Post on 21-Apr-2018
transcript
Harvard-SEAS 1'
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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
Harvard-SEAS 5'
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Outline
• Coded Bi-directional Relaying
• 3 Protocols and 4 Relaying Schemes
• Capacity Results
• Signaling Schemes
Harvard-SEAS 6'
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Coded Bi-directional Relaying
• Traditional bi-directional relaying takes place in two steps:
a b
a b
Harvard-SEAS 8'
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Coded Bi-directional Relaying
• With a relay, are 4 phases needed? NO!
a brr
a br
BETTER: 2 phases!
Harvard-SEAS 9'
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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.
Harvard-SEAS 10'
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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?
Harvard-SEAS 16'
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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?
Harvard-SEAS 17'
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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?
Harvard-SEAS 18'
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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
|
Harvard-SEAS 19'
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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
Harvard-SEAS 20'
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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
Harvard-SEAS 21'
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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)
Harvard-SEAS 22'
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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)
Harvard-SEAS 23'
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Capacity Results
• Key Ideas for Proofs
• Inner and Outer Bounds
• The Gaussian Case
• Conclusions
Harvard-SEAS 24'
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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)
Harvard-SEAS 25'
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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
Harvard-SEAS 26'
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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.
Harvard-SEAS 27'
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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.
Harvard-SEAS 28'
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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.
Harvard-SEAS 29'
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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).
Harvard-SEAS 30'
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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
Harvard-SEAS 32'
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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.
Harvard-SEAS 33'
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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
Harvard-SEAS 34'
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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.
Harvard-SEAS 35'
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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
Harvard-SEAS 36'
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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.
Harvard-SEAS 37'
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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
Harvard-SEAS 38'
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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.
Harvard-SEAS 39'
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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
Harvard-SEAS 40'
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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.
Harvard-SEAS 41'
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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
Harvard-SEAS 43'
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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 .
Harvard-SEAS 45'
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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.
Harvard-SEAS 46'
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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
Harvard-SEAS 47'
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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.
Harvard-SEAS 48'
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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).
Harvard-SEAS 49'
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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
Harvard-SEAS 50'
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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.
Harvard-SEAS 52'
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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
).
Harvard-SEAS 53'
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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
Harvard-SEAS 55'
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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 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 59'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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'
&
$
%
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.