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Capacity Bounds and Signaling Schemes forBi-directional Coded Cooperation Protocols

Vahid Tarokh

based on papers in collaboration with

Toshiaki Koike Akino, Natasha Devroye

Sang Joon Kim, Patrick Mitran and Petar Popovski

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The problem

a b

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The problem

a b

a br

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The problem

a b

a br

What happens when you add a relay?

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Outline

• Coded Bi-directional Relaying

• 3 Protocols and 4 Relaying Schemes

• Capacity Results

• Signaling Schemes

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Coded Bi-directional Relaying

• Traditional bi-directional relaying takes place in two steps:

a b

a b

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Coded Bi-directional Relaying

• With a relay, are 4 phases needed?

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Coded Bi-directional Relaying

• With a relay, are 4 phases needed? NO!

a brr

a br

BETTER: 2 phases!

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Coded bi-directional relaying

a brr

a br

wa wb

wa ⊕ wb• In particular, if the messages of a and b are wa and wb

respectively and belong to an algebraic group (such as binary

addition), then it is sufficient for the relay node to successfully

transmit wa ⊕ wb simultaneously to a and b.

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Coded Bi-directional Relaying

One possible approach we could take:

• Transmission Strategy:

– Phase 1: Node a sends wa.

– Phase 2: Node b sends wb.

– Phase 3: Node r (the relay) sends wa ⊕ wb

• Decoding

– Node a computes wa ⊕ (wa ⊕ wb) = wb, and

– Node b computes wb ⊕ (wa ⊕ wb) = wa.

• Is this the best strategy?

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Four Possible Protocols (who transmits when?)

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DT: Direct Transmission

a b

a b

ime of phase 1 and 2, respectively

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MABC: Multiple Access Broadcast

a �� �a ��wa w �

wa ⊕ wb

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TDBC: Time Division Broadcast

a �� �a ��wa

wa ⊕ wba br

wb

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HBC: Hybrid Broadcast

a �� �a ��

a ��

a ��

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Relaying Schemes (what does the relay forward?)

• The relay may process and forward the received signals

differently, depending on the different relaying capabilities or

assumptions (about the required complexity or knowledge).

a

a

wa w

What to send?

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Relaying Schemes Considered

• Amplify and Forward (AF)

• Decode and Forward (DF)

• Compress and Forward (CF)

• Mixed Forward

a

a

wa w

What to send?

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Amplify and Forward (AF)

• The relay sends a scaled version of the signal it receives.

• Very little computation is needed.

a rr

a r

wa w; Y (1)r

|

(X(2)r

;=ξ ; Y(1)r

|

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Decode and Forward (DF)

• The relay decodes both wa and wb.

• Much computation, and transmitter codebooks are needed at

the relay.

a rr

a r

wa w

wa ⊕ w

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Compress and Forward (CF)

• The relay compresses/quantizes the received signal.

• Less computation than DF and transmitter codebooks are not

needed at the relay.

a rr

a r

wa w

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Mixed Forward

• The relay decodes wa and compresses wb, combines them into a

new message wr according to a bijective function, which it

encodes and transmits.

a rr

a r

wa w �Bijective function B forms s wr = B(wa, wr0)

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Comparison of protocols

" $

Protocol Side information Phase Interference

MABC not present 2 present

TDBC present 3 not present

HBC present 4 present

Relaying Complexity Noise Relay needs

AF very low carried nothing

DF high eliminated full codebooks

CF low distortion p(yr)

Mixed moderate partially carried a codebook, p(yr)

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Capacity Results

• Key Ideas for Proofs

• Inner and Outer Bounds

• The Gaussian Case

• Conclusions

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Our Contributions

• Comprehensive treatment of 8 possible half-duplex

bi-directional relaying protocols in Gaussian noise: achievable

rate regions, outer bounds, and their relative performance

under different SNR and relay geometries.

• Surprisingly, the four phase hybrid protocol is sometimes

strictly better than the two or three phase protocols

previously introduced in the literature.

r

CF, AF, DF, Mixed relaying schemes (4 possibilities)

MABC, TDBC protocols (2 possibilities)

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Our Contributions

• DF MABC: exact capacity region.

� brr

� br

w wb

wa ⊕ wb• Our regions/bounds take into account node side information

that a node may acquire when it is not transmitting.

a brr

a br

wa

wa ⊕ wba br

wb

Side-information

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Some conclusions drawn

• For the MABC protocol, DF or CF is the optimal scheme,

depending on the given channel and SNR regime.

• In the TDBC protocol, in most cases the relative performance

of the forwarding schemes agrees with the amount of

information and complexity available at the relay, that is, in

order of increasing complexity, (and performance), the

protocols are AF, CF, Mixed and DF.

• In general, the MABC protocol outperforms the TDBC

protocol in the low SNR regime, while the reverse is true in the

high SNR regime.

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Outline

• We first clarify our notation and assumptions.

• We then outline the ideas used in constructing the achievable

rate regions and outer bound.

• We state the theorems for different protocols and relaying

schemes.

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Notation and Assumptions

• Each terminal node, a (resp. b), has its own message, wa (resp.

wb), that it wishes to send to the other terminal node, node b

(resp. a), with the help of the relay node r.

• No node can simultaneously transmit and receive.

• X(j)i : the encoded output of of node i during phase j.

• Y(j)i : the noisy received signal at node i during phase j

• Y(ℓ)r : the quantizer output at the relay during phase ℓ.

• Ra and Rb: transmitted data rates of the terminal node a and

b respectively.

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Key ideas: Cut-set bound for outer bound

If the rates {R(ij)} are achievable with a protocol P and

RΣ(S → Sc) denotes the total rate of independent information sent

from set S to set Sc then for all sets S:

RΣ(S → Sc) ≤∑

i

∆iI(X(i)(S); Y

(i)(Sc)

|X(i)(Sc), Q).

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Key Ideas: side information for inner bounds

• In TDBC nodes a and b decode using 2 phases each.

• The presence or lack of presence of side information

differentiates protocols.

• This is a key issue in our computation of the underlying

theoretical limits and has many implications on system design.

a brr

a br

wa

wa ⊕ wba br

wb

Side-information

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Decode and forward (DF)

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The DF MABC Protocol: capacity region

• Theorem 1: The capacity region of the half-duplex

bi-directional relay channel with the MABC protocol is the

union of

Ra < min{

∆1I(X(1)a

; Y (1)r

|X(1)b

, Q), ∆2I(X(2)r

; Y(2)b

|Q)}

Rb < min{

∆1I(X(1)b

; Y (1)r

|X(1)a

, Q), ∆2I(X(2)r

; Y (2)a

|Q)}

Ra + Rb < ∆1I(X(1)a

, X(1)b

; Y (1)r

|Q)

over all joint distributions p(q)p(1)(xa|q)p(1)(xb|q)p(2)(xr|q)with |Q| ≤ 5.

• Remark: If the relay is not required to decode both messages,

then the region above is still achievable, and removing the

constraint on the sum-rate Ra + Rb yields an outer bound.

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Ra < min{

∆1I(X(1)a

; Y (1)r

|X(1)b

, Q), ∆2I(X(2)r

; Y(2)b

|Q)}

Rb < min{

∆1I(X(1)b

; Y (1)r

|X(1)a

, Q), ∆2I(X(2)r

; Y (2)a

|Q)}

Ra + Rb < ∆1I(X(1)a

, X(1)b

; Y (1)r

|Q)

a brr

a br

wa wb

wa ⊕ wb∆1

∆2

Phase 1 MAC Phase 2 BC

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The DF TDBC Protocol: achievable rate region

Theorem 2: An achievable region of the half-duplex bi-directional

relay channel with the decode and forword TDBC protocol is the

union of

Ra ≤minn

∆1I(X(1)a ; Y (1)

r |Q), ∆1I(X(1)a ; Y

(1)b

|Q) + ∆3I(X(3)r ; Y

(3)b

|Q)o

Rb ≤minn

∆2I(X(2)b

; Y (2)r |Q), ∆2I(X

(2)b

; Y (2)a |Q) + ∆3I(X(3)

r ; Y (3)a |Q)

o

over all joint distributions p(q)p(1)(xa|q)p(2)(xb|q)p(3)(xr|q) with

|Q| ≤ 4.

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Ra ≤minn

∆1I(X(1)a

;Y (1)r

|Q),∆1I(X(1)a

; Y(1)b

|Q) + ∆3I(X(3)r ;Y

(3)b

|Q)o

Rb ≤minn

∆2I(X(2)b

;Y (2)r |Q),∆2I(X

(2)b

; Y (2)a

|Q) + ∆3I(X(3)r

;Y (3)a

|Q)o

I(X(1)a

;

; Y (2)a

3

; Y (1)r

, , Y(1)b

I(X(3)r ;

; Y(3)b

I(X(2)b; Y (2)

r

); Y (3)a

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The TDBC Protocol: Outer bound

Theorem 3: The capacity region of the half-duplex bi-directional

relay channel with the TDBC protocol is outer bounded by the

union of

Ra ≤ ∆1I(X(1)a ; Y (1)

r , Y(1)b

|Q)

Ra ≤ ∆1I(X(1)a ; Y

(1)b

|Q) + ∆3I(X(3)r ; Y

(3)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r , Y (2)

a |Q)

Rb ≤ ∆2I(X(2)b

; Y (2)a |Q) + ∆3I(X(3)

r ; Y (3)a |Q)

over joint distributions p(q)p(1)(xa|q)p(2)(xb|q)p(3)(xr|q) with

|Q| ≤ 4.

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Ra ≤ ∆1I(X(1)a

; Y (1)r

, Y(1)b

|Q)

Ra ≤ ∆1I(X(1)a

; Y(1)b

|Q) + ∆3I(X(3)r ; Y

(3)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r , Y

(2)a

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)a

|Q) + ∆3I(X(3)r

; Y (3)a

|Q)

I(X(1)a

;

; Y (2)a

3

; Y (1)r

, , Y(1)b

I(X(3)r ;

; Y(3)b

I(X(2)b; Y (2)

r

); Y (3)a

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The DF HBC Protocol: achievable rate region

Theorem 4: An achievable region of the half-duplex bi-directional

relay channel with the Decode and Forward HBC protocol is the

union of

Ra ≤ ∆1I(X(1)a ; Y (1)

r |Q) + ∆3I(X(3)a ; Y (3)

r |X(3)b

, Q)

Ra ≤ ∆1I(X(1)a ; Y

(1)b

|Q) + ∆4I(X(4)r ; Y

(4)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r |Q) + ∆3I(X

(3)b

; Y (3)r |X(3)

a , Q)

Rb ≤ ∆2I(X(2)b

; Y (2)a |Q) + ∆4I(X(4)

r ; Y (4)a |Q)

Ra + Rb ≤ ∆1I(X(1)a ; Y (1)

r |Q) + ∆2I(X(2)b

; Y (2)r |Q)+

∆3I(X(3)a , X

(3)b

; Y (3)r |Q)

over all joint distributions

p(q)p(1)(xa|q)p(2)(xb|q)p(3)(xa|q)p(3)(xb|q)p(4)(xr|q) with |Q| ≤ 5.

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Ra ≤ ∆1I(X(1)a

; Y (1)r

|Q) + ∆3I(X(3)a

; Y (3)r

|X(3)b

, Q)

Ra ≤ ∆1I(X(1)a

; Y(1)b

|Q) + ∆4I(X(4)r ; Y

(4)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r |Q) + ∆3I(X

(3)b

; Y (3)r |X(3)

a, Q)

Rb ≤ ∆2I(X(2)b

; Y (2)a

|Q) + ∆4I(X(4)r

; Y (4)a

|Q)

Ra + Rb ≤ ∆1I(X(1)a

; Y (1)r

|Q) + ∆2I(X(2)b

; Y (2)r |Q)+

∆3I(X(3)a

, X(3)b

; Y (3)r |Q)

a brr

a br

a br

a br

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The HBC Protocol: outer bound

Theorem 5: The capacity region of the half-duplex bi-directional

relay channel with the HBC protocol is outer bounded by the union

Ra ≤ ∆1I(X(1)a ; Y (1)

r , Y(1)b

|Q) + ∆3I(X(3)a ; Y (3)

r |X(3)b

, Q)

Ra ≤ ∆1I(X(1)a ; Y

(1)b

|Q) + ∆4I(X(4)r ; Y

(4)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r , Y (2)

a |Q) + ∆3I(X(3)b

; Y (3)r |X(3)

a , Q)

Rb ≤ ∆2I(X(2)b

; Y (2)a |Q) + ∆4I(X(4)

r ; Y (4)a |Q)

Ra + Rb ≤ ∆1I(X(1)a ; Y (1)

r |Q) + ∆2I(X(2)b

; Y (2)r |Q)+

∆3I(X(3)a , X

(3)b

; Y (3)r |Q)

over joint distributions

p(q)p(1)(xa|q)p(2)(xb|q)p(3)(xa, xb|q)p(4)(xr|q) with |Q| ≤ 5.

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Ra ≤ ∆1I(X(1)a

; Y (1)r

, Y(1)b

|Q) + ∆3I(X(3)a

; Y (3)r

|X(3)b

, Q)

Ra ≤ ∆1I(X(1)a

; Y(1)b

|Q) + ∆4I(X(4)r ; Y

(4)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r , Y

(2)a

|Q) + ∆3I(X(3)b

; Y (3)r |X(3)

a, Q)

Rb ≤ ∆2I(X(2)b

; Y (2)a

|Q) + ∆4I(X(4)r

; Y (4)a

|Q)

Ra + Rb ≤ ∆1I(X(1)a

; Y (1)r

|Q) + ∆2I(X(2)b

; Y (2)r |Q)+

∆3I(X(3)a

, X(3)b

; Y (3)r |Q)

a brr

a br

a br

a br

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Compress and Forward (CF)

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Key Ideas: CF

How to compress and forward

• In CF, network coding techniques such as the algebraic group

operation wa ⊕ wb cannot be used to generate wr.

• Compress to wr (and re-encode as X(2)r (wR)) if find Y

(1)r (wr)

that is jointly typical (JT) with Y(1)r .

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How to compress and forward

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Key Ideas: decoding in CF

[Two joint typicality decoders]

• Consider trying to decode wb at node a.

• After phase 2, node a has the sequences x(1)a (wa) and y

(2)a .

• To find the desired wb,

– First find wr

– Use wr to find wb.

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Decoding wb at node a

1

(1) Get wr

• Node then finds th

(x(1)a (wa), y

(1)r (wr))

wr such that

are best matched

32

1

2

3

wr such that

are best matchedd (x(2)r (wr),y

(2)a )

(2) Get wb such that (x(1)a (wa), x

(1)b

(wb), y(1)r (wr)) are best matched

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The CF MABC Protocol: achievable rate region

Theorem 6: An achievable region of the half-duplex bi-directional

relay channel with the compress and forward MABC protocol is the

union of Ra, Rb subject to

∆1I(Y (1)r ; Y (1)

r |X(1)b

, Q) ≤ ∆2I(X(2)r ; Y

(2)b

|Q)

∆1I(Y (1)r ; Y (1)

r |X(1)a , Q) ≤ ∆2I(X(2)

r ; Y (2)a |Q)

Ra ≤ ∆1I(X(1)a ; Y (1)

r |X(1)b

, Q)

Rb ≤ ∆1I(X(1)b

; Y (1)r |X(1)

a , Q)

over all joint distributions, p(q)p(1)(xa|q)p(1)(xb|q)p(1)(yr|xa, xb)p

(1)(yr|yr)p(2)(xr|q) with |Q| ≤ 7.

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Intuition behind CF MABC

Suppose we wish to decode wa at node b.

• First, to decode wr: there are I(Y(1)r ; Y

(1)r ) bits of information

in this message, of which only

I(Y(1)r ; Y

(1)r ) − I(Y

(1)r ; X

(1)b ) = I(Y

(1)r ; Y

(1)r |X(1)

b) must be sent

after using the side information at node b, X(1)b .

• ⇒ ∆1I(Y(1)r ; Y

(1)r |X(1)

b) ≤ ∆2I(X

(2)r ; Y

(2)b

)

• Once you have wr use this to obtain wa by looking for

(x(1)a (wa), x

(1)b (wb), y

(1)r (wr)) that are best matched.

• ⇒ Ra ≤ ∆1I(X(1)a ; Y

(1)r |X(1)

b).

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∆1I(Y (1)r ; Y (1)

r |X(1)b

, Q) ≤ ∆2I(X(2)r ; Y

(2)b

|Q)

∆1I(Y (1)r ; Y (1)

r |X(1)a , Q) ≤ ∆2I(X(2)

r ; Y (2)a |Q)

Ra ≤ ∆1I(X(1)a ; Y (1)

r |X(1)b

, Q)

Rb ≤ ∆1I(X(1)b

; Y (1)r |X(1)

a , Q)

a brr

a br

wa wb

∆1

∆2

Decoding wr

Decoding wa, wb with side information

(X(2)r

; Y (1)rI(X(1)

a ; |X(1)b

,

); Y(2)b; Y (2)

a

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The CF TDBC Protocol: achievable rate region

Theorem 7: An achievable region of the half-duplex bi-directional

relay channel with the compress and forward TDBC protocol is the

union of Ra, Rb subject to

∆1I(Y (1)r

; Y (1)r

|Q) + ∆2I(Y (2)r

; Y (2)r

|X(2)b

, Q) ≤ ∆3I(X(3)r

; Y(3)b

|Q)

∆2I(Y (2)r

; Y (2)r

|Q) + ∆1I(Y (1)r

; Y (1)r

|X(1)a

, Q) ≤ ∆3I(X(3)r

; Y (3)a

|Q)

Ra ≤ ∆1I(X(1)a

; Y (1)r

, Y(1)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r , Y (2)

a |Q)

over all joint distributions, p(q)p(1)(xa|q)p(1)(yr|xa)

p(1)(yr|yr)p(2)(xb|q)p(2)(yr|xb)p

(2)(yr|yr)p(3)(xr|q) with |Q| ≤ 7.

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CF TDBC steps

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Intuition behind CF TDBC

Suppose we wish to decode wa at node b.

• First, to decode wr0: there are I(Y(1)r ; Y

(1)r ) + I(Y

(2)r ; Y

(2)r ) bits

of information, of which only

I(Y(1)r ; Y

(1)r ) + I(Y

(2)r ; Y

(2)r ) − I(Y

(2)r ; X

(2)b ) =

I(Y(1)r ; Y

(1)r ) + I(Y

(2)r ; Y

(2)r |X(2)

b) must be sent after using the

side information X(2)b .

• ⇒ ∆1I(Y(1)r ; Y

(1)r ) + ∆2I(Y

(2)r ; Y

(2)r |X(2)

b) ≤ ∆3I(X

(3)r ; Y

(3)b

)

• Once you have wr0 use this to obtain wa by looking for

(x(1)a (wa), y

(1)r (wr0), y

(1)b

(wa)) that are best matched.

• ⇒ Ra ≤ ∆1I(X(1)a ; Y

(1)r , Y

(1)b

).

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∆1I(Y (1)r

; Y (1)r

|Q) + ∆2I(Y (2)r

; Y (2)r

|X(2)b

, Q) ≤ ∆3I(X(3)r

; Y(3)b

|Q)

∆2I(Y (2)r

; Y (2)r

|Q) + ∆1I(Y (1)r

; Y (1)r

|X(1)a

, Q) ≤ ∆3I(X(3)r

; Y (3)a

|Q)

Ra ≤ ∆1I(X(1)a

; Y (1)r

, Y(1)b

|Q)

Rb ≤ ∆2I(X(2)b

; Y (2)r , Y (2)

a |Q)

I(X(1)a ;

; Y (2)a

3

; Y (1)r , , Y

(1)b

I(X(3)r ;

; Y(3)b

I(X(2)b; Y (2)

r

); Y (3)a

Decoding wr

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Mixed forward

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Mixed MABC Protocol: achievable rate region

Theorem 8: An achievable region of the half-duplex bi-directional

relay channel with the mixed forward MABC protocol is the union

of Ra, Rb subject to

∆1I(Y (1)r

; Y (1)r

|X(1)a

, Q) ≤ ∆2I(X(2)r

; Y (2)a

|Q)

Ra ≤ min{

∆1I(X(1)a ; Y (1)

r |Q),[

∆2I(X(2)r

; Y(2)b

|Q) − ∆1I(Y (1)r

; Y (1)r

|X(1)b

, Q)]+

}

Rb ≤ ∆1I(X(1)b

; Y (1)r

|X(1)a

, Q)

over all joint distributions, p(q)p(1)(xa|q)p(1)(xb|q)p(1)(yr|xa, xb)p

(1)(yr|yr)p(2)(xr|q) with |Q| ≤ 6.

Harvard-SEAS 56'

&

$

%

Mixed MABC steps

Harvard-SEAS 57'

&

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%

Mixed TDBC Protocol: achievable rate region

Theorem 9: An achievable region for the half-duplex

bi-directional relay channel with a mixed TDBC protocol is the

union of Ra, Rb subject to

∆2I(Y (2)r

; Y (2)r

|Q) ≤ ∆3I(X(3)r

; Y (3)a

|Q)

Ra ≤ min{

∆1I(X(1)a

; Y (1)r

|Q), ∆1I(X(1)a

; Y(1)b

|Q)+

[

∆3I(X(3)r ; Y

(3)b

|Q) − ∆2I(Y (2)r ; Y (2)

r |X(2)b

, Q)]+

}

Rb ≤ ∆2I(X(2)b

; Y (2)r

, Y (2)a

|Q)

over all joint distributions, p(q)p(1)(xa|q)p(2)(xb|q)p(2)(yr|xb)

p(2)(yr|yr)p(3)(xr|q) with |Q| ≤ 6.

Harvard-SEAS 58'

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Mixed TDBC steps

Harvard-SEAS 59'

&

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The Gaussian Case

• We apply the previous results to the Gaussian channel.

• hij is the effective channel gain between transmitter i and

receiver j, which is modeled as a complex number. We assume

that the channel is reciprocal.

a br

N(0,1) N(0,1)

+ +har hbr

habPower Pa Power Pb

Power Pr

Harvard-SEAS 60'

&

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%

The Gaussian case

• For the analysis of the Compress and Forward scheme, we

assume Y(ℓ)r are zero mean Gaussians and define

P(ℓ)y := E[(Y

(ℓ)r )2] , P

(ℓ)y := E[(Y

(ℓ)r )2] and σ

(ℓ)y := E[Y

(ℓ)r Y

(ℓ)r ].

• We focus here on the results of the optimization.

• We consider four different relaying schemes for each MABC

and TDBC bi-directional protocol. For example, an achievable

rate region of the AF MABC protocol is given by:

Ra ≤1

2C

( |har|2|hbr|2PaPr

|har|2Pa + |hbr|2Pb + |hbr|2Pr + 1

)

Rb ≤ 1

2C

( |har|2|hbr|2PbPr

|har|2Pa + |hbr|2Pb + |har|2Pr + 1

)

.

Harvard-SEAS 61'

&

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%

Achievable Regions: some conclusions

• At low SNR, the DF MABC protocol dominates the other protocols.

• The MABC protocol in general outperforms the TDBC protocol as

the benefits of side information and reduced interference are

relatively small in this regime.

• The DF scheme outperforms the other schemes since the relatively

large amount of noise can be eliminated in the DF scheme, which

cannot be done using the other schemes.

• In contrast, the DF TDBC protocol dominates the other protocols

at high SNR since the direct link is strong enough to convey

information in this regime.

• Notably, some HBC rate pairs are strictly outside the outer bounds

of the MABC and TDBC protocols.

Harvard-SEAS 62'

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Achievable Regions

har = hbr = 1, hab = 0.2, N = 1, and P = 0 dB.

0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ra

Rb

MABCTDBCAFDFCFMixedOuter

Harvard-SEAS 63'

&

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%

Achievable Regions

har = hbr = 1, hab = 0.2, N = 1, and P = 50 dB.

0 5 10 150

2

4

6

8

10

12

14

Ra

Rb

MABCTDBCAFDFCFMixedOuter

Harvard-SEAS 64'

&

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%

Achievable Regions

Comparison of the DF scheme only in the same channel, har = hbr = 1,

hab = 0.2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ra

Rb

DTMABCTDBCHBCTDBC outer bound

Harvard-SEAS 65'

&

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The effect of the relay’s position

• We plot the maximum sum-rate Ra + Rb as a function of the

relay position dar = ζdab (0 < ζ < 1) when the relay r is located

on the line between a and b.

• We apply hab = 0.2 and Pa = Pb = Pr = 20 dB and let

|hij |2 = k/d3.8ij for k constant and a path-loss exponent of 3.8.

a br

N(0,1) N(0,1)

+ +

ar = ζdab dbr=(1-ζ)dab

Harvard-SEAS 66'

&

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The effect of the relay’s position

• Under a σ = Rb/Ra ratio restriction, we optimize the time

duration and find the maximum sum data rate. Here we have σ

unconstrained, and hab = 0.2 and Pa = Pb = Pr = 20dB and

dar = ζdab (0 < ζ < 1) .

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

ζ

Sum

−ra

te

MABCTDBCAFDFCFMixedOuter

Harvard-SEAS 67'

&

$

%

Relay Position

σ = Rb/Ra = 1, Pa = Pb = Pr = 20 dB, dar = ζdab.

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

ζ

Sum

−ra

te

MABCTDBCAFDFCFMixedOuter

Harvard-SEAS 68'

&

$

%

Relay Position

σ = Rb/Ra = 2, Pa = Pb = Pr = 20 dB, dar = ζdab.

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

ζ

Sum

−ra

te

MABCTDBCAFDFCFMixedOuter

Harvard-SEAS 69'

&

$

%

Next

• As with any information theoretic limit, the question of how to

achieve it in practice becomes important.

• We now discuss the coding schemes.

Harvard-SEAS 70'

&

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Review of MABC Relaying

Step 1: Multiple Access Stage

Step 2: Broadcast Stage

Node R

Node A Node B

Node A Node B

XA=M4(SA)

Node R

Network Coding

SR = C(SA, SB)ˆ ˆ

SA SB

XB=M4(SB)

SR

XR=MR(SR) XR=MR(SR)

HA HB

Harvard-SEAS 71'

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Phase 1: multiple access channel

• Let SA and SB be the messages from the terminals A and B,

respectively, which are drawn from a set Z4 = {0, 1, 2, 3}.

• The QPSK constellations XA and XB are used for sending the

messages: XA = M4(SA) and XB = M4(SB), where M4(·)denotes a QPSK modulation function.

• In this phase, the terminals A and B simultaneously transmit

QPSK signals XA and XB to the relay node R.

• The relay node R receives the overlapped signal as follows:

YR = HAXA + HBXB + ZR,

where YR, HA, HB and ZR are the received signal, the channel

gain to node A, to node B, and the AWGN, resp.

Harvard-SEAS 72'

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Phase 2: broadcast channel

• The relay node R generates a network-coded signal XR ∈ C from the

received signal YR.

• Suppose that the relay R employs the maximum-likelihood (ML)

estimation for (SA, SB) to get (SA, SB).

• The ML estimated messages are compressed by a network coding

function: SR = C(SA, SB).

• The network-coded data is broadcast as XR = MR(SR), where

MR(·) is a modulation function.

• One example is QPSK constellation with 4-ary XOR network code:

C(SA, SB) = SA ⊕ SB ∈ Z4, MR(·) = M4(·).

• Is the 4-ary XOR network coding good enough for all channels?

Harvard-SEAS 73'

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Design Strategy of Network Codes

• Any arbitrary network code must have the following property:

C(S′

A, SB) 6= C(S′′

A, SB), for any S′

A 6= S′′

A ∈ Z4, given SB ,

C(SA, S′

B) 6= C(SA, S′′

B), for any S′

B 6= S′′

B ∈ Z4, given SA.

• We wish to minimize the error event probability at the relay

node R after network coding: Pr(C(SA, SB) 6= C(SA, SB)).

• To this end, we will design a network code that maximizes the

distance profile (below) between replicated pairs:

d2((S′

A, S′

B), (S′′

A, S′′

B)) =∣

∣HA (M4(S′

A) −M4(S′′

A))

+ HB (M4(S′

B) −M4(S′′

B))∣

∣

2, for any C(S′

A, S′

B) 6= C(S′′

A, S′′

B).

Harvard-SEAS 74'

&

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%

Example I: Rx Replicas (HB/HA ≃ 1/√

2)

I

Q

(0,0)

C → 0

(0,1)

(0,2)(0,3)

(1,0)(1,1)

(1,2)(1,3)

(2,0)(2,1)

(2,2)

(2,3)

(3,0)(3,1)

(3,2)

(3,3)

C → 0

C → 0 C → 0

C → 1C → 1

C → 1C → 1

C → 2

C → 2

C → 2

C → 2

C → 3

C → 3C → 3

C → 3

In this nearly phase-synchronous case, 4-ary XOR network coding

provides a good minimum distance.

Harvard-SEAS 75'

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XOR Network Coding for ∠[HB/HA] ≃ 0

• The bit-wise 4-ary XOR network code C(SA, SB) = SA ⊕ SB can be

represented in a table as follows:

SA\SB 0 1 2 3

0 0 1 2 3

1 1 0 3 2

2 2 3 0 1

3 3 2 1 0

• Since all the other possible network codes with quaternary

cardinality can be given by row or column-wise permutations, the

number of possibilities is 4! × 4! = 576 at most.

• Among all the possible 4-ary codes, the 4-ary XOR coding is the

best choice for such a phase-synchronous case (∠[HB/HA] ≃ 0).

Harvard-SEAS 76'

&

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Example II: Rx Replicas (HB/HA ≃ j/√

2)

I

Q

(0,2)

C → 2

(0,0)

(0,3)(0,1)

(1,2)(1,0)

(1,3)(1,1)

(2,2)(2,0)

(2,3)

(2,1)

(3,2)(3,0)

(3,3)

(3,1)

C → 1

C → 2 C → 1

C → 0C → 3

C → 3C → 0

C → 0

C → 3

C → 3

C → 0

C → 2

C → 1C → 2

C → 1

In the quadrature-phase difference case, 4-ary XOR network coding

provides low minimum distance.

Harvard-SEAS 77'

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XOR Network Coding for ∠[HB/HA] ≃ π/2

• For the quadrature-phase case (∠[HB/HA] = π/2), the traditional

4-ary XOR operation does not ensure reliable relaying.

• Reliability may be improved slightly modifying the code using an

anti-rotation function R(·) as follows:

C(SA, SB) = SA ⊕R(SB),

R(0) = 1, R(1) = 3, R(2) = 0, R(3) = 2.

• It is now apparent that the network code C should be designed

according to the channel ratio HB/HA = γ(cos θ + j sin θ).

• We will search for codes which optimize the distance profile as a

function of the channel parameters γ and θ.

Harvard-SEAS 78'

&

$

%

Selection Rule of Best 4-ary Network Coding

γ cos θC0

γ sin θ

10.5

C0

C1

C1

We have so far illustrated two optimal codes:

C0 is the 4-ary XOR, C1 is the modified 4-ary XOR.

Harvard-SEAS 79'

&

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Degradation Due to Distance Shortening

• The pure 4-ary XOR network code should be used for

| tan θ| ≤ 1, otherwise the modified 4-ary XOR code should be

used.

• They can offer the maximum distance profile among all the

possible quaternary codes.

• However, they are significantly degraded due to a distance

shortening under some specific channel conditions close to the

selection borderline (| tan θ| = 1).

• For example, when we have HB/HA = (1 + j)/√

2, the

minimum distance becomes zero for any quaternary codes.

Harvard-SEAS 80'

&

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%

Distance Shortening at HB/HA ≃ (1 + j)/√

2

Q

(0,2)

(0,0)

(0,3)

(0,1)(1,2)

(1,0)

(1,3)

(1,1)

(2,2)

(2,0)

(2,3)

(2,1)(3,2)

(3,0)

(3,3)

(3,1)

C → 0

C → 0

C → 0

C → 0

C → 3

C → 2

C → 3

C → 1

C → 3

C → 1

C → 3

C → 2C → 1

C → 2

C → 2

C → 1

I

In this channel condition, the 4-ary XOR code (and all the other

quaternary codes) provide low minimum distance.

Harvard-SEAS 81'

&

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Zero Distance Shortening

• There are eight points in which the minimum distance becomes

zero (except when HA = 0 or HB = 0.)

• Around these points, we use a different network code that does

not constraint the minimum cardinality to be quatenary.

• In fact, we can avoid the distance shortening by using the

following quinary (5-ary) code for HB/HA = (1 + j)/√

2:

SA\SB 0 1 2 3

0 0 2 4 1

1 3 0 2 4

2 1 3 0 2

3 4 1 3 0

Harvard-SEAS 82'

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5-ary Network Coding at HB/HA ≃ (1 + j)/√

2

Q

(0,2)

(0,0)

(0,3)

(0,1)(1,2)

(1,0)

(1,3)

(1,1)

(2,2)

(2,0)

(2,3)

(2,1)(3,2)

(3,0)

(3,3)

(3,1)

C → 0

C → 0

C → 0

C → 0

C → 1

C → 1

C → 2

C → 2

C → 3

C → 3

C → 4

C → 4C → 2

C → 4

C → 1

C → 3

I

Even in such a distance shortening condition, the 5-ary network

code can be used.

Harvard-SEAS 83'

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Network Coding Design Method

• The distance profile can be improved by using the quinary

network codes.

• What is the best network code with arbitrary cardinality given

the channel parameters γ and θ?

• We have designed network codes for this scenario.

Harvard-SEAS 84'

&

$

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Designed Network Codebook

• Our design method yields the following ten codes, optimized

for all possible channel conditions, listed below.

• C0 and C1 are 4-ary codes while C2 - C9 are 5-ary network codes.

(0, 0) (0, 1) (0, 2) (0, 3) (1, 0) (1, 1) (1, 2) (1, 3) (2, 0) (2, 1) (2, 2) (2, 3) (3, 0) (3, 1) (3, 2) (3, 3) -ary

C0 0 1 2 3 1 0 3 2 2 3 0 1 3 2 1 0 4

C1 1 3 0 2 0 2 1 3 3 1 2 0 2 0 3 1 4

C2 0 2 4 1 3 0 2 4 1 3 0 2 4 1 3 0 5

C3 3 2 0 1 0 1 2 4 1 3 4 0 4 0 3 2 5

C4 2 1 0 4 0 4 3 2 3 2 1 0 1 0 4 3 5

C5 2 1 3 0 1 4 0 2 3 0 1 4 0 2 4 3 5

C6 1 4 2 0 4 2 0 3 2 0 3 1 0 3 1 4 5

C7 1 0 2 3 4 2 1 0 0 4 3 1 2 3 0 4 5

C8 4 0 1 2 2 3 4 0 0 1 2 3 3 4 0 1 5

C9 0 3 1 2 2 0 4 1 4 1 0 3 3 4 2 0 5

Harvard-SEAS 85'

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Selection Rule of Best Network Code

The best code should be selected out of the codebook according to

the channel parameters γ and θ as follows.

γ cos θC0

γ sin θ

10.5 2

C0

C1

C1

C2C2 C3

C3C4

C4

C5 C5

C7

C7

C6C6

C8

C8

C9 C9

C5

C5

C8C8 C7 C7

C2

C2

C3C3

C6

C6

C4C4

C9

C9

Harvard-SEAS 86'

&

$

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Constellation Design for 5-ary Network Coding

• The designed network codes may have 5-ary alphabet with

non-equiprobable a priori probabilities; more specifically,

Pr(SR = 0) = 4/16 and Pr(SR 6= 0) = 3/16.

• We can design an optimized quinary constellation for the

non-equiprobable symbol scenario.

Harvard-SEAS 87'

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End-to-End Throughput Evaluations

• Next, we evaluate the performance of the proposed network

coding strategy.

• Simulation parameters:

– The channel is modeled as Rician with K-factors of KR of 0 dB

and 10 dB respectively.

– Packets length = 256 symbols long.

– Noise variance = σ2.

– Average SNR = E[|HA|2 + |HB|2]/2σ2.

– The average channel power ratios are E[|HB |2]/E[|HA|2] = 0 dB

or 5 dB.

Harvard-SEAS 88'

&

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Protocols

• 5QAM Denoising: 2-stage MABC protocol. The relay

adaptively selects the best network code including both the

quinary and quaternary cardinalities by observing the channel

ratio HB/HA.

• QPSK Denoising: 2-stage MABC protocol. The relay switches

the network code between the basic 4-ary XOR and the

modified 4-ary XOR by observing the channel phase difference

| tan θ|.

• 4-Stage Protocol: During the first and second stages, two

terminals sequentially transmit the own messages to the relay.

The relay then transmits the successful messages to the

corresponding destination at the third stage and fourth stage.

Harvard-SEAS 89'

&

$

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Protocols

• Denoising with Precoding: 2-stage MABC protocol. During the

MA stage, two terminals perform phase synchronization to

achieve θ = 0. Hence, the network code is always the basic

4-ary XOR operation.

• 3-Stage Network Coding: TDBC with QPSK and 4-ary XOR

network coding.

Harvard-SEAS 90'

&

$

%

End-to-End Throughput for KR = 10 dB

0

0.5

1

1.5

2

5 10 15 20 25 30

End-t

o-E

nd T

hro

ughput (b

ps/H

z)

Average SNR (dB)

Nakagami-Rice Fading

Rician Factor: 10 dB

2-Stage Denoising

3-Stage Net Coding

4-Stage Relaying

QPSK Denoising

5QAM Denoising

Denoising w/ Precoding

3-Stage Net Coding

4-Stage Relaying

Channel Power Ratio

0 dB 5 dB

Harvard-SEAS 91'

&

$

%

End-to-End Throughput for KR = 0 dB

0

0.5

1

1.5

2

5 10 15 20 25 30

End-t

o-E

nd T

hro

ughput (b

ps/H

z)

Average SNR (dB)

Nakagami-Rice Fading

Rician Factor: 0 dB

2-Stage Denoising

3-Stage Net Coding

4-Stage Relaying

QPSK Denoising

5QAM Denoising

Denoising w/ Precoding

3-Stage Net Coding

4-Stage Relaying

Channel Power Ratio

0 dB 5 dB

Harvard-SEAS 92'

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$

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Summary

• We have derived achievable rate regions, outer bounds, as well

as signaling schemes for bi-directional relaying channels which

exploit:

– The wireless broadcast nature of the channels.

– Side information at various transmitters/receivers.

– Analytically optimized network codes.