the capacity of interference channels with partial transmitter cooperation
DESCRIPTION
The Capacity of Interference Channels with Partial Transmitter Cooperation. Ivana Marić Roy D. Yates Gerhard Kramer Stanford WINLAB, Rutgers Bell Labs. Decoder 1. Sender 1. Sender 2. Decoder 2. X 1. W 1. encoder 1. channel. ^. Y 1. W 1. - PowerPoint PPT PresentationTRANSCRIPT
The Capacity of Interference Channels The Capacity of Interference Channels with Partial Transmitter Cooperationwith Partial Transmitter Cooperation
Ivana MarićIvana Marić Roy D. Yates Gerhard Roy D. Yates Gerhard KramerKramer Stanford WINLAB, Rutgers Bell LabsStanford WINLAB, Rutgers Bell Labs
Sender 1
Sender 2
Decoder 1
Decoder 2
2
• Gaussian Channel in Standard FormGaussian Channel in Standard Form
X1
√h21
√h12X2
Y1
Y2
1
1
channelencoder 1
encoder 2
X1
X2
W1
W2
W1
^
decoder 1
decoder 2
Y1
Y2 W2
^
p(y1,y2|x1,x2)
Interference Channel Interference Channel
221122
122111
ZXXhY
ZXhXY
Z1~ N (0,1), Z2~ N (0,1)where where
• How to cope with How to cope with interferenceinterference is not well understood is not well understood• The interference channel capacity is a long standing open problemThe interference channel capacity is a long standing open problem
3
Strong Interference Strong Interference
• Capacity region known if there is strong interference:
• Capacity regionCapacity region = = Capacity region of a compound MAC in which bothboth messages are decoded at both receivers [Ahlswede, 1974]
)|;()|;(
)|;()|;(
112122
221211
XYXIXYXI
XYXIXYXI
• For the Gaussian channel in standard form, this means [Sato, Han&Kobayashi, 1981]
1
1
21
12
h
h
for all input product probability distributions [Costa&El Gamal,1987]
• Focus of this work: strong interference
4
Compound MACCompound MAC
Source 1
Source 2
Decoder 1
Decoder 2
W1
W2
(W1, W2)
(W1, W2)
5
Cooperation in Interference ChannelCooperation in Interference Channel
• Not captured in the interference channel model:• Broadcasting: sources “overhear” each other’s transmission• Cooperation
• Our goal: consider cooperation and derive capacity results• Work on more involved problems than already unsolved?• What are insightful and tractable channel models?• How do we model cooperation?
channelencoder 1
encoder 2
X1
X2
W1
W2
W1
^
decoder 1
decoder 2
Y1
Y2 W2
^
p(y1,y2|x1,x2)
6
• Full cooperation: MIMO broadcast channel• DPC optimal [Weingarten, Steinberg & Shamai, 04], [Caire & Shamai, 01], [Viswanath, Jindal & Goldsmith, 02]
• Several cooperation strategies proposed [Host-Madsen, Jindal, Mitra& Goldsmith, 03, Ng & Goldsmith, 04]
• [Jindal, Mitra& Goldsmith, Ng& Goldsmith]:•Gain from DPC when sources close together•When apart, relaying outperforms DPC•Assumptions:•Dedicated orthogonal cooperation channels •Total power constraint
Transmitter Cooperation for Gaussian ChannelsTransmitter Cooperation for Gaussian Channels
S1
S2
R1
R2
S1
S2
R1
R2
7
Transmitter Cooperation In Discrete Memoryless Channel Transmitter Cooperation In Discrete Memoryless Channel
• MAC with partially cooperating encoders [Willems, 1983]
• Capacity region CMAC (C12,C21) determined by Willems, 1983
• Communication through conference
• Realistic model
C21channel
encoder 1
encoder 2 decoder
X1
X2
W1
W2
Y (W1,^
W2)^
C12
8
• Each transmitter sends K symbols
•V1k depends on previously received
• Alphabet size of is at most NC12
• Alphabet size of is at most NC21
Cooperation through ConferenceCooperation through Conference
2,1),,...,( 1 tVV tKt
K
kk NC
1212log V
12211
2 ,..., k
k VVV
KkVWhV
VWhVk
kk
kkk
,...,1,
,1
1222
12111
1K
K
kk NC
1121log V
KV2
KV1
9
Partial Transmitter CooperationPartial Transmitter Cooperation
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1,W0
W2,W0
Y1
Y2
• In the conference: partial information about messages W1, W2 exchanged between encoders
• After the conference: • Common message at the encoders
• Encompasses scenarios in which encoders have a partial knowledge about each other’s messages
10
Transmitter CooperationTransmitter Cooperation
• Limited cooperation allows transmitters to exchange partial information about each other messages
• After cooperation• Common message known to both encoders• Private message at each encoder
• In the strong interference, the capacity region tied to the capacity region of The Compound MAC with Common Information
•Two MAC channels with a common and private messages
common
private
private
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1
W2
Y1
Y2
W0
(W1,W2, W0) ^ ^ ^
(W1,W2, W0) ^ ^ ^p(y1,y2|x1,x2)
11
Compound MAC with Common InformationCompound MAC with Common Information
Source 1
Source 2
Decoder 1
Decoder 2
W1,W0
W2,W0
(W1, W2,W0)
(W1, W2,W0)
12
The Compound MAC with Common Information
• Encoding
• Decoding
• The error probability
• (R0, R1, R2) achievable if, for any , there is an (M0, M1, M2, N, Pe) code such that
• The capacity region is the closure of the set of all achievable (R0, R1, R2)
• Easy to determine given the result by [Slepian & Wolf, 1973]
0,WWf ttt x
ttgtWtWtW Y)(ˆ),(ˆ),(ˆ021
),,()Y(),,()Y( 2102221011 WWWgWWWgPPe
2,1t
eP iNRiM 2
0
2,1,0i
13
• Capacity region [ Slepian & Wolf , 1973]
)};,(
)|;,(
),|;(
),|;(
21210
2121
122
211
YXXIRRR
UYXXIRR
UXYXIR
UXYXIR
:),,( 210 RRRMAC C
union over p(u)p(x1|u)p(x2|u)p(y|x1,x2)
common
private
private
channel
encoder 1
encoder 2
decoder
X1
X2
W1
W2
YW0 (W1,W2,W0 ) ^ ^ ^
p(y|x1,x2)
The MAC with Common Information
14
• Capacity region [ Slepian & Wolf , 1973]
)};,(
)|;,(
),|;(
),|;(
21210
2121
122
211
YXXIRRR
UYXXIRR
UXYXIR
UXYXIR
:),,( 210 RRRMAC C
union over p(u)p(x1|u)p(x2|u)p(y|x1,x2)
common
private
private
channel
encoder 1
encoder 2
decoder
X1
X2
W1
W2
YW0 (W1,W2,W0 ) ^ ^ ^
p(y|x1,x2)
The MAC with Common Information
)'( pMACR
15
The Compound MAC with Common Information
);,(),;,(min
)|;,(),|;,(min
),|;(),,|;(min
),|;(),,|;(min
221121210
22112121
1221122
2212111
YXXIYXXIRRR
UYXXIUYXXIRR
UXYXIUXYXIR
UXYXIUXYXIR
:),,( 210 RRRCMAC C
union over p(u)p(x1|u)p(x2|u)p(y1,y2|x1,x2)
• Capacity region [MYK, ISIT 2005]
common
private
private
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1
W2
Y1
Y2
W0
(W1,W2, W0) ^ ^ ^
(W1,W2, W0) ^ ^ ^p(y1,y2|x1,x2)
16
ppp
21 MACMACCMAC RR C
union over p(u)p(x1|u)p(x2|u)p(y1,y2|x1,x2)
• For each p : ( R0,R1,R2 ) is an intersection of rate regions RMACt achieved in two MACs
with common information:
2
),|,(),|( 21212111y
xxyypxxypp 1
),|,(),|( 21212122y
xxyypxxypp
The Compound MAC with Common Information
• Capacity region
17
ConverseConverse
• Error probability in MACt
• Error probability in CMAC
→ Necessary condition for :
→ Rates confined to R MAC1 (p) and R MAC2 (p) for every p
2,1)( 210 t,W,WWgPP ttet Y
2102221011 ,,)(,,)( WWWgWWWgPPe YY
eee PPP 21,max
0eP ,01 eP 02 eP
18
• From Slepian and Wolf result, choosing the rates ( R0,R1,R2 ) in
will guarantee that Pe1 and Pe2 can be made arbitrarily small
→ Pe will be arbitrarily small
21 MACMAC RR
Achievability Achievability
• The probability of error
),,()(),,()( 2102221011 WWWgWWWgPPe YY
),,()(),,()( 2102221011 WWWgPWWWgP YY
21 ee PP
19
ImplicationsImplications
• We can use this result to determine the capacity region of several channel with partial transmitter cooperation:
1. The Strong Interference Channel with Common Information
2. The Strong Interference Channel with Unidirectional Cooperation
3. The Compound MAC with Conferencing Encoders
20
• Compound MAC with common information
After The ConferenceAfter The Conference
common
private
private
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1
W2
Y1
Y2
W0
(W1,W0, ) ^ ^
• Relax the decoding constraints:• Each receiver decodes only one private message
W2
^
(W2,W0, ) ^ ^
W1
^
21
TheoremTheorem
• For the interference channel with common information satisfying
for all product input distributions the capacity region C is
);,(),;,(min
)|;,(),|;,(min
),|;(
),|;(
121121210
22112121
1222
2111
YXXIYXXIRRR
UYXXIUYXXIRR
UXYXIR
UXYXIR
:),,( 210 RRRC
union over p(u)p(x1|u)p(x2|u)p(y1,y2|x1,x2)
)|;()|;(
)|;()|;(
112122
221211
XYXIXYXI
XYXIXYXI
• Capacity region = capacity region of the compound MAC withCapacity region = capacity region of the compound MAC with
common informationcommon information
22
ProofProof
• Achievability
• Follows directly from the achievability in the Compound MAC with common information
– Decoding constraint is relaxed
• Converse
• Using Fano’s inequality
• In the interference channel with no cooperation: outer bounds rely on the independence of X1 and X2
• Due to cooperation: Codewords not independent
• Theorem conditions obtained from the converse
23
Relationship to the Strong Interference ConditionsRelationship to the Strong Interference Conditions
)|;()|;(
)|;()|;(
112122
221211
XYXIXYXI
XYXIXYXI
),|;(),|;(
),|;(),|;(
112122
221211
UXYXIUXYXI
UXYXIUXYXI
for all input distributions
for all product input distributions
• The same interference channel class satisfies two sets of conditions
• Strong interference channels conditions
)()( 21 21xpxp XX
)()()( 21 21uxpuxpup UXUXU
24
Interference Channel With Unidirectional CooperationInterference Channel With Unidirectional Cooperation
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1
W2
Y1
Y2
W1 ^
W2^
• Encoding functions
• Decoding functions
tttte WgPP )(, Y• The error probability },max{ 2,1, eee PPP 2,1tfor
2122
111
,WWf
Wf
x
x
222
111
ˆ
ˆ
Y
Y
gW
gW
• The difference from the interference channel: one encoder knows both messages
25
Cognitive Radio SettingsCognitive Radio Settings
• Cognitive Radio Channel [Devroye, Mitran, Tarokh, 2005]
• An achievable rate region• Consider simple two-transmitter, two-receiver network:
Assume that one transmitter is cognitive• It can “overhear” transmission of the primary user• It obtains partially the primary user’s message it can cooperate
Secondary user
Primary user
Decoder 1
Decoder 2
(cognitive radio)
Interference channel with unidirectional cooperation
• The assumption that the full message W1 is available at the cognitive user is an over-idealization
26
Interference Channel with Unidirectional Cooperation
• The Interference Channel with Unidirectional Cooperation [Marić, Yates & Kramer, 2005]
• Capacity in very strong interference• The Interference Channel with Degraded Message Set
[Wu, Vishwanath & Arapostathis, 2006]
• Capacity for weak interference and for Gaussian channel in weak interference
• Cognitive Radio Channel [Joviċić& Viswanath, 2006]
• Capacity for Gaussian channel in weak interference• The Interference Channel with a Cognitive Transmitter
[Marić, Goldsmith, Kramer& Shamai, 2006]
• New outer bounds and an achievable region
27
• For the interference channel with unidirectional cooperation satisfying
for all joint input distributions p(x1,x2), the capacity region C is
• Capacity region = capacity region of the Compound MAC channel with Common Information
TheoremTheorem
);,();,(
)|;()|;(
221121
112122
YXXIYXXI
XYXIXYXI
);,(
)|;(
12121
1222
YXXIRR
XYXIR
:),( 21 RRC
where the union is over p(x1,x2)p(y1,y2|x1,x2).
28
Achievability: Compound MAC with Common Information Compound MAC with Common Information
• Encode as for a Compound MAC with Common Information
• Due to unidirectional cooperation:
• W1 is a common message
• encoder 1 has no private message R’0=R1, R’1=0 choose: U=X1
common
private
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
Y1
Y2
W1
W2
(W1,W2) ^ ^
(W1,W2) ^̂
:),( 21 RRCMAC C );,(),;,(min
)|;(),|;(min
22112121
1221122
YXXIYXXIRR
XYXIXYXIR
29
ConverseConverse
• Using Fano’s inequality
• Interference channel with no cooperation: outer bounds rely on the independence of X1 and X2
• Due to cooperation: Codewords not independent
• Theorem conditions obtained from the converse
30
Converse (Converse (Continued Continued ))
• Lemma: If per-letter conditions are satisfied for all p(x1,x2), then
),|;(),|;( 11121122 WIWI XYXXYX
)|X;YI(X)|X;YI(X 112122
• Proof: Similar to proof by [Costa& El Gamal,1987] with the changes
• X1, X2 not independent
• Conditioning on W1
• We would next like
But because the situation is asymmetric, this seems difficult
N
N
nnnn N;Y,XXI
1221 )( 21 RRN
21 RRN
N
nnnn YXXI
1121 );,(• Bound
),|;(),|;( 11121122 WIWI XYXXYX • Requires
31
Converse (Converse (Continued Continued ))
• Recall that the achievable rates are
• By assumption, for all p(x1,x2)
• The rates
union over all p(x1,x2)
are thus achievable and are an outer bound
:),( 21 RRCMAC C );,(),;,(min
)|;(),|;(min
22112121
1221122
YXXIYXXIRR
XYXIXYXIR
);,();,(
)|;()|;(
221121
112122
YXXIYXXI
XYXIXYXI
:),( 21 RR);,(
)|;(
12121
1222
YXXIRR
XYXIR
32
Gaussian ChannelGaussian Channel
• Channel outputs:
• Power constraints:
iiii
iiii
zxxhy
zxhxy
221122
122111
2,11
1
2
tPxN
N
itti
Z1~ N (0,1)
Z2~ N (0,1)
• Noise:
S1
S2
x1
√h21
√h12x2
y1
y2
1
1
33
• Encoder 2: - Dedicates a portion P2 to help
-Uses superposition coding
Capacity Achieving Scheme in Strong InterferenceCapacity Achieving Scheme in Strong Interference
11
2202 X
P
PXX
1P
2P 2)1( P 22
2 )1(1log2
1PR
1212122
2121 ||21log2
1PPPhPhRR 11 ,0~ PX
220 ,0~ PX
• Encoder 1: Codebook x1(w1) 11 ,0~ PX 12,...,11NRw
2P
• Decoders reduce interference as they can decode each other’s messages
1W
2W
dec 1
dec 2
enc 1
enc 2
√h21
√h12
34
• Interference channel – no cooperation:
• Interference channel with common information:
• Unidirectional cooperation:
- more demanding conditions
• For P1= P2 sufficient conditions:
• For P1= 0 conditions never satisfied
• Channel reduces to a degraded broadcast channel from sender 2
Gaussian Channel- Strong Interference ConditionsGaussian Channel- Strong Interference Conditions
1
1
21
12
h
h
1
1
21
12
h
h
121
2112
h
hh
S1
S2
x1
√h21
√h12x2
y1
y2
1
1
221
212
221
212
21
)()1(
)()1(
1||
hh
hh
h
21 / PP
35
Gaussian Channel with Unidirectional CooperationGaussian Channel with Unidirectional Cooperation
h12
1
weak strong
interference
1/ 21 PP• Let
h21 1
•Sufficient conditions:
121
2112
h
hh
• Weak and strong interference regime solved
• Our current work: in between regime
s1 s2d2 d1
•Strong interference scenario:
36
Gaussian Channel With ConferencingGaussian Channel With Conferencing
• Channel outputs:
• Power constraints:
iiii
iiii
zxxhy
zxhxy
221212
121211
2,11
1
2
tPxN
N
itti
Z1~ N (0,1)
Z2~ N (0,1)
• Noise:
S1
S2
Z1
x1
√h12
+
+
Z2
√h21x2
y1
y2
1
1
• Note: additional resources needed for the conference
C12 C21
37
a = 0
max sum rate
c=0(no cooperation)
Gaussian Channel With ConferencingGaussian Channel With Conferencing
a increases
• Symmetric channel
21
21 ][
PP
XXEa
• a=0: X1, X2 independent
• Max sum-rate: Two sum-rate bounds equal
h12 = h21= hP1 = P2 = P
c = c12 = c21
Zt ~ N (0,1) t=1,2
38
a = 0
max sum ratec=0
(no cooperation)
c=0(no cooperation)
Gaussian Channel With ConferencingGaussian Channel With Conferencing
Sender t power, t=1,2:
Pc-conference power
a increases
N
iti Px
N 1
21
N
icti PPx
N 1
21
• Symmetric channel
)1log(2
1cPc
39
• Large c:c:
• Rate region
→→ Maximized for a=1
• Senders able to exchange indexes W1 and W2
→→ Full cooperation condition
→ → Less cooperation needed as the receivers further away
Full Cooperation in Gaussian ChannelFull Cooperation in Gaussian Channel
21 h
N
PCc
hah
N
PCRR 221
40
Interference Channel With ConferencingInterference Channel With Conferencing
• Relax the constraint → Each user decodes only one message:
Strong interference conditions?• After the conferenceAfter the conference
common
private
private
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1
W2
Y1
Y2
W0
(W1,W01 ) ^ ^
(W2,W02 ) ^ ^
• Common message contains information about Common message contains information about bothboth original messages original messages W0=(W01,W02)
• Decoders are interested in a part of the common message• Strong interference conditions?
41
• Encompasses various models:
• MIMO broadcast
• Unidirectional cooperation
• Partial cooperation
• Strong Interference conditions?
Interference Channel With ConferencingInterference Channel With Conferencing
(W1,W01) ̂^
channel
encoder 1
encoder 2
decoder 1
decoder 2
X1
X2
W1 W01 W02
Y1
Y2
W2 W01 W02 (W2,W02) ̂^
42
DiscussionDiscussion
• Introduced cooperation into the Interference Channel:
• Capacity results for scenarios of strong interference
• If the interference is strong : decoders can decode the interference
• No partial decoding at the receivers → easier problem
• Ongoing work:
• Channel models that incorporate node cooperation
– Capture characteristics of cognitive radio networks
• Capacity results and cooperation gains for more general settings
43
Ongoing WorkOngoing Work
• Determining the fundamental limits of wireless networks requires understanding the optimal ways of cooperation
• Hierarchy of cooperation
• Encoding schemes
• Optimum resource allocation
• How to cope with interference
44
• Not neededNot needed
45
a = 0
max sum rate
a increases A B
The Gaussian Channel Capacity RegionThe Gaussian Channel Capacity Region
c+R1
)(hPC
• Corner A:
• S1 acts as a relay:
•Sends at rate c
• S2 sends independent information at rate
• Corner B:
• In addition, S1 sends own data
• Sum-rate for a=0 achieved
)(hPC
)(21 PhPCRR
c
S2
S1
c=0(no cooperation)
46
Interference Channel with Confidential Messages Interference Channel with Confidential Messages
channelencoder 1
encoder 2
X1
X2
W1
W2
W1
^
decoder 1
decoder 2
Y1
Y2 W2
^
p(y1,y2|x1,x2)
Interference ChannelInterference Channel
W1
W2
47
Interference Channel with Confidential MessagesInterference Channel with Confidential Messages
• Joint work with: Ruoheng Liu, Predrag Spasojevic and Roy D. Yates • Developed inner and outer bounds
Receiver 1
Transmitter 1
encoder 1
encoder 2
decoder
X1
X2
W1
W2
W1
^
channel
p(y1, y2|x1, x2)
Y1
Transmitter 2
H(W2| Y1,W1)
Receiver 2
decoder W2
^
H(W1| Y2,W2)
Y2
48
The Compound MAC With Conferencing Encoders The Compound MAC With Conferencing Encoders
channelencoder 1
encoder 2
X1
X2
W1
W2
C21C12
decoder 1
decoder 2
Y1
Y2
(W1,W2)^ ^
(W1,W2)^ ^p(y1,y2|x1,x2)
• Adopt Willems’ cooperation model
• Decoding constraint: Both messages decoded by both receivers
• Encoding:
• Decoding:
• The probability of error:
K
K
VWf
VWf
1222
2111
,
,
x
x
ttgWW Y21ˆ,ˆ
21222111 ,)(,)( WWgWWgPPe YY
49
TheoremTheorem
• The Compound MAC capacity regioncapacity region C (C12,C21) is
union over p(u)p(x1|u)p(x2|u)p(y1,y2|x1,x2) denoted p
• For each p : Intersection between rate regions of two MACs with partially cooperating encoders
);,(
)|;,(
),|;(
),|;(
12121
211212121
211122
122111
YXXIRR
CCUYXXIRR
CUXYXIR
CUXYXIR
)};,(
)|;,(
),|;(
),|;(
22121
211222121
211222
122211
YXXIRR
CCUYXXIRR
CUXYXIR
CUXYXIR
R MAC1 (p,C12,C21) R MAC2 (p,C12,C21)
:),(),( 212112 RRCC C
21122211212112 ,,,,),( CCCCCC MACMAC ppp
RR C
50
Gaussian Channel With ConferencingGaussian Channel With Conferencing
)2(min0
)(min0
)(min0
)(min0
:),(),(
22112211}2,1{
21
21122211}2,1{
21
2122}2,1{
2
1211}2,1{
1
212112
bPhaPhPhPhCRR
CCPbhPahCRR
CPbhCR
CPahCR
RRCC
jjjjj
jjj
jj
jj
C
• Union over all
• 10,10 ba
bbaahh 1,1,12211
• Use the maximum entropy theorem• The Gaussian C-MAC capacity region
• For C12=C21=0: optimum choice is a=0, b=0