Broadcast Channels with Cooperation: Capacity and Duality for the Semi-Deterministic Case Ziv Goldfeld, Haim H. Permuter and Gerhard Kramer Ben Gurion University and Technische Universit¨ at M¨ unchen IEEE Information Theory Wrokshop April-May, 2015 Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 1 / 12
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Broadcast Channels with Cooperation: Capacity
and Duality for the Semi-Deterministic Case
Ziv Goldfeld, Haim H. Permuter and Gerhard Kramer
Ben Gurion University and Technische Universitat Munchen
IEEE Information Theory Wrokshop
April-May, 2015
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 1 / 12
Outline
Channel-source duality for BCs
Semi-deterministic BC with decoder cooperation
Source coding dual
Capacity results
Summary
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 2 / 12
Duality - Preface
“There is a curious and provocative duality between the properties of a
source with a distortion measure and those of a channel...”
(C. E. Shannon, 1959)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 3 / 12
Duality - Preface
“There is a curious and provocative duality between the properties of a
source with a distortion measure and those of a channel...”
(C. E. Shannon, 1959)
PTP Duality: [Shannon, 1959], [Cover and Chiang, 2002], [Pradhan et
al., 2003], [Gupta and Verdu, 2011].
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 3 / 12
Duality - Preface
“There is a curious and provocative duality between the properties of a
source with a distortion measure and those of a channel...”
(C. E. Shannon, 1959)
PTP Duality: [Shannon, 1959], [Cover and Chiang, 2002], [Pradhan et
al., 2003], [Gupta and Verdu, 2011].
The solutions are dual - Information measures coincide.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 3 / 12
Duality - Preface
“There is a curious and provocative duality between the properties of a
source with a distortion measure and those of a channel...”
(C. E. Shannon, 1959)
PTP Duality: [Shannon, 1959], [Cover and Chiang, 2002], [Pradhan et
al., 2003], [Gupta and Verdu, 2011].
The solutions are dual - Information measures coincide.
A formal proof of duality is still absent.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 3 / 12
Duality - Preface
“There is a curious and provocative duality between the properties of a
source with a distortion measure and those of a channel...”
(C. E. Shannon, 1959)
PTP Duality: [Shannon, 1959], [Cover and Chiang, 2002], [Pradhan et
al., 2003], [Gupta and Verdu, 2011].
The solutions are dual - Information measures coincide.
A formal proof of duality is still absent.
Solving one problem =⇒ Valuable insight into solving dual.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 3 / 12
Duality - Preface
Point-to-Point Case:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
Empirical Coordination: (X, Y) ∈ T(n)
ǫ (PXP ⋆Y |X)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
Empirical Coordination: (X, Y) ∈ T(n)
ǫ (PXP ⋆Y |X)
Fixed-Type Code: (X, Y) ∈ T(n)
ǫ (P ⋆XPY |X)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
R⋆ = I(X;Y )
C = I(X;Y )
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Duality - Preface
Point-to-Point Case:
Source CodingChannel Coding
XEncoder
TDecoder
Y
MEncoder
XPY |X
YDecoder
M
R⋆ = I(X;Y )
C = I(X;Y )
0 I(X ; Y ) R
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 4 / 12
Multi-User Duality - Broadcast Channels
(M1, M2)Encoder
X PY1,Y2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1(Y1)
M2(Y2)
X1
X2
Encoder 1
Encoder 2T2(X2)
T1(X1)
YDecoder
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 5 / 12
Multi-User Duality - Broadcast Channels
(M1, M2)Encoder
X PY1,Y2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1(Y1)
M2(Y2)
X1
X2
Encoder 1
Encoder 2T2(X2)
T1(X1)
YDecoder
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 5 / 12
Multi-User Duality - Broadcast Channels
(M1, M2)Encoder
X PY1,Y2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1(Y1)
M2(Y2)
X1
X2
Encoder 1
Encoder 2T2(X2)
T1(X1)
YDecoder
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 5 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Broadcast Channel
(X, Y1, Y2) ∈ T(n)
ǫ
(
P ⋆XPY1,Y2|X
)
Dual Source Coding Setting
(X1, X2, Y) ∈ T(n)
ǫ
(
PX1,X2P ⋆
Y |X1,X2
)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Broadcast Channel
(X, Y1, Y2) ∈ T(n)
ǫ
(
P ⋆XPY1,Y2|X
)
Dual Source Coding Setting
(X1, X2, Y) ∈ T(n)
ǫ
(
PX1,X2P ⋆
Y |X1,X2
)
e.g., Markov relations, deterministic functions, etc.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Broadcast Channel
(X, Y1, Y2) ∈ T(n)
ǫ
(
P ⋆XPY1,Y2|X
)
Dual Source Coding Setting
(X1, X2, Y) ∈ T(n)
ǫ
(
PX1,X2P ⋆
Y |X1,X2
)
e.g., Markov relations, deterministic functions, etc.
Additional Principles:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Broadcast Channel
(X, Y1, Y2) ∈ T(n)
ǫ
(
P ⋆XPY1,Y2|X
)
Dual Source Coding Setting
(X1, X2, Y) ∈ T(n)
ǫ
(
PX1,X2P ⋆
Y |X1,X2
)
e.g., Markov relations, deterministic functions, etc.
Additional Principles:
Causal/non-causal encoder CSI←→ Causal/non-causal decoder SI
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Broadcast Channel
(X, Y1, Y2) ∈ T(n)
ǫ
(
P ⋆XPY1,Y2|X
)
Dual Source Coding Setting
(X1, X2, Y) ∈ T(n)
ǫ
(
PX1,X2P ⋆
Y |X1,X2
)
e.g., Markov relations, deterministic functions, etc.
Additional Principles:
Causal/non-causal encoder CSI←→ Causal/non-causal decoder SI
Decoder cooperation←→ Encoder cooperation
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Multi-User Duality - Broadcast Channels
Probabilistic relations are preserved:
Broadcast Channel
(X, Y1, Y2) ∈ T(n)
ǫ
(
P ⋆XPY1,Y2|X
)
Dual Source Coding Setting
(X1, X2, Y) ∈ T(n)
ǫ
(
PX1,X2P ⋆
Y |X1,X2
)
e.g., Markov relations, deterministic functions, etc.
Additional Principles:
Causal/non-causal encoder CSI←→ Causal/non-causal decoder SI
Decoder cooperation←→ Encoder cooperation
⋆ Result Duality: Information measures at the corner points coincide! ⋆
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 6 / 12
Cooperative SD-BC vs. Cooperative WAK ProblemWithout cooperation: [Gelfand vs. Pinsker, 1980] and [Wyner, 1975]&[Ahlswede-Korner, 1975]
(M1, M2)Encoder
X 1{Y1=f(X)}
×PY2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1
M2
M12(Y1) ∈ [1 :2nR12 ]
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 7 / 12
Cooperative SD-BC vs. Cooperative WAK ProblemWithout cooperation: [Gelfand vs. Pinsker, 1980] and [Wyner, 1975]&[Ahlswede-Korner, 1975]
(M1, M2)Encoder
X 1{Y1=f(X)}
×PY2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1
M2
M12(Y1) ∈ [1 :2nR12 ]
BCs with Cooperation:
Physicaly degraded (PD) [Dabora and Servetto, 2006].
Relay-BC [Liang and Kramer, 2007].
State-dependent PD [Dikstein, Permuter and Steinberg, 2014].
Degraded message sets / PD with parallel conf. [Steinberg, 2015].
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 7 / 12
Cooperative SD-BC vs. Cooperative WAK ProblemWithout cooperation: [Gelfand vs. Pinsker, 1980] and [Wyner, 1975]&[Ahlswede-Korner, 1975]
(M1, M2)Encoder
X 1{Y1=f(X)}
×PY2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1
M2
M12(Y1) ∈ [1 :2nR12 ]
X1
X2
Encoder 1
Encoder 2
[1 :2nR12 ] ∋ T12(X1)
T2
T1
YDecoder
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 7 / 12
Cooperative SD-BC vs. Cooperative WAK ProblemWithout cooperation: [Gelfand vs. Pinsker, 1980] and [Wyner, 1975]&[Ahlswede-Korner, 1975]
(M1, M2)Encoder
X 1{Y1=f(X)}
×PY2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1
M2
M12(Y1) ∈ [1 :2nR12 ]
X1
X2
Encoder 1
Encoder 2
[1 :2nR12 ] ∋ T12(X1)
T2
T1
YDecoder
Semi-Deterministic BC
(X, Y1, Y2) ∈ T(n)
ǫ (P ⋆X1{Y1=f(X)}PY2|X)
WAK Problem
(Y, X1, X2) ∈ T(n)
ǫ (PY 1{X1=f(Y )}P ⋆X2|Y )
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 7 / 12
Cooperative SD-BC vs. Cooperative WAK ProblemWithout cooperation: [Gelfand vs. Pinsker, 1980] and [Wyner, 1975]&[Ahlswede-Korner, 1975]
(M1, M2)Encoder
X 1{Y1=f(X)}
×PY2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1
M2
M12(Y1) ∈ [1 :2nR12 ]
X1
X2
Encoder 1
Encoder 2
[1 :2nR12 ] ∋ T12(X1)
T2
T1
YDecoder
Semi-Deterministic BC
(X, Y1, Y2) ∈ T(n)
ǫ (P ⋆X1{Y1=f(X)}PY2|X)
WAK Problem
(Y, X1, X2) ∈ T(n)
ǫ (PY 1{X1=f(Y )}P ⋆X2|Y )
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 7 / 12
Cooperative SD-BC vs. Cooperative WAK ProblemWithout cooperation: [Gelfand vs. Pinsker, 1980] and [Wyner, 1975]&[Ahlswede-Korner, 1975]
(M1, M2)Encoder
X 1{Y1=f(X)}
×PY2|X
ChannelY1
Y2
Decoder 1
Decoder 2
M1
M2
M12(Y1) ∈ [1 :2nR12 ]
X1
X2
Encoder 1
Encoder 2
[1 :2nR12 ] ∋ T12(X1)
T2
T1
YDecoder
Semi-Deterministic BC
(X, Y1, Y2) ∈ T(n)
ǫ (P ⋆X1{Y1=f(X)}PY2|X)
WAK Problem
(Y, X1, X2) ∈ T(n)
ǫ (PY 1{X1=f(Y )}P ⋆X2|Y )
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 7 / 12