for the 802.16 (WirelessMAN/WiMAX)I j - multihop relay specificationI m - advanced air interface


Let’s begin


Memoryless Point-to-Point Channels

I Gaussian channel

I zi - bandlimited AWGN, i.i.d., E{|zi |2} = N


Memoryless Point-to-Point Channels

I Gaussian channel

I zi - bandlimited AWGN, i.i.d., E{|zi |2} = N

I Discrete channel: xi ∈ X , yi ∈ Y


The Memoryless Point-to-Point Channel Model

I A channel is characterized by the conditional distribution ofits output at time i :

p(yi |y i−1, x i ), xi ∈ X and yi ∈ Y,

i = 1, 2, ...


The Memoryless Point-to-Point Channel Model

I A channel is characterized by the conditional distribution ofits output at time i :

p(yi |y i−1, x i ), xi ∈ X and yi ∈ Y,

i = 1, 2, ...

I p(yi |y i−1, x i ) takes into account all the effects of signalprocessing: time synchronization, frequency synchronization,PLL, equalizer,...

I A channel is called memoryless if p(yi |y i−1, x i ) = p(yi |xi )

X Y p 1

p 2 p 3

q 1 q 2

q 3


The Memoryless Point-to-Point Channel: BSC

I |X | = 2, |Y| = 2

I xi - BPSK signal

I Decoding takes place after a 2-level quantization at thereceiver with threshold at zero


The Memoryless Point-to-Point Channel: BSC

I |X | = 2, |Y| = 2

I xi - BPSK signal

I Decoding takes place after a 2-level quantization at thereceiver with threshold at zero

⇔ Binary symmetric channel

X Y p

p

1-p

1-p


Channel Capacity

X n Y n p(y n |x n ) Encoder Decoder W W ^

I R denotes the information rate in bits/symbol

I In 1948 Claude E. Shannon showed that transmittinginformation over a (memoryless) PtP channel p(y |x) can bedone with an arbitrarily small probability of error as long as

R ≤ maxp(x)

I (X ;Y )


Channel Capacity




R ≤ maxp(x)

I (X ;Y )

I The capacity achieving code is characterized by thedistribution p(x)


Channel Capacity




R ≤ maxp(x)

I (X ;Y )


I Average probability of error: P(n)e = Pr(W 6= W )


Channel Capacity




R ≤ maxp(x)

I (X ;Y )



I Codebook: Generate 2nR i.i.d. codewordsPr(xn) =

∏n

i=1 pX (xi )


Channel Capacity




R ≤ maxp(x)

I (X ;Y )



I Codebook: Generate 2nR i.i.d. codewordsPr(xn) =

∏n

i=1 pX (xi )⇒ ∀ε > 0 we can find n large enough s.t. ∃ at least one

codebook for which P(n)e ≤ ε


Channel Capacity: Converse


R ≤ maxp(x)

I (X ;Y )




R ≤ maxp(x)

I (X ;Y )

I Conversely, if R > maxp(x) I (X ;Y ) then the averageprobability of error achieved by any code is bounded awayfrom zero for any n




R ≤ maxp(x)

I (X ;Y )

I Conversely, if R > maxp(x) I (X ;Y ) then the averageprobability of error achieved by any code is bounded awayfrom zero for any n

I Definition: The Capacity of a channel is the maximal ratefor which reliable communication can be achieved


AWGN Channel Capacity

I Let P be an average power constraint on the channel input:

1

n

n∑

i=1

|xi(w)|2 ≤ P , w ∈ W

I The problem: Find the input distribution p(x) thatmaximizes I(X;Y) subject to average input powerconstraint P

I The solution: X ∼ CN (0,P)I The capacity:

C = log2

(

1 + |g |2 P

N

)

bits/transmission


Network Capacity

I A network is a collection of K nodes (sources, sinks) anddirected edges (links).


Network Capacity


I Assume symbol time synchronization of all elements in thenetwork

I Analyze the network throughput for a block of n symbols


Network Capacity




I Let node k, k = 1, 2, ...,K send message Wk ∈ Wk

I The rate is Rk =log2 |Wk |

n


Network Capacity






n

I Let Dk be the set of nodes that decode Wk

I W jk , j ∈ Dk


Network Capacity






n

I Let Dk be the set of nodes that decode Wk

I W jk , j ∈ Dk

I The probability of error for network transmission is

P(n)e = Pr

K⋃

k=1

⋃

j∈Dk

{

W jk 6= Wk

}


Network Capacity Region

I The capacity region is the set of all rate vectors

(R1,R2, ...,RK ) such that the probability of error P(n)e can

be made arbitrarily small by taking n large enough

I Why it is important?

1. It is the theoretical upper bound2. Determines optimal communication strategies3. Leads to practical designs

T3

R1

R3

R2T2

T1

R1

R2

I Very hard to find


Multiuser Networks: The Multiple Access Channel

I The MAC: p(y |x1, x2)I p(x1, x2) = p(x1)p(x2)I Introduced by Shannon in 1961I Capacity known for both discrete and Gaussian channels

I Capacity [Ahlswede’71, Liao’72]I MIMO [Telatar’99]I Fading [Gallager’94, Shamai & Wyner’97, Tse & Hanly’98]


Multiuser Networks: The Broadcast Channel

I The BC: p(y1, y2|x)

I Introduced by Cover in 1972I Capacity known only for special cases

I Degraded channels [Bergmans’73,74, Gallager’74]I General BC with degraded message sets [Korner & Maron’77]I MIMO BC [Weingarten, Steinberg & Shamai’06]

I Best achievable region due to Marton’79I Best upper bound due to Nair & El-Gamal’07


Multiuser Networks: The Relay Channel

I The relay channel: p(y , y1|x , x1)

I The most basic form of cooperation

I Introduced by van der Meulen in 1968I Capacity known only for special cases

I Physically degraded channels [Cover & El-Gamal’79]I Gaussian relay channel with SNR → ∞ [Kramer’05]I Stochastically degraded relay channel with deterministic link

[Zhang’88]

I Fundamental schemes introduced by Cover & El-Gamal’79I Decode-and-forwardI Compress-and-forward


Multiuser Networks: The Interference Channel

I The interference channel: p(y1, y2|x1, x2)

I The building block for multiple-pairs communication

I Introduced by Shannon in 1961I Capacity known only for special cases

I Strong interference [Carleial’75, Sato’81, Costa & El-Gamal’87]I Gaussian IC with very weak interference [Shang, Kramer &

Chen’08, Motahari & Khandani’08]I Cognitive Gaussian ICI No interference

I Best achievable region due to Han & Kobayashi’81


The Interference Channel: Strong Interference vs. Weak

InterferenceWeak interference

I At least one of the cross-links isworse than the its respectivedirect link

Tx 2

Rx 2

Tx 1 Rx

1

I Decoding W1 at Rx2 reducesthe maximum rate of W1

I No single-letter expression forthe capacity region

I Capacity known for special cases

Strong interference

I The cross-links are better thanthe direct links

Tx 2

Rx 2

Tx 1 Rx

1

I Decoding W1 at Rx2 does notconstrain the maximum rate ofW1

I The capacity achieving schemeis known [Sato’81], [Han andKobayashi’81]

• The rates with weak interference are generally less than therates with (very) strong interference


The Cut-Set Upper Bound

I A general tool for outer bounding the capacity region of anetwork

I Let V = {1, 2, ...,K} index the network nodes

I Let R ij denote the information rate from node i to node j

I A cut is a partition of V into two sets S and S = V \ STheorem (∼Aref’80)

If the information rates{R ij

}are achievable then there exists

a joint distribution p(x(1), x(2), ..., x(K)

)such that for every cut

(S, S)∑

i∈S,j∈S

R ij ≤ I(

X(S);Y(S)|X(S)

)


Operating Regimes: Half-Duplex vs. Full-Duplex

I We shall compare results for different operating regimes

I In full-duplex the nodes receive and transmit simultaneously

I In half-duplex a node can either receive or transmitI Often encountered in wireless systems in practice


Time-Varying Channels: Fast vs. Slow Fading

I When the channel is time varying, the received signal isgiven by

yr ,i = htr ,ixt,i + zi

I htr,i is the channel gain between the transmitter and receiverat time i

I htr ,i models a Rayleigh fading channel: htr ,i ∼ CN (0, 1)

I There are three types of Rayleigh fadingI Fast fading: htr,i ∼ CN (0, 1), i.i.d., i = 1, 2, ..., nI block fading: htr,i = htr , i = 1, 2, ..., nI Slow fading: htr,i = htr


Time-Varying Channels: Channel State Information

I CSI is the knowledge a network node has on the channels

I Two types: transmitter CSI (CSIT) and receiver CSI (CSIR)

I Let H denote the random channel state and let the channelbe defined by p(y |x , h).

I There are four possible CSI configurations:

CSIT CSIR Capacity

No No maxp(x) I (X ;Y )

p(y |x) =∑

h p(y |x , h)p(h)No Yes maxp(x) I (X ;Y |H)

Yes No variesYes Yes EH{maxp(x |h) I (X ;Y |h)}


Time-Varying Channels: Outage Capacity

I Shannon’s capacity measure is also called ergodic capacityI Assumes that the channel is information stable (ex. i.i.d.

fading)I Application is delay tolerant

I For slow fading Rayleigh channels, the mutual informationI (X ;Y |h) is a random variable

I depends on the channel realization hI The channel is non-ergodic

I Note that for every R > 0, Pr (I (X ;Y |h) < R) > 0

⇒ The Shannon capacity is zero

I The event {h : I (X ;Y |h) < R} is called outage

I Outage capacity is the maximum rate that can beguaranteed for a given outage probability Pout:

supR

Pr(I (X ;Y |h) < R) ≤ Pout


Diversity

I Transmitting signals carrying the same information overdifferent paths in time, frequency or space

I Cooperation diversity is achieved when nodes forward toother nodes information

I Enhance desired informationI Facilitate interference cancellation


Diversity




I Diversity reduces outage probability


Diversity




I Diversity reduces outage probability

I Transmitter cooperation, receiver cooperation

I When the transmitters cooperate and also the receiverscooperate the system resembles a MIMO system

⇒ Distributed MIMO

I Differences between MIMO and distributed MIMO:I Messages known only at source nodesI Cannot perform antenna power allocationI Nodes may have half-duplex constraints


Time-Varying Channels: Finite-State Models

I Slow fading, fast fading are extreme cases

I An alternative model for time-varying channels with memoryI Correlated fading, multipathI Filters (pulse shape, IF and RF filters)I AGC, Timing, PLL, equalizer

I The finite-state channel (FSC) was introduced as early as1953 [McMillan’53]

I Time variations are represented by correlated states


Finite-State Channels

I Memory is captured by the state of the channel at the endof the previous symbol’s transmission

I Si is the channel state at time iI s0 is the initial channel state

p(yi , si |xi , xi−1, y i−1, s i−1, s0) = p(yi , si |xi , si−1)

I Si−1 contains all the history information for time iI S is finite

I ISI channel: Si−1 =(Xi−1,Xi−2, ...,Xi−J )

n i

x i

A/D

y i

ISI Channel

k

h(k)


Analysis of Large Networks: Scaling Laws

I Finding the capacity region of even small networks is a verydifficult task

I Many possibilities for cooperation

I Scaling laws allow us to obtain insights on the performanceof large scale networks.

I Pioneering work of Gupta and Kumar’00

I NotationI f (n) = O(g(n)) ⇔ limn→∞

∣∣∣f (n)g(n)

∣∣∣ < ∞

I f (n) = Θ(g(n)) ⇔ f (n) = O(g(n)) and g(n) = O(f (n))


Analysis of Large Networks: Scaling Laws

I Definitions:I Assume a network that consists of n nodes that form m

source-destination pairs.I Let dl denote the distance between source and destination

for pair l ≤ mI The transport capacity is defined as

CT = sup(R1,R2,...,Rm) feasible

m∑

l=1

Rldl

I The transport capacityI Provides a single number which summarizes what a network

can deliverI Follows a scaling law such as

CT (n) = Θ(√

n),O(n) bit-meters/second

I Does not provide explicit information on the individual rates


Some Important Questions

I How to incorporate relaying into the design of a network?I Compare performance of different schemesI Under what conditions capacity is achievedI What are the maximum rate gains we can expect from

adding relays to the network?

I Different aspects of relaying that arise when consideringmultiple communicating pairs

I Do not exist in the classic relay channel

I Understand the fundamental performance tradeoffsassociated with node cooperation

I Analysis of cooperation in large scale networks


Cooperative Strategies


In This Section...

I Decode-and-forward

I Compress-and-forward

I Amplify-and-forward

I Capacity upper bound

I Performance comparison



T3

R1

R3

R2T2

T1



T3

R1

R3

R2T2

T1

I Traditional approach: multihop routing

I Many point-to-point links

I Intermediate nodes store and forward packets


BroadcastI Wireless networks are inherently broadcast

I Any transmission is overheard by neighboring nodes


BroadcastI Wireless networks are inherently broadcast

I Any transmission is overheard by neighboring nodes

T3

R1

R3

R2T2

T1

I Interference is harmful for current wireless network designsI Cooperative strategies exploit broadcast


Relay Channel

I Message W ∈ {1, . . . ,M} sent at rate R

I Encoding at the source: X n1 = f1(W )

I At the relay at time i : X2,i = f2,i(Yi−12 ), i = 2, . . . , n

I Decoding: W = g(Y n3 )

I R = log2 M/n


Decode-and-Forward

I Exploit broadcast transmission at the source


Decode-and-Forward


I Source and relay transmit simultaneously


Decode-and-Forward



I Messages sent in blocks: w1,w2, . . . ,wb, . . .


Decode-and-Forward



I Messages sent in blocks: w1,w2, . . . ,wb, . . .

I Two random codebooks: xn1 , xn

2 generated withp(x2)p(x1|x2)


Superposition Coding

I Random codebooks xn1 , xn

2 generated with p(x2)p(x1|x2)

In block b:

I The source: xn1 (hb(wb−1),wb) block Markov encoding

I The relay: xn2 (hb(wb−1))


Decode-and-Forward Strategies

I Irregular encoding, successive decodingI Codebooks xn

1 , xn2 have different sizes

I Regular encoding, sliding-window decodingI Decoding over two block

I Regular encoding, backward decodingI Decoding starts from the last received block


Decode-and-Forward

R ≤ I (X1;Y2|X2)

R ≤ I (X1;Y3|X2) + I (X2;Y3) = I (X1,X2;Y3)

R = maxp(x1,x2)

min{I (X1;Y2|X2), I (X1,X2;Y3)}


Decode-and-Forward in AWGN Channels

I Choose: Gaussian p(x1, x2)I E [|X1|2] = P1, E [|X2|2] = P2, E [X1X

∗2 ] = ρ

√P1P2

I Superposition codebook:

I Gen. symbols: X10 ∼ CN (0, (1 − ρ2)P1), X2 ∼ CN (0,P2)

I In block b: xn10(wb), xn

2 (wb−1)

xn1 (wb−1,wb) = xn

10(wb) + ρ√

P1P2

xn2 (wb−1)

Y2 = h12X1 + Z2

Y3 = h13X1 + h23X2 + Z3


DF Rate in AWGN Channels

R = maxρ

min

{

log2

(

1 +|h12|2(1 − |ρ|2)P1

N

)

,

log2

(

1 +|h13|2P1

N+

|h23|2P2

N+

2Re{ρh13h∗23}

√P1P2

N

)}

I Signals coherently-combinedI Relay signal perfectly phase-aligned with the source signalI Not practicalI Decoding constraint at the relay can be severeI DF optimal for |h12| → ∞: source and relay act as two

transmit antennasI DF performs well when the relay is close to the source


Antenna-Clustering Capacity

I Generalizes to multiple relays

I Relays act as a multiple-transmit antenna


Classic Multihop

R = min

{

T log2(1 +|h12|2P1

TN), T log2(1 +

|h23|2P2

TN)

}

I For α = 2, performs worse then using no relay at all

I Gains for α > 2 and for half-duplex relaysI α-path-loss exponent


DF in Half-Duplex Relay Channel

I All nodes know a priori when a relay listens/talks




I Mode modulation: data modulates listen/talk interval




I Mode modulation: data modulates listen/talk interval

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5 cut−set bound for afixed slot strategy

DF, fixed

DF, random

Pr(M2=L) for DF

Pr(M2=L) for cut−set bound

relay off

d

Rat

e [b

it/us

e]


Compress-and-Forward

I Relay does not decode the source message

I Relay quantizes Y n2 into quantization codeword Y n

2I By finding a jointly typical yn

2 with received yn2

I Three codebooks: xn1 (wb), y

n2 (sb−1, zb), x

n2 (sb−1)

How does relay operate?

I In block b: knows sb−1, decides on zb thru quantization

I Obtains y2(sb−1, zb)

What does it send?

I Binning: each z randomly assigned to bin s

I In block b + 1 : sends x2(sb) such that zb ∈ sb



Destination in block b + 1:

I Decodes sbI Determines zb ∈ sbI Knows y2(sb−1, zb), x2(sb−1)

I Decodes wb using (y2(sb−1, zb), y3,b)



Destination in block b + 1:

I Decodes sb → RQ ≤ I (X2;Y3)

I Determines zb ∈ sbI Knows y2(sb−1, zb), x2(sb−1)

I Decodes wb using (y2(sb−1, zb), y3,b) R ≤ I (X1; Y2,Y3|X2)


Compress-and-Forward Rate

R = I (X1; Y2,Y3|X2)

subject toI (Y2;Y2|Y3X2) ≤ I (X2;Y3)

for p(x1)p(x2)p(y2|x2, y2)p(y1, y2|x1, x2)

I R is single-user rate when receiver has two antennas


Compress-and-Forward in AWGN Channels

I Choose: Y2 = Y2 + Z2 Z2 ∼ CN (0, N2)

I For smallest N2 choose: I (Y2; Y2|X2Y3) = I (X2;Y3)

N2 = NP1(|h12|2 + |h13|2) + N

P2|h23|2

R = log2

(

1 +P1|h12|2N + N2

+P1|h13|2

N

)

I Optimal for |h23| → ∞: relay and destination act astwo-receive antenna

I CF performs well when the relay is close to destination



I Generalizes to multiple relays

I Relays act as a multiple-receive antenna



I Two closely spaced clusters: DF and CF

I Achieves optimal scaling behavior


Amplify-and-Forward

I In discrete channel: X2,i = Y2,i−1, Y ⊆ X


Amplify-and-Forward

I In discrete channel: X2,i = Y2,i−1, Y ⊆ XI In Gaussian channel: X2,i = aiY2,i−1 i = 1, . . . , n

I ai chosen to satisfy power constraint


Amplify-and-Forward



I At the destination channel with ISI:

Y3,i = h13X1,i + h23X2,i + Z3,i

= h13X1,i + ah12h23X1,i−1 + Z ′3,i


Amplify-and-Forward



I At the destination channel with ISI:

Y3,i = h13X1,i + h23X2,i + Z3,i

= h13X1,i + ah12h23X1,i−1 + Z ′3,i

I Waterfilling optimization of the spectrum of X n1

I Relay should not always transmit with maximum power

I In low-SNR: bursty AF improves performance


AF Scaling Capacity

I Optimal scaling as number of relays increases

I Requires coherent combining of relay signals


Cut-Set Bound on Capacity

I Cut: partition of the set of nodes into two sets: (S, S)

I W (S)- set of messages with source in S and sink in SI Choose encoders (inputs): PX .

I Denote as R(PX ,S) set of rates that satisfies:∑

w∈W (S)

Rw ≤ I (XS ;YS |XS) (1)

I Cut-set bound for fixed PX :

R(PX ) =⋂

S

R(PX ,S)

I Cut-set bound:

R =⋃

PX

R(PX )


Cut-Set Bound ExamplesI Point-to-point channel

R =⋃

PX

I (X ;Y )

I Relay Channel

R =⋃

PX1X2

min{I (X1;Y2,Y3|X2), I (X1,X2;Y3)}


Performance Comparison

P1 = P2 = 10, N=1,α = 2

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10

1

2

3

4

5

6

upper boundDF

ρ for DF

CF

relay off

AF

d

Ra

te [b

it/u

se]


Cooperative Strategies: Summary

I DF: when relay is close to source

I CF: when relay is close to destination

I Generalize to multiple relays

I Capacity results are rare


Conferencing and Feedback


In This Section...

I Cooperation via conferencing

I FeedbackI Fundamental results for memoryless channelsI Finite-state channels

I Summary


Conferencing

C 21 C 12

Rx 1

Rx 2

Tx

Y 1 n

Y 2 n

I Conferencing refers to two users interactively helping eachother decode their messages:

I The transmission over the wireless medium is typicallyreceived by users in the vicinity of the target user

I Users have dedicated (orthogonal) links between them, overwhich they communicate


Multi-Step Conferencing

Tx

W 21

(1)

Rx 1

Rx 2

W 12

(1)

Step 1

W 21

(2)

Rx 1

Rx 2

W 12

(2)

Step 2

W 21

(K)

Rx 1

Rx 2

W 12

(K)

Step K

W 2

W 1 ^

^

I A conference can span several cyclesI At each cycle receivers use more refined knowledge on the

other receiver’s channel outputI Decoding takes place after the last cycleI Admissible conference: the total rates of the conference

messages is less than the capacity of the conference links

1

n

K∑

k=1

log2

∣∣W(k)

ij

∣∣ ≤ Cij , (i , j) ∈

{(1, 2), (2, 1)

}


A Conference: Formal Definition

Tx

W 21

(1)

Rx 1

Rx 2

W 12

(1)

Step 1

W 21

(2)

Rx 1

Rx 2

W 12

(2)

Step 2

W 21

(K)

Rx 1

Rx 2

W 12

(K)

Step K

W 2

W 1 ^

^

I An (C12,C21)-admissible K -cycle conference between Rx1

and Rx2 consists ofI K message sets from node i to node j , (i , j) =

{(1, 2), (2, 1)

}

W(k)ij =

{

1, 2, ..., 2nR(k)ij

}

, k = 1, 2, ..., K .

I K pairs of mapping functions

h(k)12 : Yn

1 ×W(1)21 ×W(2)

21 × · · · ×W(k−1)21 7→ W(k)

12

h(k)21 : Yn

2 ×W(1)12 ×W(2)

12 × · · · ×W(k)12 7→ W(k)

21


Full Cooperation

C 21 C 12

Rx 1

Rx 2

Tx

Y 1 n

Y 2 n

I Full cooperation: When each receiver sends his channeloutput to the other receiver

I Y n1 becomes available at Rx2



Full Cooperation

C 21 C 12

Rx 1

Rx 2

Tx

Y 1 n

Y 2 n




I Full cooperation can be achieved with a single cycle ifI C12 = H(Y1|Y2) and C21 = H(Y2|Y1)


Full Cooperation

C 21 C 12

Rx 1

Rx 2

Tx

Y 1 n

Y 2 n





⇒ In one step Rx2 can send to Rx1 enough information thatwill allow Rx1 to recover Y n

2I Using a scheme by Slepian & Wolf’73I Rate I (X ; Y1, Y2) is achievable at Rx1


Full Cooperation

C 21 C 12

Rx 1

Rx 2

Tx

Y 1 n

Y 2 n





⇒ In one step Rx2 can send to Rx1 enough information thatwill allow Rx1 to recover Y n

2I Using a scheme by Slepian & Wolf’73I Rate I (X ; Y1, Y2) is achievable at Rx1

I We will focus on results for partial cooperation


Conferencing: MAC

I The encoders exchange messages prior to transmission

I The capacity region [Willems’83]

R1 ≤ I (X1;Y |X2,U)+C12

R2 ≤ I (X2;Y |X1,U)+C21

R1 + R2 ≤ min{I (X1,X2;Y |U)+C12 + C21, I (X1,X2;Y )

}

for p(u)p(x1|u)p(x2|u)

I This is achieved with a single conference step


Conferencing: Relay Channel

I Compare two schemes (C = Crd + Cdr ):I Single step (classic relaying, Cdr = 0)I Single cycle with

I Step 1: CF from destination to relayI Step 2: DF from relay to destination


Conferencing: Broadcast Channel

1I(X;Y )

2I(X;Y )

2I(X;Y )+C

1I(X;Y )

R2

R1

C12

12

I When the channel is physically degraded, a single conferencestep achieves capacity

x

y 1

p(y 1 |x)

y 2

p(y 2 |y 1 )

I It is enough to let the strong receiver assist the weak receiver


Conferencing: Broadcast Channel

1I(X;Y )

2I(X;Y )

2I(X;Y )+C

1I(X;Y )

R2

R1

C12

12

I When the channel is physically degraded, a single conferencestep achieves capacity

x

y 1

p(y 1 |x)

y 2

p(y 2 |y 1 )

I It is enough to let the strong receiver assist the weak receiver

I For the general BCI It is still an open question whether higher rates can be

achieved with multiple stepsI Can design a K -cycle conference using K − 1 CF cycles and

a final DF step


Feedback

I PtP channel: the receiver sends back information to thetransmitter

I Allows transmitter to adapt its signal to the channel

Xi = fi (W ,Y i−1)

I Network: the wireless medium is a broadcast mediumI Signals received at nodes in the vicinity of the destination are

correlated with the signal at the destination node


Feedback in Multiuser Scenarios

I In Multiuser scenarios feedback facilitates both direct andindirect cooperation

I Direct: Feedback sent from the destination receiverI Indirect: Feedback sent from neighbouring receivers

I Consider for example the BCI Feedback from one receiver can increase the rate to both

receivers


Memoryless Multiuser Scenarios

I Sometimes feedback does not helpI The PtP DMC (p(yn|xn) =

∏n

i=1 p(yi |xi ))I The physically degraded DMBC [El-Gamal’78,81]

x

y 1

p(y 1 |x)

y 2

p(y 2 |y 1 )


Memoryless Multiuser Scenarios

I Sometimes feedback does not helpI The PtP DMC (p(yn|xn) =

∏n

i=1 p(yi |xi ))I The physically degraded DMBC [El-Gamal’78,81]

x

y 1

p(y 1 |x)

y 2

p(y 2 |y 1 )

I Feedback does help in the following scenarios:I The discrete, memoryless MAC [Gaarder & Wolf’75]I The discrete, memoryless relay channel

I Feedback achieves the cut-set bound [Cover & El-Gamal’79]

I The general BC [Ozarow’79]I Including the stochastically degraded channel


Channels with Memory: Finite-State Channels

I The memory for time i is represented by the state Si−1

I The PtP-FSC:





I The PtP-FSC:


I The FS-MAC:

p(yi , si |x1,i , x2,i , xi−11,1 , x i−1

2,1 , y i−1, s i−1, s0) = p(yi , si |x1,i , x2,i , si−1)




I The PtP-FSC:


I The FS-MAC:

p(yi , si |x1,i , x2,i , xi−11,1 , x i−1

2,1 , y i−1, s i−1, s0) = p(yi , si |x1,i , x2,i , si−1)

I The FSBC:

p(yi , zi , si |xi , xi−1, y i−1, z i−1, s i−1, s0) = p(yi , zi , si |xi , si−1)



I Can model effects beyond the physical propagation mediumI Filters, loops

I Example: to incorporate the effect of a K -tap equalizer, thestate Si−1 can also be a function of Y i−1

i−K

I Si−1 = f (Y i−1i−K )





i−K

I Si−1 = f (Y i−1i−K )

I Useful for analyzing correlated fading between the twoextremes of fast (i.i.d.) and slow





i−K

I Si−1 = f (Y i−1i−K )


I Notation: Directed Mutual Information

I (X n → Y n) =

n∑

i=1

I (X i ;Yi |Y i−1)





i−K

I Si−1 = f (Y i−1i−K )


I Notation: Directed Mutual Information

I (X n → Y n) =

n∑

i=1

I (X i ;Yi |Y i−1)︸︷︷︸

H(X i |Y i−1)−H(X i |Y i )


Finite-State Channels: Capacity of the PtP-FSC

I The capacity of a channel with memory is usually given by alimiting expression as the blocklength n → ∞

I We must verify that the limit exists and is finiteI Otherwise the channel does not support reliable

communication in the Shannon sense

I We assume no CSI

I Capacity without feedback [Gallager’68]

C = limn→∞

maxp(xn)

mins0∈S

1

nI (X n;Y n|s0)


Finite-State Channels: Capacity of the PtP-FSC

I The capacity of a channel with memory is usually given by alimiting expression as the blocklength n → ∞

I We must verify that the limit exists and is finiteI Otherwise the channel does not support reliable

communication in the Shannon sense

I We assume no CSI


C = limn→∞

maxp(xn)

mins0∈S

1

nI (X n;Y n|s0)

I Capacity with feedback [Permuter, Weissman &Goldsmith’08]

CFB = limn→∞

max∏ni=1 p(xi |x i−1,y i−1)

mins0∈S

1

nI (X n → Y n|s0)


Remark


C = limn→∞

maxp(xn)

mins0∈S

1

nI (X n;Y n|s0)

I Capacity with feedback

CFB = limn→∞

max∏ni=1 p(xi |x i−1,y i−1)

mins0∈S

1

nI (X n → Y n|s0)

I Feedback increases the capacity of the PtP-FSC [Permuteret al.’08]

I In contrast to the DMC


The Finite-State Broadcast Channel with Feedback and

Cooperation

Encoder W

1

W 2

W 0

Broadcast Channel

p(z,y,s|x,s’)

X i

Z i Decoder 2

C

Y i Decoder 1

Z i-1

W 0

W 0 ^

^ ^

W 2 ^

W 1 ^

D A

Z i-1

Y i-1

B D

D

I 8 possible configurations

I Switch C facilitates full cooperation



Cooperation: Conclusions

I When all switches are openI FSBC without feedback/cooperation





I When the FSBC is physically degraded capacity is achievedusing a superposition codebook with memory






I Feedback can help the physically degraded FSBCI Although it does not help the physically degraded DMBC







I When switch C is closed the channel behaves as a physicallydegraded channel

I Capacity is achieved with a superposition codebook for allfeedback configurations







I When switch C is closed the channel behaves as a physicallydegraded channel

I Capacity is achieved with a superposition codebook for allfeedback configurations

I When switch C is open and feedback comes from one useronly

I Capacity achieved if the channel is physically degraded andthe strong user is sending feedback


Summary

I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node


Summary


I For the MAC a single cycle achieves capacity


Summary


I For the MAC a single cycle achieves capacityI For the physically degraded BC a single step achieves

capacity


Summary



capacityI For the relay channel:

I When C/g is high ⇒ single CF stepI When C/g is low ⇒ single DF stepI For intermediate values of C/g ⇒ iterative decoding


Summary





I Feedback allows the transmitter to adapt its transmissionaccording to the received signal


Summary





I Feedback allows the transmitter to adapt its transmissionaccording to the received signal

I Finite-state channels allow modelling of propagation as wellas implementation aspects


Summary

I In general, feedback is useful in network scenariosI For PtP-DMC feedback does not help


Summary


I When the channel has memory ⇒ feedback is, in general,useful also for PtP links


Summary



I In broadcast scenarios with full cooperationI Capacity achieving schemes can be derived both with and

without feedback


Summary



I In broadcast scenarios with full cooperationI Capacity achieving schemes can be derived both with and

without feedback

I In multiuser scenarios feedback from a neighbouring nodecan help other nodes to communicate


Summary First Half


Summary First Half

I Cooperation is an important tool for coping with the designchallenges of future wireless networks


Summary First Half


I Channel modelsI Discrete, AWGNI Fading: fast, block, slow

I Diversity

I Finite-state: correlated time variations


Summary First Half



I Diversity


I CSI


Summary First Half



I Diversity


I CSI

I Performance metricsI Channel CapacityI Transport capacity

I Scaling laws


Summary First Half

I RelayingI Decode-and-forwardI Compress-and-forwardI Amplify-and-forward


Summary First Half


I ConferencingI Interactive decoding


Summary First Half


I ConferencingI Interactive decoding

I FeedbackI Cooperation between transmitters and receiversI Useful in network scenarios

I Feedback does not have to come from the target receiver


Networks with Multiple Source-Destination Pairs


In This Section...

I Differences when relaying for multiple pairs

I Cooperation in interference channel with a relay

I Cooperation in cognitive radio networks


Relaying

I Relay strategies well developedI decode, compress, amplify -and-forward

I Capture broadcast

I No interferenceI One flow


Relaying for Multiple Sources?

I The smallest network: interference channel with a relay


Relaying for Multiple Sources?

I The smallest network: interference channel with a relay

I Simple approach: multihop routing

I Relay time-shares in helping sources


Multihop


Multihop

I How can we do better?

I No combining of bits, symbols or packets at the relay


Generalized Relaying

I Joint encoding and forwarding of multiple data streams


Network Coding

Butterfly network:

Routing achieves(R1,R2) = (β, 1 − β),for any β ∈ [0, 1]


Network Coding

Butterfly network:

Routing achieves(R1,R2) = (β, 1 − β),for any β ∈ [0, 1]

Network coding: relaycombines packets. Achieves(R1,R2) = (1, 1)


Joint Encoding of Messages

Network Coding idea:

Generalized relaying:


Encoding Elements From...

I Relay channel: generalized amplify, quantize, decode-and-forward

I MAC channel: interference cancellation

I Interference channel: rate-splitting

I Broadcast channel: binning, dirty paper coding

I Many encoding strategies can be applied

I Evaluation is difficult

I Goal: Develop strategies that can be applied to largernetworks and can bring gains


Simple Joint Encoding Strategies: Gaussian Channel

Y3 = h13X1 + h23X2 + Z3

Yj =

3∑

i=1

hijXi + Zj

I Amplify-and-Forward (analog network coding):

X3 = cY3 = c(h13X1 + h23X2 + Z3)

I Decode-and-Forward:

X3 =√

P3(√

cV1(W1) +√

cV2(W2))

I Can outperform time-sharing


DF with Network Coding

Y1 = h31X3 + Z1

Y2 = h32X3 + Z2

Y3 = h13X1 + h23X2 + Z3

I MAC to relay, BC to sources



Y1 = h31X3 + Z1

Y2 = h32X3 + Z2

Y3 = h13X1 + h23X2 + Z3


Relay broadcasts:

I For R1 = R2 : xn3 (w1 ⊕ w2)



Y1 = h31X3 + Z1

Y2 = h32X3 + Z2

Y3 = h13X1 + h23X2 + Z3


Relay broadcasts:

I For R1 = R2 : xn3 (w1 ⊕ w2)

I For R1 ≥ R2 : xn3 (w11,w12 ⊕ w2)

where he splits w1 = (w11,w12) at (R ′1,R2)



Y1 = h31X3 + Z1

Y2 = h32X3 + Z2

Y3 = h13X1 + h23X2 + Z3


Relay broadcasts:

I For R1 = R2 : xn3 (w1 ⊕ w2)

I For R1 ≥ R2 : xn3 (w11,w12 ⊕ w2)


I AF: x3 = a(h13x1 + h23x2 + z3), under power constraint



Y1 = h31X3 + Z1

Y2 = h32X3 + Z2

Y3 = h13X1 + h23X2 + Z3


Relay broadcasts:

I For R1 = R2 : xn3 (w1 ⊕ w2)

I For R1 ≥ R2 : xn3 (w11,w12 ⊕ w2)


I AF: x3 = a(h13x1 + h23x2 + z3), under power constraint

I xn3 (w1,w2)


Differences when Relaying for Multiple Sources


Differences when Relaying for Multiple Sources

I Joint relaying of multiple data streams

I Interference:

I Sources create interference

I Relaying one message increases interference to other users


Interference Channel

I No relay

I Capacity region unknown


Interference Channel

I No relay

I Capacity region unknown

I except...


In Strong Interference

Gaussian channel:

Y1 = X1 + aX2 + Z1

Y2 = bX1 + X2 + Z2

I Cross-link is ’stronger’ than direct: a, b ≥ 1

I Optimal: jointly decode both messages

I Multiaccess channel to each receiver

I Gains from interference cancellation


In General: Rate-Splitting

I If interference not strong: unwanted messages cannot bedecoded

I To reduce interference: partial decoding

I An encoder splits message into two messages

I Decoder decodes one unwanted message and cancelsinterference


Interference Forwarding

I Relay observes signals from both sources

I Relay can use some of its power to forward interference

I Increase interference to cancel it


Special Case Scenario

I No source1-relay link

I Can forwarding interference W2 help both receivers?

I Increases rate R2 but increases interference at destination 1


Encoding

I No rate-splitting nor binning

I Block-Markov, regular encoding

I Decoding: sliding-window or backward


Gaussian Channel

Y1 = X1 + h12X2 + h13X3 + Z1

Y2 = h21X1 + X2 + h23X3 + Z2

Y3 = h32X2 + Z3


No Relaying

I No relay: strong interference regime

I

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R1

R2

Rate Regions of Gaussian Channels

without relay

h12 = 1, h221 = 2, h2

23 = 0.15, h232 = 12


Relaying


I With relay, no interference forwarding

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R1

R2


without relay

with relay, h13

=0

h12 = 1, h221 = 2, h2

23 = 0.15, h232 = 12


Relaying and Interference Forwarding


I With relay, and interference forwarding

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R1

R2


without relay

with relay, h13

=0

with relay, h13

=2

h12 = 1, h221 = 2, h2

23 = 0.15, h232 = 12


Interference Forwarding

Relay can...

I help decoder by interference forwardingI Interference cancelation

I hurt decoder by increasing interferenceI Interference rate becomes too large

Interference forwarding:

I through decode,compress -and-forward

I More general schemes


Virtual (Distributed) MIMO

I No dedicated relay

I Transmitter cooperation

I Transmitters need knowledge about each other’s messages

I Obtained through:

1. Cooperative strategies2. Dedicated orthogonal links; conferencing3. Feedback4. Cognition


Gains From Virtual MIMOI Orthogonal links for cooperation

0 10 20 30 40 500.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Exp

ecte

dsu

mra

tes

(bps

)

Cooperation channel gain G (dB)

CMIMO

CBC, CMAC

RTX-RX

RTX

RRX

CNC


Cognitive Radio Networks

I Motivation: bandwidth gridlock

I Wireless spectrum is crowded

I Licensed band not efficiently used

I Its inefficient use led to spectrum holes

From slides by

B. Brodersen,

BWRC cognitive

radio workshop


Cognitive Radios

I Co-exist with oblivious users without impacting their service

I Sense the environment

I Use the obtained side information to adaptively transmit


Interweave (Opportunistic) Approach

From slides by

B. Brodersen,

BWRC cognitive

radio workshop

I Dynamic spectrum access

I Sense the environment

I Transmit in a spectrum hole


Underlay Approach

I Share the bandwidth; created interference below a threshold


Cognition and Cooperation

I Why not use obtained information for cooperation?

I In cooperation: a helper needs knowledge about relayedmessage

I Assistance of the source nodeI Listening to the channel

I Cognitive node can obtain similar information throughcognition

I Overlay paradigm: share the band and compensate forinterference by cooperation


Overlay Approach

I What is the optimal cognitive strategy?


It All Hinges on...Side Information

I Interweave: users’ activity

I Underlay: channel gains

I Overlay: channel gains, codebooks and (partial) messages


How Can Side Information be Obtained?

I Interweave: users’ activityI Detection of spectrum holesI Holes common to the transmitter and receiver

I Underlay: channel gains

I If there is a channelreciprocity or feedback

I Overlay: channel gains, codebooks and (partial) messagesI Codebooks: through protocolI Messages via: retransmission; cooperation; decoding


Cognitive Radio Channel Model

I Two messages: Wk ∈ {1, . . . ,Mk} sent at rates Rk

I Encoding: X n1 = f1(W1,W2), X n

2 = f2(W2)

I Decoding: Wk = gk(Y nk )


Elements of Cognitive Encoding Strategy

I Opportunistic approach: interference avoidance


Elements of Cognitive Encoding Strategy

I Opportunistic approach: interference avoidance

I Utilize techniques developed from many canonical models

1. Cooperative strategiesTo increase rate at oblivious receiver

2. Rate-splittingTo allow oblivious decoder to cancel part of interference

3. Precoding against interferenceTo eliminate interference at cognitive receiver


Cooperation

I To increase rate for the oblivious receiver

I Cognitive radio acts as a relay

X n1 = f1(W1,W2)

I Dedicates some power to transmit the other user’s message

I Increases interference to its own receiver


Rate-Splitting

I To reduce interference

I Without cognition: interference channel


Precoding against Interference

I To eliminate interference at cognitive receiver

I Full cognition: MIMO broadcast channel

I Strategy: precoding against interference[Gel’fand and Pinsker, 1979]

I Gaussian channels: Dirty-paper coding (DPC) [Costa, 1981]

I Achieves capacity


Capacity Results for Gaussian Channels

Y1 = X1 + aX2 + Z1

Y2 = bX1 + X2 + Z2

inteference

��

��

��

��

1

b1

stronginteference

weak

Wu et.al.

a I Regions for which capacity is known:

I Strong interference, a > b > 1Cooperation achieves capacity

I Weak interference, b ≤ 1Dirty paper coding and cooperationachieve capacity


Insights

1. Orthogonalizing transmissions is suboptimal

2. Canceling strong interference is beneficial

3. Rate-splitting can be used for partially canceling interference


Insights

I Side information in cognitive radio networks can be used for:

I Cooperation

I Precoding against interference

I In considered network: cooperation and GP precodingcapacity-achieving in some regimes

I Delay should be considered


Relaying for Multiple Sources

I Jointly encode messages

I Exploit broadcast

I Relays forward messages and interference

I Create virtual MIMO

joint

jointencodingexploit

broadcast

interferenceforwarding

encoding

f(W1, W3)

W1

W2

W3


Cooperation in Fading Channels


In This Section...

I Examples

I Diversity-multiplexing tradeoff for the PtP MIMO channel

I DMT for cooperative systems

I Summary


Fading Channels

I Rayleigh fading: Many scatterers, no LOSI Communication in dense urban areas

I Rician Fading: Many scatterers with LOSI Satellite communications

0 0.2 0.4 0.6 0.8 1−14

−12

−10

−8

−6

−4

−2

0

2

Time, seconds

Rel

ativ

e P

ower

, dB

, Rel

ativ

e to

RM

S

K = 5


Channel Model

Consider a point-to-point (PtP) MIMO channel:

H sd

Source (m) Destination (n)

I m transmit antennas

I n receive antennas

I Consider codes with blocklength l


Channel Model

H sd


I Block-fading model: H is constant for the entire block oflength l .

I

Y =

√

SNR

mHX + Z

X ∈ Cm×l , H ∈ Cn×m, Y ∈ Cn×l , Z ∈ Cn×l

I hi ,j ∼ CN (0, 1), i.i.d.; zi ,j ∼ CN (0, 1), i.i.d.

I Power constraint: 2−Rl∑2Rl

i=1 ||X(i)||2F ≤ ml


MIMO Systems in Fading Environments: Diversity Gain

I Diversity: Sending the same information through severalpaths.

I Each path is subject to independent fading ⇒ increase thereliability of reception.

I m = 1 ⇒ Pe(SNR).= SNR−n

I Definition: Diversity gain d

d = − limSNR→∞

log Pe(SNR)

log SNR

I For i.i.d. Rayleigh fading the maximal diversity gain is mn

I Probability of error dominated by the outage event


MIMO Systems in Fading Environments: Multiplexing Gain

I Spatial Multiplexing: Sending independent information overparallel spatial channels.

I Each Tx-Rx pair is fading independently thus creating aparallel channel.

I At high SNR, under i.i.d. Rayleigh fading assumption, the(ergodic) channel capacity is given by

C (SNR) = min{m, n} log SNR + O(1)

I Definition: Multiplexing gain r

r = limSNR→∞

R(SNR)

log SNR

I For i.i.d. Rayleigh fading the maximal multiplexing gain ismin{m, n}


Example: Diversity-Multiplexing Relationship for

Alamouti’s Scheme

I 2 × 2 system

I R = r log SNR

I X =

[x1 −x∗

2

x2 x∗1

]

I Y =√

SNR2 HX + Z

I Using Alamouti’s scheme we arrive at the equivalent channel

yi =

√

SNR||H||2F2

xi + wi

I Pout.= Pr

(

||H||2F ≤ SNR−(1−r)+)

I d(r) = 4(1 − r)+


Remarks

I The blocklength l is fixed

I Analysis for SNR goes to infinity

I A scheme S is a collection of codes {C(SNR)}, one for eachSNR

I Rate is R(SNR)

I Both data rate and error probability scale with SNR

I Each scheme is characterized by the two parameters (d , r)

I Define for a fixed r the function d∗(r) as

d∗(r) , supS with the same r

d


Diversity Gain vs. Multiplexing Gain - The Fundamental

Result

I Characterized by Zheng and Tse in 2003.

Theorem (Zheng & Tse’03)

When l ≥ m + n − 1, the optimal tradeoff curve d∗(r) is thepiecewise linear function connecting the points (k, d∗(k)), k =0, 1, ...,min{m, n}, where

d∗(k) = (m − k)(n − k).

I d∗max = mn

I r∗max = min{m, n}


Diversity Gain vs. Multiplexing Gain - The Fundamental

Result

I Figure from Zheng & Tse’03.


Cooperative Networks

I Cooperation can be used to create a virtual MIMO:I Several single antenna nodes cooperate in sending/receiving

informationI CSI assumptions

I Which cooperation strategy is DMT superior?


Half Duplex Relay Channel

Relay (1)

Source (1) Destination (1)

H sr

H sd

H rd

I MISO upper bound: d∗(r) = 2(1 − r)

I Orthogonal DF: d∗(r) = 2 − 4r

I Dynamic DF

d∗(r) =

{2(1 − r), , 0 ≤ r ≤ 0.51−rr

, 0.5 ≤ r ≤ 1

I Non-orthogonal AF: d∗(r) = (1 − r) + (1 − 2r)+

I CF with relay CSIT: d∗(r) = 2(1 − r)


Half Duplex Relay Channel

I Figure from Kramer, Maric & Yates’06.


Full Duplex MIMO Relay Channel: The Optimal DMTRelay (k)


H sr

H sd

H rd

Yr = HsrXs + Zr

Yd = HsdXs + HrdXr + Zd

Theorem (Yuksel & Erkip’07)

The optimal DMT is equal to

d∗(r) = min{dm(n+k)(r), d(m+k)n(r)}.

I The best DMT is achieved by the CF schemeI d∗

max = min{m(n + k), (m + k)n}I r∗max = min{min{m, (n + k)},min{n, (m + k)}}

= min{m, n}Ivana Maric and Ron Dabora Cooperation in Wireless Networks 142

Full Duplex MIMO Relay Channel: DMT Analysis of DFRelay (k)


H sr

H sd

H rd

Theorem (Yuksel & Erkip’07)

The DMT achieved by DF is given by

d∗DF (r) =

min{d(m+k)n(r),

dmn(r) + dmk(r)}, 0 ≤ r ≤ min{m, n, k}dmn(r) ,min{m, n, k} ≤ r ≤ min{m, n}

I If m = 1 or n = 1, DF is optimalI if k < min{m, n} the relay cannot help at high r

I Relay cannot decode if the multiplexing gain is too highI Additional outage event due to decoding at the relay


Multiple Relays: Non-Clustered

I Single antenna nodes

I Optimal DMT:d∗(r) = d13(r)

I DMT is achieved using DF at both relays


Multiple Relays: Clustered

I Single antenna nodes

I Clustered nodes: the channel is AWGN (no fading)

I Optimal DMT:

d∗(r) =

{d22(r) , r ≤ 1

0 , r > 1

I DMT is achieved using DF at Relay 1 and CF at Relay 2


Clustered vs. Non-Clustered

I Figure from Yuksel & Erkip’07.


DMT of Multi-hop Relaying

I Full duplex: DMT achieved by DF

d∗FD(r) = min

{dmk(r), dkn(r)

}




d∗FD(r) = min

{dmk(r), dkn(r)

}

I Half duplex with fixed time allocation (α, 1 − α)

d∗HD-fixed(r) = min

{

dmk

( r

α

)

, dkm

(r

1 − α

)}




d∗FD(r) = min

{dmk(r), dkn(r)

}

I Half duplex with fixed time allocation (α, 1 − α)

d∗HD-fixed(r) = min

{

dmk

( r

α

)

, dkm

(r

1 − α

)}

I Half duplex DDF (m, k, n) = (2, 2, 2)

d∗HD(r) =

8−10r2−r

, r ∈ [0, 1/2)3−4r1−r

, r ∈ (1/2, 2/3)

41−r2−r

, r ∈ (2/3, 1]


Degrees of Freedom of Cooperative Networks

I The term degrees of freedom is sometimes used instead ofmultiplexing gain.

I The capacity at high SNR:

C (SNR) = min{m, n} logSNR

m+

max{m,n}∑

i=|m−n|+1

E{

log χ22i

}+ o(1)

I For a MIMO channel, when the matrix is full rank, weachieve the maximal degrees of freedom.

I For Rayleigh fading with CSIR the MIMO channel matrix isfull rank

I Cooperation does not increase the DOF


Degrees of Freedom with Cognitive Cooperation

I When cooperation is based on a cognitive relay node theneach pair can achieve DOF 1

I The sum-rate DOF is 2

⇒ A cognitive relay increases the DOF of the system

I Achieved by instantaneous interference cancellation


Summary

I DMT is a performance metric for fading channelsI Asymptotically high SNR

I Outage is the dominating error event

I Finite blocklength

I Soft information (CF) is important for achieving themaximum DMT

I With a single relay:I CF achieves the optimal DMTI With full-duplex single antenna nodes: DF also achieves the

optimal DMT

I Clustering improves diversity but does not improve DOFI Source cluster ⇒ use DFI Destination cluster ⇒ use CF


Scaling Laws


Large Network Analysis

I Initiated by Gupta and Kumar




I n source-destinations: n-dimensional capacity region





I We cannot solve even for n=2!






I Assumption:

I Each pair wants to communicate at rate: R(n) bits/s

I Total throughput: T (n) = nR(n)






I Assumption:

I Each pair wants to communicate at rate: R(n) bits/s

I Total throughput: T (n) = nR(n)

I Max achievable scaling?


Scaling in Multihop Networks

Upper bound: T (n) ≤ O(√

n)




n)

I Rate for each pair goes to zero




n)


I Nearest neighbor communication and spatial reuse optimal

I Number of retransmissions increases




n)




I System is interference limited




n)





Communication scheme:

I Multihop routing

I Treat all unwanted signals as noise

I Achievability proved using percolation theory




n)





Communication scheme:

I Multihop routing

I Treat all unwanted signals as noise

I Achievability proved using percolation theory

I Can cooperative encoding schemes change the scaling?


Is Multihop Optimal?

I Dense vs. extended networks

Extended Networks:

For α > 4: multihop is order-optimal

I α-path-loss exponent

I Attenuation is too large for cooperation to help




Extended Networks:




For α ≤ 4?




Extended Networks:




For α ≤ 4?

I 2 ≤ α ≤ 3: n2−α/2

α > 3: n1/2

I For α = 2: linear scaling

I For α > 3: multihop is optimal




Extended Networks:




For α ≤ 4?

I 2 ≤ α ≤ 3: n2−α/2

α > 3: n1/2

I For α = 2: linear scaling

I For α > 3: multihop is optimal

I Dense networks, α ≥ 2: T (n) = O(n1−ε) is achievable

I Arbitrarily close to linear scaling


Motivation for Cooperative Strategy

I Nearest neighbor communications are not enough

I Linear scaling: in MIMO system

I With n transmit and receive antennas, in high-SNR:n log(SNR)

I We already saw gains from clustering and mimicking MIMO

I This requires cooperation within clusters


Three-Phase Cooperative Scheme

I Form M-node clusters

I Sources in clustercooperate

I MIMO long-rangetransmissions

I Destinations in clustercooperate


Within Each Stage

I Transmit cluster: information bits exchanged betweensources

I Receive cluster: quantized observations exchanged betweendestinations

I Spatial reuse: non-adjacent clusters send simultaneously

I TDMA long-range communications between clusters


Hierarchial Cooperation

I Perform multiple stages of the three-phase scheme

I Each stage improves throughput

I After h stages:T (n) = O(nh/(h+1))

I Choose h s.t.h

h + 1≥ 1 − ε

to obtainT (n) = O(n1−ε)


Summary

I Lots of progress

I Different behavior for dense vs. extended networks

I Cooperation can change scaling

I For extended networks:

I For α ≥ 3:√

n, multihop is optimal

I For 2 ≤ α ≤ 3: n2−α/2

I Linear scaling only for α = 2

I Dense networks: capacity scales linearly

I Design of practical systems requires more detailed analysis


Summary and Challenges


Summary

Relaying for Multiple Sources

I Gains from simple joint encoding strategies

I Canceling strong interference is beneficial

I Relays forward messages and interference

I Gains from virtual MIMO

I Gains from cognitive encoding techniques

Cooperation brings diversity-multiplexing gains

I Compress-and-forward achieves the optimal DMT in singlerelay channel

Cooperative communications can change capacity scaling


Conventional Network Architecture

I Network protocol layers

I Store-and-forward routing via a sequence of links

I Point-to-point transmissions on the path

I Network layer: decides on the next node, modifies the header

I PHY/Link layer: discards a packet in error

T3

R1

R3

R2T2

T1


Network Architecture: Cooperative Protocols

Exploit broadcast

I Nodes collect erroneous packets

I A link is not necessarily point-to-point

Allow for encoding at the nodes

I Relaying, joint encoding of messages, network coding



I Decoder: soft combining of packets

I Protocol: provide for relaying and routing





I For example:

I Routing on the network layer





I For example:

I Routing on the network layer

I Sequence of cooperative links


Demonstrated Gains from Cooperative Communications

For small networks

I Rates in relay channel

I Diversity-multiplexing gains

I Rate regions for multiple sources

For large networks

I Scaling law O(√

n) → O(n)

−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10

1

2

3

4

5

6

upper boundDF

ρ for DF

CF

relay off

AF

d

Rat

e [b

it/u

se]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

multiplexing gain r

div

ers

ity g

ain

d dynamic DF

CF andMISO upper bound

nonorthogonalAF

orthogonal AF andorthogonal DF

no cooperation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R1

R2


without relay

with relay, h13

=0

with relay, h13

=2


Challenges

I Capacity results for canonical models

I Many encoding possibilities at the relays

I Practical cooperative schemes

I Cooperative protocols

joint

jointencodingexploit

broadcast

interferenceforwarding

encoding

f(W1, W3)

W1

W2

W3


Bibliography

I T. Cover and A. El Gamal, ”Capacity Theorems for the RelayChannel” IEEE Trans. Inf. Th., vol. 25, no. 5, Sept. 1979.

I A. B. Carleial, ”Multiple- Access Channels with DifferentGeneralized Feedback Signals” IEEE Trans. Inf. Th., vol. 28, no.6, Nov. 1982.

I F.M.J. Willems, ”Informationtheoretical Results for the DiscreteMemoryless Multiple Access Channels”. Ph.D. dissertation, Oct1982.

I G. Kramer. M. Gastpar and P. Gupta, ”Cooperative Strategiesand Capacity Theorems for Relay Networks” IEEE Trans. Inf. Th.,vol. 51, no. 9, Sept. 2005.

I G. Kramer, I. Maric and R. Yates, ”CooperativeCommunications,” , Foundations and Trends in Networking,Hanover, MA: NOW Publishers Inc., vol. 1, no. 3-4, 2006


Bibliography

I L. Zheng and D. Tse, ”Diversity and Multiplexing: A FundamentalTradeoff in Multiple Antenna Channels” IEEE Trans. Inf. Th.,Vol. 49(5), May 2003.

I M. Yuksel and E. Erkip. Multi-antenna Cooperative WirelessSystems: A Diversity-multiplexing Tradeoff Perspective, IEEETrans. Inf. Th., Special Issue on Models, Theory, and Codes forRelaying and Cooperation in Communication Networks, vol. 53,no. 10, pp. 3371-3393, October 2007.

I F. Xue and P.R. Kumar, ”Scaling Laws for Ad Hoc WirelessNetworks: An Information Theoretic Approach” NOW Publishers,2006, and references therein.

I A. Ozgur, O. Leveque and D. Tse, ”Hierarchical Cooperationachieves Optimal Capacity Scaling in Ad Hoc Networks” IEEETrans. Inf. Th., vol. 53, no. 10, Oct. 2007.


Cooperation in Wireless Networks - Wireless Systems...

Documents

Transcript of Cooperation in Wireless Networks - Wireless Systems...