Cours Abdellatif ZAIDI

Cooperative Techniques in Networks

Abdellatif Zaidi

Universite Paris-Est Marne La Vallee, France

[email protected]

Spring School on 5G CommunicationsHammamet, March 2014

Abdellatif Zaidi Cooperative Techniques in Networks 1 / 90

Tutorial Goals

Review and discuss some important aspects of cooperative communications innetworks

Provide intuition for basic concepts

Connect with recent results

Encourage research activity

Contribute to networking practice


Recurring Themes and Take-Aways

1 Many approaches, benefits, and challenges to utilizing relaying and cooperativecommunications in networks.

2 Relaying also includes, or should also include, multihop (store-and-forward routing)and network coding.

3 The capacity of relay systems is difficult to analyze, but the theory is surprisinglyflexible and diverse.

4 Although generalizations to networks with many nodes are in general not easy,there are schemes which scale appropriately.

5 In networks with interference, relaying can help, not only by adding power/energyspatially, but also by allowing distributed interference cancellation; and in generalthis boots network capacity.

6 To cope with network interference efficiently, classic relaying schemes as such ingeneral do not suffice, and need be combined carefully with other appropriatetechniques.


Caveats

1 I assume familiarity with: entropy, mutual information, capacity of discretememoryless and additive white Gaussian noise channels, rate-distortion theory

2 References are not provided in the main slides; only selected references are given atthe end. These, along with the references therein, point to many other recentpapers on relaying and cooperative communications in networks.

3 Time constraints limit the scope to less than originally planned.

4 I may go fast. Please feel free to stop me, whenever you want to.


Tutorial Outline

Part 1: Basics on Cooperation

Part 2: Cooperation in Presence of Interference

Part 3: Interaction and Computation


Basics on Cooperation

�� Part I : Basics on Cooperation


Basics on Cooperation

Outline: Part 1

1 Introduction and Models

2 Protocols

Amplify-and-Forward (AF)Decode-and-Forward (DF)Compress-and-Forward (CF)

3 Information Rates


Basics on Cooperation Introduction and Models

Wireless Network

A communication network has devices andchannels

Network purpose: enable message exchangebetween nodes

Main features of wireless networks:

- Fading: electromagnetic scattering,

absorption, node mobility, humidity

- Broadcasting: nodes overhear wireless

transmissions (creates interference)

W1 → (X1,Y1)

WN → (XN,YN)

p(y1, . . . ,yN|x1, . . . ,xN)

Wk → (Xk,Yk)



Fast and Slow Fading

Space-time models for huv:

- Deterministic: electromagnetic wave propagation equations

- Random: {huv,i}ni=1 is a realization of an integer-time stochastic

process {Huv,i}ni=1

(The random model admits uncertainty and is simpler).

Marginal distributions of the Huv,i:

- Assume the Huv,i, i = 1, . . . , n, have the same marginal distribution

Huv during a given communication session

- No fading: Huv is a known constant

- Rayleigh fading: Huv is complex, Gaussian, 0-mean, unit var.

Temporal correlation: two extremes

- Fast fading: huv,i are independent realizations of Huv

- Slow fading: Huv,i = Huv for all i



Discrete Memoryless Network Model

There are M source messages Wm,m = 1, 2, . . . ,M .

Message Wm is estimated as message Wm(u)at certain nodes u

W1 → (X1,Y1)

WN → (XN,YN)

p(y1, . . . ,yN|x1, . . . ,xN)

Wk → (Xk,Yk)

Source model:

- Messages are statistically independent

- Sources are not bursty

Device model:

- Node u has one input variable Xu and one output variable Yu

- Causality: Xu,i = fu,i(local messages, Y i−1u ), i = 1, . . . , n

- Cost constraint example: E[f(Xu,1 . . . , Xu,n)] ≤ Pu



Discrete Memoryless Network Model (Cont.)

W1 → (X1,Y1)

WN → (XN,YN)

p(y1, . . . ,yN|x1, . . . ,xN)

Wk → (Xk,Yk)

Channel model:

- A network clock governs operations: node u transmits Xu,i betweenclock tick (i− 1) and tick i, and receives Yu,i at tick i

- Memoryless: Yu,i generated by Xu,i, all u, via the channel

PY1Y2Y3...|X1X2X3...(·)

Capacity region: closure of set of all (R1, R2, . . . , RM ) for which

Pr[∪m,u

{Wm 6= Wm

}]can be made close to zero for large n



Node Coding/Broadcasting

Traditional approach:

- Channels treated as point-to-point links

- Data packets traverse paths (sequences of nodes)

Other possibilities:

Broadcasting: nodes overhear wireless transmissionsNode coding: nodes process

- reliable message or packet bits (network coding)- reliable or unreliable symbols (relaying/cooperation)

These concepts already appear in 3-node networks


Basics on Cooperation Protocols

Building Block

1

2

3 WW

Direct transmission from Node 1 to Node 3

Multihop transmission through Node 2

Amplify-and-Forward (AF)Decode-and-Forward (DF)Compress-and-Forward (CF)

Capacity still unknown

Multiaccess problem: from Nodes 1 and 2 to Node 3Broadcast problem: from Node 1 to Nodes 2 and 3Feedback problem: output at Node 2 is a form of feedback

Superposition coding, binning, feedback techniques, etc.



Direct Transmission

Encoder Decoder WXn

1 Y n3

Xn2 = bn

PY3|X1,X2(y3|x1, x2)W

Relay does not participate, i.e., X2,i = b (often 0) for all i

A standard random coding argument shows that rates

R < Rdir := maxPX1|X2

(·|b),bI(X1;Y3|X2 = b)

are achievable

Rdir is in fact capacity if the relay channel is reversely degraded, i.e.,

PY2Y3|X1X2(·) = PY3|X1X2

(·)PY2|Y3(·)

(Think: Y2 is a noisy version of Y3)



Multihop Transmission

Node 1 transmits to Node 2, and Node 2 transmits to Node 3

Motivated by a cascade of two channels, i.e.,

PY n2 Y

n3 |X

n1 X

n2

(·) = PY n2 |X

n1

(·)PY n3 |X

n2

(·)

Question: What should Node 2 transmit ?

Non-Regenerative / Amplify-and-Forward (AF)Node 2 sets X2,i = βY2,i−k for some k ≥ 1Compress-and-Forward (CF) / Estimate-Forward (EF)Node 2 conveys a quantized version of Y n2 to Node 3Regenerative / Decode-and-Forward (DF)Node 2 decodes W and re-encodes into a codeword Xn

2

Transmissions usually occur in a pipeline of two (or more) blocks (potentially ofvarying sizes)



Block Pipeline

Relay

Destination

Source

· · ·

· · ·

· · ·

Block BBlock B-1Block 2

· · ·

R R

y2[B − 1]

Block 1

x1(w1)

y2[1] y2[2] y2[B]

y3[1] y3[2] y3[B]y3[B − 1]

R 0

a

b

x1(w2) x1(wB−1)

f(B−1)2 ({y2[i]}B−2

i=1 )f(2)2 (y2[1]) f

(B)2 ({y2[i]}B−1

i=1 )

B blocks, each of length n channel uses. Message W = (w1, w2, . . . , wB)

Block 1 fills the pipeline; and block B empties the pipelineBlocks 2 through B can blend broadcast and multiaccess

Extremes

n = 1, B large: Like an intersymbol interference (ISI) channeln large, B = 2: No interblock interference, half-duplex

n,B large: Effective rate (B−1)nRnB = (B−1)

B R→ R

Memory within and among input blocks allowed through f(i)2

({y2[j]

}i−1

j=1

)Abdellatif Zaidi Cooperative Techniques in Networks 16 / 90

Basics on Cooperation Protocols : Amplify-and-Forward (AF)

Amplify-and-Forward (AF)

Relay

Destination

Source

· · ·

· · ·

· · ·


· · ·

R R

y2[B − 1]

Block 1

x1(w1)

y2[1] y2[2] y2[B]

y3[1] y3[2] y3[B]y3[B − 1]

R 0

a

b

x1(w2) x1(wB−1)

y2[1] y2[B − 1]y2[B − 2]

Choose f(i)2 (·) to be a linear function:

Discrete Channels

Often x2[i] , f(i)2

({y2[j]

}i−1

j=1

):= y2[i− 1]

Requires Y2 ⊆ X2

Continuous Channels

Often x2[i] , f(i)2

({y2[j]

}i−1

j=1

):= βy2[i− 1]

β chosen subject to a power constraint



Multihop AF, B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

y2[1]

y3[1] y3[2]

a

b

x1(w)

If Node 3 decodes using only Block 2, R achievable if

R < Rmaf := maxPX1,1|X2,1

(·|b),a,b

1

2I(X1,1;Y3,2|X2,1 = b,X1,2 = a)

I(X1,1;Y3,2|X2,1 = b,X1,2 = a) computed for the effective channelPY3,2|X1,1X1,2X2,1

(·|·, a, b) in which X2,2 = Y2,1

Also known as ”non-regenerative repeating”Abdellatif Zaidi Cooperative Techniques in Networks 18 / 90


Diversity AF, B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

y2[1]

y3[1] y3[2]

a

b

x1(w)

If Node 3 decodes using both Blocks 1 and 2, then R achievable for

R < Rdaf := maxPX1,1|X2,1

(·|b),a,b

1

2I(X1,1;Y3,1Y3,2|X2,1 = b,X1,2 = a)

Similar to repetition coding, except X2,2 = Y2,1 is a corrupted version of X1,1



Non-Orthogonal AF (NAF), B = 2

Relay

Destination

Source

Block 2

R R

Block 1

y2[1] y2[2]

y2[1]

y3[1] y3[2]

b

x1(w1) x1(w2)

If Node 1 sends new information in Block 2, and Node 3 decodes using bothBlocks 1 and 2, then R achievable for

R < Rndaf := max1

2I(X1,1X1,2;Y3,1Y3,2|X2,1 = b)

with max over PX1,1|X2,1(·|b)PX1,2|X2,1

(·|b) and b

A combination of DAF and Direct from Node 1 to Node 3



Intersymbol Interference AF (IAF)

Pipeline with n = 1, B →∞ creates an effective intersymbol interference (ISI)channel

Input memory important in this case (”waterfilling”)

Additional improvements with ”bursty AF”, mainly at low SNR



AF Summary

Schemes discussed so far are all special cases of Block Pipeline AF.

GenerallyRmaf < Rdaf < Rndaf < Riaf

but coding and decoding grows increasingly complex

MAF, DAF, and NDAF with B = 2 are useful for half-duplex systems

IAF with B →∞ is useful for full-duplex systems


Basics on Cooperation Protocols : Decode-and-Forward (DF)

Decode-and-Forward (DF)

Relay

Destination

Source

· · ·

· · ·

· · ·


· · ·x1(w1, w2)

R R

y2[B − 1]

Block 1

x1(1, w1) x1(wB−1, 1)

y2[1] y2[2] y2[B]

x2(1) x2(w1) x2(wB−2) x2(wB−1)

y3[1] y3[2] y3[B]y3[B − 1]

x1(wB−2, wB−1)

R 0

f(i)2 (·): in block i, Relay uses y2[i− 1] to decode message wi−1, and re-encodes it

into x2(wi−1)

Joint typicality decoding: look at wi−1 s.t. x1(wi−2, wi−1), y2[i− 1] is jointlytypical given x2(wi−2)

Multi-user codebooks designed jointly

x1[i] := x1(wi−1, wi) and x2[i] := x2(wi−1) both depend upon wi−1

x1[i] := x1(wi−1, wi) also depends upon wiJoint distributions PX2

(·)PX1|X2(·)



Multihop DF, B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

x2(w)

y3[1] y3[2]

a

b

x1(w)

Let q be the fractional length of Block 1 (Block 2 of fractional length q := 1− q)

At the end of Block 1, Node 2 decodes message w reliably if

R < q×I(X1;Y2|X2 = b)

and w = w with high probability

If Node 3 decodes using only Block 2, R achievable if

R < q×I(X2;Y3|X1 = a)



Multihop DF (MDF), B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

x2(w)

y3[1] y3[2]

a

b

x1(w)

Thus, a rate R is achievable if

R < Rmdf := max0≤q≤1

min{q× maxPX1|X2

(·|b),bI(X1;Y2|X2 = b),

= q× maxPX2|X1

(·|a),aI(X2;Y3|X1 = a)}

Routing: Time-sharing between Direct from Node 1 to Node 2 and Direct fromNode 2 to Node 3

Also known as ”regenerative repeating”



Diversity DF (DDF), B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

x2(w)

y3[1] y3[2]

a

b

x1(w)

If Node 3 decodes using both Blocks 1 and 2, then R achievable for

R < Rddf := max min{q×I(X1,1;Y2,1|X2,1 = b),

q×I(X1,1;Y3,1|X2,1 = b)

+ q×I(X2,2;Y3,2|X1,2 = a)}with max over PX1,1|X2,1

(·|b)PX2,2|X1,2(·|a), a, b, and 0 ≤ q ≤ 1



Non-Orthogonal DDF (NDDF), B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

b x2(w1)

y3[1] y3[2]

x1(w1) x1(w2)

If Node 1 sends new information in Block 2, and Node 3 decodes using bothBlocks 1 and 2, then R achievable for

R < Rnddf,2 := max min{q×I(X1,1;Y2,1|X2,1 = b),

q×I(X1,1;Y3,1|X2,1 = b)

+ q×I(X1,2X2,2;Y3,2)}with max over PX1,1|X2,1

(·|b)PX1,2X2,2(·), a, b, and 0 ≤ q ≤ 1Abdellatif Zaidi Cooperative Techniques in Networks 27 / 90


Non-Orthogonal DDF (NDDF), B →∞

Relay

Destination

Source

· · ·

· · ·

· · ·


· · ·x1(w1, w2)

R R

y2[B − 1]

Block 1

x1(1, w1) x1(wB−1, 1)

y2[1] y2[2] y2[B]

x2(1) x2(w1) x2(wB−2) x2(wB−1)

y3[1] y3[2] y3[B]y3[B − 1]

x1(wB−2, wB−1)

R 0

All blocks of length n

Three encoding and decoding algorithms: differ in complexity and delay

requirements.

Regular enc., sliding-window dec.: R < I(X2;Y3) + I(X1;Y3|X2)Regular enc., backward dec.: R < I(X1, X2;Y3)Irregular enc., successive dec.: binning at encoding (Cover, El Gamal)



Non-Orthogonal DDF (NDDF), B →∞

Relay

Destination

Source

· · ·

· · ·

· · ·


· · ·x1(w1, w2)

R R

y2[B − 1]

Block 1

x1(1, w1) x1(wB−1, 1)

y2[1] y2[2] y2[B]

x2(1) x2(w1) x2(wB−2) x2(wB−1)

y3[1] y3[2] y3[B]y3[B − 1]

x1(wB−2, wB−1)

R 0

All blocks of length n

Rate R is achievable if

R < Rnddf,∞ := max min{I(X1;Y2|X2), I(X1X2;Y3)}

with max over PX1X2(·)Rnddf,∞ is in fact capacity if relay channel is physically degraded, i.e,

PY2Y3|X1X2(·) = PY2|X1X2

(·)PY3|Y2X2(·)



DF Summary

Schemes discussed so far are all special cases of Block Pipeline DF.

GenerallyRmdf < Rddf < Rnddf,2 < Rnddf,∞

but coding and decoding grows increasingly complex

MDF, DDF, and NDDF with B = 2 are useful for half-duplex systems

NDDF with B →∞ is useful for full-duplex systems


Basics on Cooperation Protocols : Compress-and-Forward (CF)

Multihop CF, B = 2

Relay

Destination

Source

Block 2

R 0

Block 1

y2[1] y2[2]

x2(s1)

y3[1] y3[2]

a

b

x1(w)

s1 is s.t. y2[s1] := y2[1]

Basic Idea: in block i, relay quantizes (scalar or vector) y2[i− 1] andcommunicates it to the destination.

Details

Fix distributions PX1|X2(·|b) and PY2|Y2X2

, to be optimized later

Generate d2nRe quantization codewords y2[s] independently and i.i.d.

according to the marginal PY2|X2(y2|b), s = 1, 2, . . . , 2nR.



Multihop CF

Details (cont.)

Upon receiving y2[1], relay quantizes it by finding a joint typical y2[1] in thequantization codebook. This is likely for n large if R > I(Y2; Y2|X2 = b)

The destination first utilizes y3[2] to decode the compression index s1. This canbe done with no error if

R < maxPX2|X1

(·|a),aI(X2;Y3|X1 = a)

Then, the destination utilizes y2[s1] := y2[1] to decode message w. This can bedone with no error if

R < Rmcf := max I(X1; Y2|X2 = b)

with max over PX1|X2(·|b), PY2|Y2X2

(·|·, b), a and b such that

I(Y2; Y2|X2 = b) < maxPX2|X1(·|a) I(X2;Y3|X1 = a)



CF Wyner-Ziv Compression

Y n3Xn

1 WW

Y n2 : Xn

2

Y n2

Send B − 1 independent messages over B blocks (each of length n)

At the end of block i, relay chooses a description yn2 [i] of yn2 [i]

Since the receiver has side information yn3 [i] about yn2 [i], we use Wyner-Ziv codingto reduce rate necessary to send yn2 [i]

R > I(Y2; Y2|X2, Y3)

= I(Y2; Y2|X2)− I(Y3; Y2|X2)

The bin index is sent to the receiver in block i+ 1 via xn2 [i+ 1]



CF Wyner-Ziv Compression (Cont.)

Y n3Xn

1 WW

Y n2 : Xn

2

Y n2

At the end of block i+ 1, the receiver first decodes xn2 [i+ 1] from which it findsyn2 [i]

R < I(X2;Y3)

It then finds unique wi s.t. xn1 (wi), xn2 [i], yn2 [i], yn3 [i] are jointly typical

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

Compress-Forward rate

RCF = maxPX1

PX2PY2|Y2,X2

I(X1; Y2, Y3|X2)

subject to I(Y2; Y2|X2, Y3) ≤ I(X2;Y3)



Summary

What we covered in this section:

Summarized basic elements of relay channels, including direct, multihop,broadcast, and multiaccess transmission

Introduced mechanics of various kinds of relay processing, includingamplify-and-forward, decode-and-forward, and compress-and-forward


Basics on Cooperation Information Rates

Information Rates

The purpose of this section is to refine the above analysis, study numericalexamples, and develop insight based on rate.

Sc

S

The capacity region C is usually difficult to compute. A useful outer bound on C isthe cut-set bound.

Let S ⊆ N and let Sc be the complement of S in N. A cut separating Wm fromone of its estimates Wm(u) is a pair (S, Sc) where Wm is connected (immediately)to a node in S but not in Sc, and where u ∈ Sc.


Basics on Cooperation Information Rates : Cut-set Bound

Cut-Set Bound

Let XS = {Xu : u ∈ S}Consider any choice of encoders, and compute

PXNYN(a, b) =

[ 1

n

n∑i=1

PXN,i(a)]PYN|XN

(b|a)

where PXN,i(·) is the marginal input distribution at time i.

Let M(S) be the set of messages separated from one of their sinks by the cut(S, Sc).

Cut bound: any (R1, R2, . . . , RM ) ∈ C satisfies∑m∈M(S)

Rm ≤ I(XS;YSc |XSc)

Cut-set bound for fixed PXN(·): intersection over all S of (R1, . . . , RM ) satisfying

the above bounds

Cut-set bound: union over PXN(·) of all such regions


Basics on Cooperation Information Rates : Cut-set Bound

Cut-Set Bound Examples

Point-to-point channel:C ⊆ ∪PX (·)I(X;Y )

Relay channel:

C ⊆ ∪PX1X2(·) min{I(X1;Y2Y3|X2), I(X1X2;Y3)}

The cut-set bound is usually loose, e.g., for two-way channels, broadcast channels,relay channels, etc.

Multiple access Broadcast

X1Y3

X2

X1

Y2 : X2

Y3


Basics on Cooperation Information Rates : Wireless Geometry

Wireless Geometry

Relay is a full-duplex device

Powers and Noise

E[X2u] ≤ Pu, u = 1, 2

Zi, i = 2, 3, ind. Gaussian

E[Z2i ] = N , i = 2, 3

Source and Dest. kept fixed. Relaymoves on the circle. The model is:

Y2 =H12

|d|α/2X1 + Z2

Y3 = H13X1 +H23

√1− d2

α/2X2 + Z3

1

d

√1− d2

2

1 3

To compare rates, we will consider two settings:

No Fading: Huv is a known constant, CSIR + CSITFast Uniform-phase fading: Huv = ej2πΦuv where Φuv is uniform in[0, 2π), with CSIR, No CSIT


Basics on Cooperation Information Rates : CF

CF Rates for AWGN Channels

Recall the Compress-Forward Lower Bound in the DM Case

C ≥ maxPX1

PX2PY2|Y2,X2

I(X1; Y2, Y3|X2)

subject to I(Y2; Y2|X2, Y3) ≤ I(X2;Y3)

For AWGN channels, a natural choice is

Y2 = Y2 + Z2

where Z2 is Gaussian with variance N2.

X1 and X2 are chosen as independent, Gaussian, and with variances P1 and P2,respectively.

The smallest possible N2 is when

I(Y2; Y2|X2Y3) = I(X2;Y3)

which gives

N2 = NP1

(|H12|2/dα12 + |H13|2/dα13

)+N

P2|H23|2/dα23


Basics on Cooperation Information Rates : CF

CF Rates for AWGN Channels

For full-duplex relays

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

= log2

(1 +

P1|H12|2

dα12(N + N2)+P1|H13|2

dα13N

)bits/use

Comments:

As SNR23 := |H23|2P2/dα23N →∞ we have N2 → 0 and Y2 → Y2 and

R becomesR = max

PX1X2(·)I(X1;Y2Y3|X2)

This is a cut rate so CF is optimal as SNR23 →∞.Important insight: use CF when the relay is near the destination, andnot AF or DF (see the rate figure).


Basics on Cooperation Information Rates : DF

DF Rates for AWGN Channels

Recall the DF block structure where x2(wb−1) is generated by PX2 , andx1(wb−1, wb) by PX1|X2

(·|x2,i(wb−1)) for all i.

After block b, the DF relay decodes wb at rate

R < I(X1;Y2|X2)

Backward decoder rate: decode wb after block b+ 1 by using y3[B],b = B,B − 1, . . . , 1, at rate

R < I(X1X2;Y3)

Sliding-window decoder rate: decode wb after block b+ 1 by using y3[b] andy3[b+ 1], b = 1, 2, . . . , B − 1, at rate

R < I(X1;Y3|X2) + I(X2;Y3) = I(X1X2;Y3)

where the first information term is due to y3[b], and I(X2;Y3) is due to y3[b+ 1](treat x1(wb, wb+1) as interference).

DF rate:R = max

PX1X2

min {I(X1;Y3|X2), I(X1X2;Y3)}


Basics on Cooperation Information Rates : DF

DF Rates for AWGN Channels

For AWGN channels, choose Gaussian PX1X2 with E[|X1|2] = P1, E[|X2|2] = P2

and ρ = E[X1X?2 ]/√P1P2.

The DF rate is

R = maxρ

min{

log2

(1 +

P1|H12|2(1− |ρ|2)

dα12N

),

log2

(1 +

P1|H13|2

dα13N+P2|H23|2

dα23N+

√P1P2Re{ρH13H

?23}

dα13dα23N

)}Comments:

As SNR12 := |H12|2P1/dα12N →∞, optimal ρ→ 1 and R becomes

R = maxPX1X2

I(X1X2;Y3)

This is a cut rate so DF is optimal as SNR12 →∞.Important insight: use DF when the relay is near the source (see therate figure).


Basics on Cooperation Information Rates : AF

AF Rates for AWGN Channels

AF processing: set

X2,i = aY2,i−1 = a(H12,i−1

dα12

X1,i−1 + Z2,i−1

)where a is chosen to satisfy the relay power constraint.

Destination output:

Y3,i =H13,i

dα13

X1,i + aH23,i

dα23

(H12,i−1

dα12

X1,i−1 + Z2,i−1

)+ Z3,i

AF effectively converts the channel into a unit-memory inter-symbol interferencechannel. The transmitter should thus perform a water-filling optimization of thespectrum of Xn

1 .

It turns out the relay should not always transmit with maximum power.

More generally:

X2,i = ~a[Y2,i−1, Y2,i−2, · · · , Y2,i−D

]TAmounts to ”filtering-and-forwarding”.



Rates For AWGN Channels, No Fading

P1/N = 10 dB, P2/N = 20 dB, α = 1, Huv = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

2

2.5

3

3.5

4

distance d

Rat

e

cut−set bound

Rate DF

Rate CF

Relay off



Rates For Fast Uniform Phase Fading, CSIR and No CSIT

For CF, choose Y2 = Y2e−jΦ12 + Z2 where Z2 is Gaussian with 0-mean and var.

N2.

→ Straightforward algebra leads to same CF rate as for AWGN relay channels.

For DF, Straightforward algebra leads to

R = maxρ

min{

log2

(1 +

P1(1− |ρ|2)

dα12N

),

E[

log2

(1 +

P1

dα13N+

P2

dα23N+ρej(Φ12−Φ13)

√P1P2

dα13dα23N

)]}By Jensen’s inequality and E[ej(Φ12−Φ13)] = 0, the best ρ is zero!

Intuition: Without phase knowledge, source and relay transmissions cannotcombine coherently.

Important insight: Coherent combining requires CSIT at either the source or relaynode and seems unrealistic in mobile environments



Rates For Fast Uniform Phase Fading Channels, CSIR andNo CSIT

P1/N = 10 dB, P2/N = 20 dB, α = 2, Huv = ej2πΦuv

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

2

2.5

3

3.5

4

distance d

Rat

e

cut−set bound

Rate DF

Rate CF

Relay off



Summary

What we covered in Part I:

1 Summarized basic elements of relay channels, including direct, multihop,broadcast, and multiaccess transmission

2 Introduced mechanics of various kinds of relay processing, includingamplify-and-forward, decode-and-forward, and compress-and-forward

3 Reviewed a cut-set bound

4 Reviewed information theory for cooperative protocols, including AF, CF, DF.

5 Examples of rate gains for a wireless relay channel

6 Gave insight on protocol choice based on geometry and constraints


Cooperation in Presence of Interference

�� Part II: Cooperation in Presence of Interference



Goal

The purpose of this section is to give a high level overview of some issues thatarise in interference relay networks.

Show, through examples, that classic relaying techniques in general do not sufficeas such in such networks, and need be combined appropriately with more advancedtechniques.

Discuss what roles relays can play in such networks, in addition to adding power,reducing path-loss, combating fading and increasing coverage.



Outline: Part 2

1 CF Generalization to Networks / Noisy Network Coding

2 Relaying in Presence of Additive Outside Interference

- Interference Not Known

- Interference Known Only to Relay

- Interference Known Only to Source

3 Generalisation: Channels with States


Cooperation in Presence of Interference Noisy Network Coding

Wyner-Ziv Compression for Two Receivers or More

?

Y n3Xn

1 WW

Y n2 : Xn

2

Y n4 W

R = I(Y2; Y2|X2Y4) R = I(Y2; Y2|X2Y3)

CF is a good candidate for relaying in networks with no CSIT, such as mobileenvironments.

In networks, which side information to take into account ?

- Multiple-description coding not optimal and complex for many users!

Binning rate (and so, the overall rate) depends on the considered side information

More generally, Wyner-Ziv type compressions require the quantized version yn2 [i] tobe tailored for a specific receiver

The problem is even more complex in networks with more than two receivers!



Noisy Network Coding

W1 → (X1,Y1)

WN → (XN,YN)

p(y1, . . . ,yN|x1, . . . ,xN)

Wk → (Xk,Yk)

Alternate compression: noisy network coding (Kim, El Gamal 2011)

Standard compression, i.e., no binning!

Every message is transmitted b times

Receiver decodes using all blocks without explicitly decoding the compressionindices



Noisy Network Coding: Outline of Proof

Source node sends same message b times; relays use compress-forward; decodersuse simultaneous decoding

No binning; dont require decoding compression indices correctly!

For simplicity, consider proof for relay channel

The relay uses independently generated compression codebooks:

Bj = {yn2 (lj |lj−1) lj , lj−1 ∈ [1 : 2nR2 ]}, j ∈ [1 : b]

lj−1 is compression index of Y n2 (j − 1) sent by the relay in block j

The senders use independently generated transmission codebooks:

Cj = {(xn1 (j,m), xn2 (lj−1))m ∈ [1 : 2nbR], lj , lj−1 ∈ [1 : 2nR2 ]}, j ∈ [1 : b]

Encoding: Sender transmits Xn1 (j,m) in block j ∈ [1 : b] Upon receiving Y n2 (j)

and knowing Xn2 (lj−1), the relay finds jointly typical Y n2 (lj |lj−1), and sends

Xn2 (lj) in block j + 1.



Example: Interference Channel with Intermittent Feedback

Encoder 1

Encoder 2

Decoder 1

Decoder 2

DM

IC

−G

F

Xn1

Xn2 Y n

4 W2

Y n3

Y n2

Y n1

WY1,Y

2,Y

3,Y

4|X

1,X

2,SW1

W2

S1

W1

S2

Feedback provided only intermittently: at time i:

- Feedback-event with proba. p1 on Dec. 1 → Enc.1, and p2 on Dec. 2 → Enc.2

Pr{Y1[i] = Y3[i]} = p1 and Pr{Y2[i] = Y4[i]} = p2

- Erasure-event with proba. p1 on Dec. 1 → Enc.1, and p2 on Dec. 2 → Enc.2

Pr{Y1[i] = ∅} = 1− p1 and Pr{Y2[i] = ∅} = 1− p2

? Can model this type of FB using a memoryless state-pair (S1, S2), with

S1 ∼ Bern-p1 and S2 ∼ Bern-p2, and (S1, S2) ∼ pS1,S2(s1, s2)

Capacity region is unknown in general, even without feedback!

Classic partial-DF schemes inefficient here, because of the intermittence



Example: IC with Intermittent Feedback (Cont.)

Encoder 1

Encoder 2

Decoder 1

Decoder 2

DM

IC

−G

F

Xn1

Xn2 Y n

4 W2

Y n3

Y n2

Y n1

WY1,Y

2,Y

3,Y

4|X

1,X

2,SW1

W2

S1

W1

S2

Key idea: each transmitter compresses, a-la noisy network coding, its output FBand sends it to both receivers.

Optimal for linear deterministic IC model

Y3[i] = H11X1[i] + H12X2[i]

Y4[i] = H22X2[i] + H21X1[i]

Recovers known results on linear deterministic IC (Tse et al.) if p1 = p2 = 1

More generally, optimal for Costa-El Gamal injective deterministic IC model

- Details in [Zaidi ”Interference channels with generalized and intermittentfeedback”, IEEE Trans. IT, 2014]


Cooperation in Presence of Interference Interference Amplification

RC with Unknown Outside Interferer

Node 4 is an unknown interferer

E[X24 ] = Q

E[Z2i ] = 1, i = 2, 3

E[X2i ] = 1, i = 1, 2

1

2

3 WW

4

We focus on the shown geometry. Node 4 interferes on both links, to the relay anddestination. The model is

Y2 = H12X1 +H42X4 + Z2

Y3 = H13X1 +H23X2 +H43X4 + Z3

Previous schemes may all perform poor if transmit power at Node 4 is too large(strong interferer).



Amplifying the Interference

Important insight:

Interference X4 is different from noise (has a structure !)

→ Treat the interference as desired information, and amplify it instead ofcombating it!

→ The destination first decodes the interference, cancels its effect, and thendecodes message W interference-free

Rationale:

Consider the following IC, powers and noise variances set to unity for simplicity.

SNR1 , |g11|2, SNR2 , |g22|2

INR1 , |g21|2, INR2 , |g12|2

W1W1

W2W2

g11

g22

g21

g12

Strong interference regime: INR1 ≥ SNR1 and INR2 ≥ SNR2

Decoding interference is optimal in the strong interference regime.

→ The relay steers the network to the strong interference regime in which theinterference can be decoded and so its effect canceled out!



Amplifying the Interference

Important insight:

Interference X4 is different from noise (has a structure !)

→ Treat the interference as desired information, and amplify it instead ofcombating it!

→ The destination first decodes the interference, cancels its effect, and thendecodes message W interference-free

Rationale:

Consider the following IC, powers and noise variances set to unity for simplicity.

SNR1 , |g11|2, SNR2 , |g22|2

INR1 , |g21|2, INR2 , |g12|2

W1W1

W2W2

g11

g22

g21

g12

Strong interference regime: INR1 ≥ SNR1 and INR2 ≥ SNR2

Decoding interference is optimal in the strong interference regime.

→ The relay steers the network to the strong interference regime in which theinterference can be decoded and so its effect canceled out!


Cooperation in Presence of Interference Binning

Partially Known Interferer

To gain intuition, consider the Gaussian model

Y2 = X1 + S + Z2

Y3 = X1 +X2 + S + Z3

E[S2] = Q, E[X2i ] = Pi, i = 1, 2

E[Z2i ] = Ni, i = 2, 3

1

2

3 WW

4

The interference S can be known to all or only a subset of the nodes

Node k, k = 1, 2, 3, knows the interference from Node 4

- strictly-causally: if, at time i, it knows Si−1 , (S1, . . . , Si−1)

- causally: if, at time i, it knows Si , (S1, . . . , Si−1, Si)

- non-causally: if, at time i, it knows Sn , (S1, . . . , Si−1, Si, . . . , Sn)

? In all cases, interference can be learned, e.g., through relaying or by

means of cognition.

In general, asymmetric models, are more difficult to solve!



Collaborative Binning Against Interference

Recall Costa’s Dirty Paper Coding for a point-to-point AWGN channel

- Input-output relation: Y = X + S + Z

E[X2] ≤ P , E[S2] = Q, E[Z2] = N

S known non-causally to Tx, but not to Rx

-Optimal precoding: Tx sends X = U − αS, with α = P/(P +N)

Capacity C = I(U ;Y )− I(U ;S) = log2(1 + P/N)

Intuition: Y = U + (1− α)S +N , 1− α→ 0

Symmetric case: S is known to both source and relay (non-causally)

X2 = (1− λ)(U1 − α1S), λ =

√βαP1/

√P (1),

X1 = λ(U1 − α1S) + (U2 − α2S) +X ′1, X ′1 ∼ N(0, γP1)

with P (1) = (√βαP1 +

√P2)2, P (2) = βαP1 and

αk =P (k)

P (1) + P (2) + (αP1 +N2), k = 1, 2.

→ eliminates completely the effect of the interference S! (optimal if channel isphysically degraded).



Collaborative Binning Against Interference

Recall Costa’s Dirty Paper Coding for a point-to-point AWGN channel

- Input-output relation: Y = X + S + Z

E[X2] ≤ P , E[S2] = Q, E[Z2] = N

S known non-causally to Tx, but not to Rx

-Optimal precoding: Tx sends X = U − αS, with α = P/(P +N)

Capacity C = I(U ;Y )− I(U ;S) = log2(1 + P/N)

Intuition: Y = U + (1− α)S +N , 1− α→ 0

Symmetric case: S is known to both source and relay (non-causally)

X2 = (1− λ)(U1 − α1S), λ =

√βαP1/

√P (1),

X1 = λ(U1 − α1S) + (U2 − α2S) +X ′1, X ′1 ∼ N(0, γP1)

with P (1) = (√βαP1 +

√P2)2, P (2) = βαP1 and

αk =P (k)

P (1) + P (2) + (αP1 +N2), k = 1, 2.

→ eliminates completely the effect of the interference S! (optimal if channel isphysically degraded).


Cooperation in Presence of Interference Cognitive Relay

Interference Known Only at Relay

Case of no interference, with DF

The source knows the relay input Xn2

The source can therefore jointly design Xn1 through PX1,X2(x1, x2)

This ensures some coherence gain as in multi-antenna transmission

R < I(X1, X2;Y3)

Case of interference known only at relay, with DF

Issue: coherent transmission is difficult to obtainRelay should exploit the known Sn

X2,i = φ2,i(Yi−12 , Sn)

Source does not know Sn, and therefore X2,i



Coding

Complete interference mitigation is impossible, due to the asymmetry

Main idea:

decompose relay input as

X2 = U1 + X2

X2 : zero mean Gaussian with variance θP2, θ ∈ [0, 1]U1 : zero mean Gaussian with variance θP2, θ = 1− θU1 is independent of S and correlated with X1

X2 is correlated with S and independent of X1

E[U1X1] = ρ′12

√θP1P2, E[X2S] = ρ′2s

√θP2Q

X2 is generated using a Generalized DPC (ρ′2s ≤ 0)

U2 = X2 + αS



Coding

Complete interference mitigation is impossible, due to the asymmetry

Main idea:

decompose relay input as

X2 = U1 + X2

X2 : zero mean Gaussian with variance θP2, θ ∈ [0, 1]U1 : zero mean Gaussian with variance θP2, θ = 1− θU1 is independent of S and correlated with X1

X2 is correlated with S and independent of X1

E[U1X1] = ρ′12

√θP1P2, E[X2S] = ρ′2s

√θP2Q

X2 is generated using a Generalized DPC (ρ′2s ≤ 0)

U2 = X2 + αS



Information Rate

Theorem

The capacity of the general Gaussian RC with interference known non-causally only atthe relay is lower-bounded by

RinG = max min

{1

2log(

1 +P1 + θP2 + 2ρ′12

√θP1P2

θP2 +Q+N3 + 2ρ′2s√θP2Q

)+

1

2log(

1 +θP2(1− ρ′22s)

N3

),

1

2log(

1 +P1(1− ρ′212)

N2

)}

where the maximization is over parameters θ ∈ [0, 1], ρ′12 ∈ [0, 1], ρ′2s ∈ [−1, 0], andθ = 1− θ.



Upper Bounding Technique

Source → Relay

R < I(X1;Y2, Y3, S|X2)

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

(Source,Relay) → Destination

R < I(X1, X2;Y3|S)

The term I(X1;S|Y3) is the rate loss due to not knowing the interference at thesource as well

Has connection with MAC with asymmetric CSI

But, with a different proof technique



Upper Bounding Technique

Source → Relay

R < I(X1;Y2, Y3, S|X2)

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

(Source,Relay) → Destination

R < I(X1, X2;Y3|S)− I(X1;S|Y3)

The term I(X1;S|Y3) is the rate loss due to not knowing the interference at thesource as well

Has connection with MAC with asymmetric CSI

But, with a different proof technique



How Tight is the Lower Bound ?

Proposition

For the physically degraded Gaussian RC, we have:1) If P1, P2, Q, N2, N3 satisfy

N2 ≥ maxζ∈[−1,0]

P1N3(P2 +Q+N3 + 2ζ√P2Q)

P1N3 + P2(1− ζ2)(P1 + P2 +Q+N3 + 2ζ√P2Q)

,

then channel capacity is given by

CDG =1

2log(1 +

P1

N2).

2) If the maximum for the upper bound is attained at the boundary ρ212 + ρ2

2s = 1, thenthe lower bound is tight.



How Tight is the Lower Bound ? (cont.)

Bounds for GRC also meet in some extreme cases:

Arbitrarily strong interference, i.e., Q→∞

C =

{min{ 1

2log(1 + P1

N2), 1

2log(1 + P2

N3)} (DG RC)

12

log(1 + P2N3

), if P2N3≤ P1

N2(General GRC)

Zero-power at the relay, i.e., P2 = 0

C =

{12

log(1 + P1Q+N3

) (DG RC)12

log(1 + P1Q+N3

), if P1Q+N3

≤ P1N2

(General GRC)

No interference, i.e., Q = 0 (DG RC)



The Deaf Helper Problem

What if the relay cannot hear the source ?

Finite interference

subtract as much as possible from Snot optimum in general

Arbitrarily stong interference

Constructed a dammed codeword X2 (independent of X1) by DPC,X2 = U2 − SAt the destination: decode U2 first and then X1

Cleans-up the channel for X1 if

I(U2;Y3)− (U2;S) > 0.

Transmission at

I(X1;Y3|S) =1

2log2(1 +

P1

N3)



Example : Degraded Gaussian RC

P1 = P2 = Q = N3 = 10 dB

0 5 10 15 20 25 300.2

0.4

0.6

0.8

1

1.2

P1/N

2 [dB]

Rate

Lower bound

Trivial upper bound

Upper bound

Trivial lower bound

0 5 10 15 20 25 300

0.5

1

ρ2 12+

ρ2 2s


Cooperation in Presence of Interference Cognitive Source

RC with Interference Known Only at Source

Model

Y2 = X1 + S + Z2

Y3 = X1 +X2 + S + Z3

E[S2] = Q, E[X2i ] = Pi, i = 1, 2

E[Z2i ] = Ni, i = 2, 3

1

2

3 WW

4

The interference S is known non-causally to only the source



Coding

Two different techniques:

1 Lower bound 1: reveal the interference to the relay

interference exploitation (binning) is performed also at relay

(share message and interference)

2 Lower bound 2: reveal to the relay just what the relay would send had the relay

known the interference

interference exploitation (binning) is performed only at source

(share X = Θ(V, S), suitable for oblivious relaying)



Coding

Two different techniques:

1 Lower bound 1: reveal the interference to the relay

interference exploitation (binning) is performed also at relay

(share message and interference)

2 Lower bound 2: reveal to the relay just what the relay would send had the relay

known the interference

interference exploitation (binning) is performed only at source

(share X = Θ(V, S), suitable for oblivious relaying)



Lower Bound 2

If the interference were also known at relay

Beginning of block i:

Source sends x1[i] := x1(wi−1, wi) | v(wi−1, j?V ),u(wi−1, wi, j

?U ), s[i]

Relay sends x[i] | v(wi−1, j?V ), s[i]

In our case (interference at only source)

The source knows wi−1 and s[i], and so x[i]

The source transmits x[i] to the relay, ahead of time, in block i− 1

The relay estimates x[i] from y2[i− 1], and sends x2[i] ≡ x[i] in block i



Lower Bound 2

Outline:

Beginning of block i:

Source looks for u(wi, j?i ) such that (u(wi, j

?i ), s[i]) ∈ Tnε

Source looks for u(wi+1, j?i+1) such that (u(wi+1, j

?i+1), s[i+ 1]) ∈ Tnε ; and then

computes x[i+ 1] | u(wi+1, j?i+1), s[i+ 1]

Source quantizes x[i+ 1] into x[mi]

uR(mi, j⋆Ri)

u(wi, j⋆i )

Marton’s coding Superposition coding

(wi, mi)

u(wi, j⋆Ui)

uR(mi, j⋆Ri)

v(wi−1)



Lower Bound 2: Marton’s Coding

Theorem

The capacity of the DM RC with interference known only at source is lower bounded by

Rlo = max I(U ;Y3)− I(U ;S)

subject to the constraint

I(X; X) < I(UR;Y2)− I(UR;S)− I(UR;U |S)

where maximization is over all joint measures onS× U× UR × X1 × X2 × X× X× Y2 × Y3 of the form

PS,U,UR,X1,X2,X,X,Y2,Y3

= QSPU,UR|SPX1|U,UR,SPX|U,SPX|X1X2=XWY2,Y3|X1,X2,S .



Lower Bound 2 (Gaussian Case)

Test channel:

X = aX + X, a := 1−D/P2, X ∼ N(0, D(1−D/P2)), 0 ≤ D ≤ P2

X ∼ N(0, P2), with E[XX] = E[XS] = E[XS] = 0

X1R ∼ N(0, γP1), with E[X1RS] = 0, 0 ≤ γ ≤ 1

U =(√ γP1

P2+

√P2 −DP2

)X + αS

UR = X1R + αR(S +

√γP1√

γP1 +√P2 −D

X),

with

α =(√γP1 +

√P2 −D)2

(√γP1 +

√P2 −D)2 + (N3 +D + γP1)

αR =γP1

γP1 +N2.



Lower Bound

Theorem

The capacity of the Gaussian RC with interference known only at source is lowerbounded by

RloG = max

1

2log(

1 +(√γP1 +

√P2 −D)2

N3 +D + γP1

),

where

D := P2N2

N2 + γP1

and the maximization is over γ ∈ [0, 1], with γ := 1− γ.



How Tight are the Bounds ? (Cont’d)

Bounds for GRC meet in some extreme cases:

Arbitrarily small noise at relay, i.e., N2 −→ 0,

CG =1

2log(

1 +(√P1 +

√P2)2

N3

)− o(1)

where o(1) −→ 0 as N2 −→ 0.

Arbitrarily strong noise at relay, i.e., N2 −→∞,

RupG =

1

2log(1 +

P1

N3)

RloG =

1

2log(1 +

P1

N3 + P2).

If P1 −→∞, bounds meet asymptotically in the power at the source if P2 � P1,yielding

CG-orth =1

2log(1 +

P1

N3) + o(1)



Example

P1 = Q = N3 = 10 dB, P2 = 20 dB

−20 −10 0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

P1/N

2 [dB]

Rate

Lower bound (Theorem 4)

Lower bound (Theorem 5)

Upper bound (Theorem 3)

Upper bound (Cut−set bound)

Trivial lower bound


Cooperation in Presence of Interference General Framework

RC with States

More generally, S may represent any information about the channel (fading, activity,..)

SOURCE DESTINATION

RELAYX2

Y3X1 W ∈ WWY2,Y3|X1,X2,SW ∈ W

Y2

Sn

RELAY

DESTINATIONSOURCEX1

Y2 X2

Y3 W ∈ WW ∈ W

Sn

WY2,Y3|X1,X2,S

Thourough results, as well as strictly-causal CSI case, in:

- Zaidi et al., ”Bounds on the Capacity of the Relay Channel with Noncausal Stateat Source”, EEE Trans. on Inf. Theory, Vol. 59, No. 5, May 2013, pp. 2639-2672.

- Zaidi et al. ”Cooperative Relaying with State Available Non-Causally at theRelay”, EEE Trans. Inf. Theory, vol. 5, no. 56, pp. 2272-2298, May 2010.

- Zaidi et al., ”Capacity Region of Cooperative Multiple Access Channel withStates”, IEEE Trans. on Inf. Theory, Vol. 59, No. 10, 2013, pp. 6153-6174



MAC with Delayed CSI

MACDecoder

Encoder 2

Encoder 1

Si−1

Wc

Xn2

Xn1

Y n

WY |X1,X2,S(Wc, W1)

W1

Si−1

Both encoders send Wc. Encoder 1 also sends W1

Both encoders know the states only strictly causally

φ1,i : Wc×W1 × Si−1 −→ X1, i = 1, . . . , n

φ2,i : Wc×Si−1 −→ X2, i = 1, . . . , n

Decoder:ψ : Yn −→Wc×W1



Main Results

1 CSI given with delay to only transmitters increases the capacity region

- Zaidi et al., Cooperative MAC with states, IT 2013

2 Gains obtained through a Block-Markov coding in which the encoders jointlycompress the CSI of the last block, using an appropriate compression scheme

3 Reminiscient of quantizing-and-transmitting noise in a non-degraded BC examplewith common noise at receivers by Dueck (Cf: Partial feedback for two-way andBC, Inf. Contr. 1980)

- MAT scheme (Maddah Ali, Tse): interference ∼ S here. In block i,Transmitter sends a linear function of (S[i− 1],X[i− 1]). This can be seenas a compressed version S = V = f(X,S) ∼ PV |X,S

- Lapidoth et al., MAC with causal and strictly causal CSI, IT 2013

- Li et al., MAC with states known strictly causally, IT 2013



Capacity Region in Some Special Cases

Let DsymMAC be the class of discrete memoryless two-user cooperative MACs, denoted by

DsymMAC, in which the channel state S, assumed to be revealed strictly causally to both

encoders, can be obtained as a deterministic function of the channel inputs X1 and X2

and the channel output Y , asS = f(X1, X2, Y ).

Theorem

For any MAC in the class DsymMAC defined above, the capacity region Cs-c is given by the

set of all rate pairs (Rc, R1) satisfying

R1 ≤ I(X1;Y |X2, S)

Rc +R1 ≤ I(X1, X2;Y )

for some measurePS,X1,X2,Y = QSPX1,X2WY |S,X1,X2

.

Example: model Y = X1 +X2 + S. Capacity Rc +R1 ≤ log(1 + (√P1 +

√P2)2/Q).



Delayed CSI Not Always Helps !

Proposition

Delayed CSI at the encoders does not increase the sum capacity

max(Rc,R1) ∈ Cs-c

Rc +R1 = maxp(x1,x2)

I(X1, X2;Y ).

Proposition

Delayed CSI at only the encoder that sends both messages does not increase thecapacity region of the cooperative MAC.


Interaction and Computation

�� Part III: Interaction and Computation



Outline: Part 3

1 Interaction for Sources Reproduction

2 Interaction for Function Computation



Setup

A B

M1

M2

...

Mt

fA(X, Y ) fB(X, Y )

YX

Under what conditions is interaction useful ?

How useful is interaction ?

What is the best way to interact ?



Interaction for Sources Reproduction Losslessly

Discrete memoryless multi-source (X1, Y1), . . . , (Xn, Yn) i.i.d. pX,Y (x, y)

Goal: Reproduce X = (X1, X2, . . . , Xn) at B with probability 1 as n→∞

A BM1

R1 = H(X|Y )

(X1, . . . , Xn) (Y1, . . . , Yn)

(X1, . . . , Xn)

One round Slepian-Wolf Coding is optimal



Interaction for Lossy Sources Reproduction


Goal: Reproduce X = (X1, X2, . . . , Xn), with E[d(X, X)] ≤ DX as n→∞

A B

M1

M2

...

Mt

(Y1, . . . , Yn)(X1, . . . , Xn)

X = (X1, . . . , Xn)Y = (Y1, . . . , Yn)

R2

Rt

R1

Interaction is useful

Rsum := R1 +R2 + . . .+Rt ≤ minPU1|X

I(U1;X|Y ) + minPU2|Y

I(U1;Y |X)



Interaction for Fonction Computation


Goal: Reproduce fA = fA(X,Y), with E[d( fA(X,Y), fA(X,Y))] ≤ DA asn→∞

A B

M1

M2

...

Mt

(Y1, . . . , Yn)(X1, . . . , Xn)

R2

Rt

R1

fB(X,Y)fA(X,Y)

Interaction is useful


Wrap-Up

Wrap-Up

1 Many approaches, benefits, and challenges to utilizing relaying and cooperativecommunications in networks.

2 Relaying also includes, or should also include, multihop (store-and-forward routing)and network coding.

3 The capacity of relay systems is difficult to analyze, but the theory is surprisinglyflexible and diverse.

4 Although generalizations to networks with many nodes are in general not easy,there are schemes which scale appropriately.

5 In networks with interference, relaying can help, not only by adding power/energyspatially, but also by allowing distributed interference cancellation; and in generalthis boots network capacity.

6 To cope with network interference efficiently, classic relaying schemes as such ingeneral do not suffice, and need be combined carefully with other appropriatetechniques.


Wrap-Up

Selected References

- T. Cover and A. El Gamal, ”Capacity theorems for the relay channel”, IEEE Trans.Inf. Theory, Vol. 25, Sep. 1979, pp. 572-584.

- G. Kramer, M. Gastpar and P. Gupta, ”Cooperative strategies and capacitytheorems for relay networks”, IEEE Trans. Inf. Theory, Sep. 2005, pp. 3037-3037.

- N. Laneman, D. Tse and G. Wornell, ”Cooperative diversity in wireless networks:Efficient protocols and outage behaviour”, IEEE Trans. Inf. Theory, vol. 50, Dec.2004, pp. 3062-3080.

- A. Zaidi, S. Kotagiri and N. Laneman, ”Cooperative relaying with State AvailableNon-Causally at the Relay”, EEE Trans. Inf. Theory, vol. 5, no. 56, pp.2272-2298, May 2010.

- A. Zaidi et al., ”Bounds on the capacity of the relay channel with noncausal stateat source”, EEE Trans. on Inf. Theory, Vol. 59, No. 5, May 2013, pp. 2639-2672.

- A. Zaidi et al., ”Capacity region of cooperative multiple access channel withstates”, IEEE Trans. on Inf. Theory, vol. 59, No. 10, 2013, pp. 6153-6174

- S.-H. Lim, Y.-H. Kim and A. El Gamal, ”Noisy Network Coding”, IEEE Trans. Inf.Theory, vol. 57, May 2011, pp. 3132-3152.

- A. Avestimeher, S. Diggavi and D. Tse, ”Wireless network information flow: adetermenistic approach”, IEEE Trans. Inf. Theory, May 2011, pp. 3132-3152.

- A. El Gamal and Y.H-. Kim, ”Network information theory”, Cambridge UniversityPress, 2011.Abdellatif Zaidi Cooperative Techniques in Networks 90 / 90

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