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
Cooperative WAK Problem - Solution
Theorem (Coordination-Capacity Region)
For a desired coordination PMF PX2PY |X2
1{X1=f(Y )}:
CWAK =⋃
R12 ≥ I(V ; X1) − I(V ; X2)R1 ≥ H(X1|V, U)R2 ≥ I(U ; X2|V ) − I(U ; X1|V )
R1 + R2 ≥ H(X1|V, U) + I(V, U ; X1, X2)
where the union is over all PX1,X2PV |X1
PU|X2,V PY |X1,U,V with
PX2PY |X2
1{X1=f(Y )} as marginal.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 8 / 12
Cooperative WAK Problem - Solution
Theorem (Coordination-Capacity Region)
For a desired coordination PMF PX2PY |X2
1{X1=f(Y )}:
CWAK =⋃
R12 ≥ I(V ; X1) − I(V ; X2)R1 ≥ H(X1|V, U)R2 ≥ I(U ; X2|V ) − I(U ; X1|V )
R1 + R2 ≥ H(X1|V, U) + I(V, U ; X1, X2)
where the union is over all PX1,X2PV |X1
PU|X2,V PY |X1,U,V with
PX2PY |X2
1{X1=f(Y )} as marginal.
Achievability via Wyner-Ziv coding, superposition coding andSlepian-Wolf binning.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 8 / 12
Cooperative WAK Problem - Solution
Theorem (Coordination-Capacity Region)
For a desired coordination PMF PX2PY |X2
1{X1=f(Y )}:
CWAK =⋃
R12 ≥ I(V ;X1)− I(V ;X2)R1 ≥ H(X1|V, U)R2 ≥ I(U ;X2|V )− I(U ;X1|V )
R1 + R2 ≥ H(X1|V, U) + I(V, U ; X1, X2)
where the union is over all PX1,X2PV |X1
PU|X2,V PY |X1,U,V with
PX2PY |X2
1{X1=f(Y )} as marginal.
Achievability via Wyner-Ziv coding, superposition coding andSlepian-Wolf binning.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 8 / 12
Cooperative WAK Problem - Solution
Theorem (Coordination-Capacity Region)
For a desired coordination PMF PX2PY |X2
1{X1=f(Y )}:
CWAK =⋃
R12 ≥ I(V ; X1) − I(V ; X2)R1 ≥ H(X1|V, U)R2 ≥ I(U ;X2|V )− I(U ;X1|V )
R1 + R2 ≥ H(X1|V, U) + I(V, U ; X1, X2)
where the union is over all PX1,X2PV |X1
PU|X2,V PY |X1,U,V with
PX2PY |X2
1{X1=f(Y )} as marginal.
Achievability via Wyner-Ziv coding, superposition coding andSlepian-Wolf binning.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 8 / 12
Cooperative WAK Problem - Solution
Theorem (Coordination-Capacity Region)
For a desired coordination PMF PX2PY |X2
1{X1=f(Y )}:
CWAK =⋃
R12 ≥ I(V ; X1) − I(V ; X2)R1 ≥ H(X1|V,U)R2 ≥ I(U ; X2|V ) − I(U ; X1|V )
R1 + R2 ≥ H(X1|V, U) + I(V, U ; X1, X2)
where the union is over all PX1,X2PV |X1
PU|X2,V PY |X1,U,V with
PX2PY |X2
1{X1=f(Y )} as marginal.
Achievability via Wyner-Ziv coding, superposition coding andSlepian-Wolf binning.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 8 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ; X2|V )
+I(V ; X1)
H(X1|V, U)
I(U ; X2|V )
−I(U ; X1|V )
H(X1) R1
R2
0
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ; X2|V )
+I(V ; X1)
H(X1|V, U)
I(U ; X2|V )
−I(U ; X1|V )
H(X1) R1
R2
0
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ; X2|V ) − I(U ; X1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V, U) , I(U ; X2|V ) + I(V ; X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ; X2|V )
+I(V ; X1)
H(X1|V, U)
I(U ; X2|V )
−I(U ; X1|V )
H(X1) R1
R2
0
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ;X1)− I(V ;X2) R12 = I(V ;Y1)− I(V ;Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ; X2|V ) − I(U ; X1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V, U) , I(U ; X2|V ) + I(V ; X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ;X2|V )
+I(V ; X1)
H(X1|V, U)
I(U ; X2|V )
−I(U ;X1|V )
H(X1) R1
R2
0
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2) R12 = I(V ; Y1) − I(V ; Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ;X2|V )− I(U ;X1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V, U) , I(U ; X2|V ) + I(V ; X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ;X2|V )
+I(V ; X1)
H(X1|V, U)
I(U ; X2|V )
−I(U ;X1|V )
H(X1) R1
R2
0
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2) R12 = I(V ; Y1) − I(V ; Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ;X2|V )− I(U ;X1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V, U) , I(U ; X2|V ) + I(V ; X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ;X2|V )
+I(V ; X1)
H(X1|V, U)
I(U ; X2|V )
−I(U ;X1|V )
H(X1) R1
R2
0
I(U ;Y2|V )
−I(U ;Y1|V )
H(Y1)
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2) R12 = I(V ; Y1) − I(V ; Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ;X2|V )− I(U ;X1|V )) (
H(Y1) , I(U ;Y2|V )− I(U ;Y1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V, U) , I(U ; X2|V ) + I(V ; X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ; X2|V )
+I(V ;X1)
H(X1|V,U)
I(U ;X2|V )
−I(U ; X1|V )
H(X1) R1
R2
0
I(U ; Y2|V )
−I(U ; Y1|V )
H(Y1)
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2) R12 = I(V ; Y1) − I(V ; Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ; X2|V ) − I(U ; X1|V )) (
H(Y1) , I(U ; Y2|V ) − I(U ; Y1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V,U) , I(U ;X2|V ) + I(V ;X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ; X2|V )
+I(V ;X1)
H(X1|V,U)
I(U ;X2|V )
−I(U ; X1|V )
H(X1) R1
R2
0
I(U ; Y2|V )
−I(U ; Y1|V )
H(Y1)
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2) R12 = I(V ; Y1) − I(V ; Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ; X2|V ) − I(U ; X1|V )) (
H(Y1) , I(U ; Y2|V ) − I(U ; Y1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V,U) , I(U ;X2|V ) + I(V ;X1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Corner Point Correspondence
For fixed joint PMFs and R12:
R1
R2
0
I(U ; X2|V )
+I(V ;X1)
H(X1|V,U)
I(U ;X2|V )
−I(U ; X1|V )
H(X1) R1
R2
0
I(U ; Y2|V )
+I(V ;Y1)
H(Y1|V,U)
I(U ;Y2|V )
−I(U ; Y1|V )
H(Y1)
Cooperative WAK Problem Cooperative Semi-Deterministic BC
R12 = I(V ; X1) − I(V ; X2) R12 = I(V ; Y1) − I(V ; Y2)
(R1, R2) at Lower Corner Point: (R1, R2) at Lower Corner Point:(
H(X1) , I(U ; X2|V ) − I(U ; X1|V )) (
H(Y1) , I(U ; Y2|V ) − I(U ; Y1|V ))
(R1, R2) at Upper Corner Point: (R1, R2) at Upper Corner Point:(
H(X1|V,U) , I(U ;X2|V ) + I(V ;X1)) (
H(Y1|V,U) , I(U ;Y2|V ) + I(V ;Y1))
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 9 / 12
Semi-Deterministic BC with Cooperation - Solution
Theorem (Capacity Region)
The capacity region is:
CBC =⋃
R12 ≥ I(V ; Y1) − I(V ; Y2)R1 ≤ H(Y1)R2 ≤ I(V, U ; Y2) + R12
R1 + R2 ≤ H(Y1|V, U) + I(U ; Y2|V ) + I(V ; Y1)
where the union is over all PV,U,Y1,XPY2|X1{Y1=f(X)}.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 10 / 12
Semi-Deterministic BC with Cooperation - Solution
Theorem (Capacity Region)
The capacity region is:
CBC =⋃
R12 ≥ I(V ; Y1) − I(V ; Y2)R1 ≤ H(Y1)R2 ≤ I(V, U ; Y2) + R12
R1 + R2 ≤ H(Y1|V, U) + I(U ; Y2|V ) + I(V ; Y1)
where the union is over all PV,U,Y1,XPY2|X1{Y1=f(X)}.
Later: Achievability and converse proofs for an alternative region.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 10 / 12
Semi-Deterministic BC with Cooperation - Solution
Theorem (Capacity Region)
The capacity region is:
CBC =⋃
R12 ≥ I(V ; Y1) − I(V ; Y2)R1 ≤ H(Y1)R2 ≤ I(V, U ; Y2) + R12
R1 + R2 ≤ H(Y1|V, U) + I(U ; Y2|V ) + I(V ; Y1)
where the union is over all PV,U,Y1,XPY2|X1{Y1=f(X)}.
Later: Achievability and converse proofs for an alternative region.
CBC emphasizes duality.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 10 / 12
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
1. Partition common message c.b. into 2nR12 bins.
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
1. Partition common message c.b. into 2nR12 bins.
2. Convey bin number via link.
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
1. Partition common message c.b. into 2nR12 bins.
2. Convey bin number via link.
User 2 Gain:v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
1. Partition common message c.b. into 2nR12 bins.
2. Convey bin number via link.
User 2 Gain: Reduced search space of commonmessage c.w. by R12.
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
1. Partition common message c.b. into 2nR12 bins.
2. Convey bin number via link.
User 2 Gain: Reduced search space of commonmessage c.w. by R12.
=⇒ More channel resources for private message.
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Cooperative Semi-Deterministic BC - Achievability Outline
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 11 / 12
EncX
Channel
Y1
Y2
Dec 1
Dec 2
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Common message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton (with commonmessage).
Cooperation:
1. Partition common message c.b. into 2nR12 bins.
2. Convey bin number via link.
User 2 Gain: Reduced search space of commonmessage c.w. by R12.
=⇒ More channel resources for private message.
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Summary
Channel-source duality for BCs.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Summary
Channel-source duality for BCs.
Cooperative semi-deterministic BCs:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Summary
Channel-source duality for BCs.
Cooperative semi-deterministic BCs:◮ Source coding dual - Cooperative WAK problem.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Summary
Channel-source duality for BCs.
Cooperative semi-deterministic BCs:◮ Source coding dual - Cooperative WAK problem.◮ Corner point correspondence.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Summary
Channel-source duality for BCs.
Cooperative semi-deterministic BCs:◮ Source coding dual - Cooperative WAK problem.◮ Corner point correspondence.
Achievability via Marton coding with a common message.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Summary
Channel-source duality for BCs.
Cooperative semi-deterministic BCs:◮ Source coding dual - Cooperative WAK problem.◮ Corner point correspondence.
Achievability via Marton coding with a common message.
Full version undergoing review for IEEE Trans. Inf. Theory;available on ArXiV at http://arxiv.org/abs/1405.7812.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Summary
Channel-source duality for BCs.
Cooperative semi-deterministic BCs:◮ Source coding dual - Cooperative WAK problem.◮ Corner point correspondence.
Achievability via Marton coding with a common message.
Full version undergoing review for IEEE Trans. Inf. Theory;available on ArXiV at http://arxiv.org/abs/1405.7812.
Thank you!
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
State-Dependant Semi-Deterministic BC vs. Dual:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
State-Dependant Semi-Deterministic BC vs. Dual:
(M1, M2)Enc
X 1{Y1=f(X,S)}
×PY2|X,S
Y1
Y2
Dec 1
Dec 2
M1
M2
Channel
S
X1
X2
Enc 1
Enc 2
T1
T2
DecY
Z
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
State-Dependant Semi-Deterministic BC vs. Dual:
(M1, M2)Enc
X 1{Y1=f(X,S)}
×PY2|X,S
Y1
Y2
Dec 1
Dec 2
M1
M2
Channel
S
X1
X2
Enc 1
Enc 2
T1
T2
DecY
Z
R1
R2
0
I(U ; X2) − I(U ; Z)
−I(U ; Z)
−I(U ; X1|Z)
H(X1|Z, U)
I(U ; X2)
H(X1|Z)R1
R2
0
I(U ; Y2) − I(U ; S)
−I(U ; S)
−I(U ; Y1|S)
H(Y1|S, U)
I(U ; Y2)
H(Y1|S)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
State-Dependant Output-Degraded BC vs. Dual:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
State-Dependant Output-Degraded BC vs. Dual:
(M1, M2)Enc
XPY1,Y2|X,S
Y1
Y2
Dec 1
Dec 2
M1
M2
Channel
S
X1
X2
Enc 1
Enc 2
T1
T2
DecY
Z
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
State-Dependant Output-Degraded BC vs. Dual:
(M1, M2)Enc
XPY1,Y2|X,S
Y1
Y2
Dec 1
Dec 2
M1
M2
Channel
S
X1
X2
Enc 1
Enc 2
T1
T2
DecY
Z
R1
R2
0
I(U ; Y2)
−I(U ; S)
I(Y ; X1, X2|U, Z)R1
R2
0
I(U ; X2)
−I(U ; Z)
I(X ; Y1, Y2|U, S)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
Action-Dependant Output-Degraded BC vs. Dual:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
Action-Dependant Output-Degraded BC vs. Dual:
(M1, M2)Enc
Xn
PY1,Y2|X,S
Y n1
Y n2
Dec 1
Dec 2
M1
M2
Channel
Si
Si
Xn1
Xn2
Enc 1
Enc 2
T1
T2
DecY n
An(M1, M2)
PS|A
An(T1, T2) Zi
PZ|X1,X2,A
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Multi-User Duality - Additional Examples
Action-Dependant Output-Degraded BC vs. Dual:
(M1, M2)Enc
Xn
PY1,Y2|X,S
Y n1
Y n2
Dec 1
Dec 2
M1
M2
Channel
Si
Si
Xn1
Xn2
Enc 1
Enc 2
T1
T2
DecY n
An(M1, M2)
PS|A
An(T1, T2) Zi
PZ|X1,X2,A
R1
R2
0
I(U ; Y2)
I(V, A; X1, X2|U)R1
R2
0
I(U ; X2)
I(V, A; Y1, Y2|U)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Achievability Outline
X1
X2
Encoder 1
Encoder 2
T12(X1)
T2(T12, X2)
T1(X1)
YDecoder
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Achievability Outline
X1
X2
Encoder 1
Encoder 2
T12(X1)
T2(T12, X2)
T1(X1)
YDecoder
Rate Corner Point 1 Corner Point 2
R12 I(V ; X1) − I(V ; X2) I(V ; X1) − I(V ; X2)
R1 H(X1) H(X1|V, U)
R2 I(U ; X2|V ) − I(U ; X1|V ) I(U ; X2|V ) + I(V ; X1)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Achievability Outline
X1
X2
Encoder 1
Encoder 2
T12(X1)
T2(T12, X2)
T1(X1)
YDecoder
Rate Corner Point 1 Corner Point 2
R12 I(V ; X1) − I(V ; X2) I(V ; X1) − I(V ; X2)
R1 H(X1) H(X1|V, U)
R2 I(U ; X2|V ) − I(U ; X1|V ) I(U ; X2|V ) + I(V ; X1)
Cooperation: Wyner-Ziv scheme to convey V via cooperation link.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Achievability Outline
X1
X2
Encoder 1
Encoder 2
T12(X1)
T2(T12, X2)
T1(X1)
YDecoder
Rate Corner Point 1 Corner Point 2
R12 I(V ; X1) − I(V ; X2) I(V ; X1) − I(V ; X2)
R1 H(X1) H(X1|V, U)
R2 I(U ; X2|V ) − I(U ; X1|V ) I(U ; X2|V ) + I(V ; X1)
Cooperation: Wyner-Ziv scheme to convey V via cooperation link.
Corner Point 1: V is transmitted to dec. by Enc. 1 within X1.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Achievability Outline
X1
X2
Encoder 1
Encoder 2
T12(X1)
T2(T12, X2)
T1(X1)
YDecoder
Rate Corner Point 1 Corner Point 2
R12 I(V ; X1) − I(V ; X2) I(V ; X1) − I(V ; X2)
R1 H(X1) H(X1|V, U)
R2 I(U ; X2|V ) − I(U ; X1|V ) I(U ; X2|V ) + I(V ; X1)
Cooperation: Wyner-Ziv scheme to convey V via cooperation link.
Corner Point 1: V is transmitted to dec. by Enc. 1 within X1.
Corner Point 2: V is explicitly transmitted to dec. by Enc. 2.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Proof Outline
Converse:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Proof Outline
Converse:
Standard techniques while defining
Vi = (T12, Xn\i1 , Xn
2,i+1),
Ui = T2,
for every 1 ≤ i ≤ n.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
AK Problem with Cooperation - Proof Outline
Converse:
Standard techniques while defining
Vi = (T12, Xn\i1 , Xn
2,i+1),
Ui = T2,
for every 1 ≤ i ≤ n.
Time mixing properties.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
1. M11 −→ Y1;
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
1. M11 −→ Y1;2. M22 −→ U.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
1. M11 −→ Y1;2. M22 −→ U.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
1. M11 −→ Y1;2. M22 −→ U.
Decoding: Joint typicality decoding.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
1. M11 −→ Y1;2. M22 −→ U.
Decoding: Joint typicality decoding.
Cooperation: Bin number of V n - 2nR12 bins.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Semi-Deterministic BC with Cooperation - Achievability
Outline
Rate Splitting: Mj = (Mj0, Mjj), j = 1, 2:◮ (M10, M20) - Public message;◮ (M11, M22) - Private messages.
Codebook Structure: Marton:◮ Public Message: (M10, M20) −→ V.◮ Private Messages - Superposed on V:
1. M11 −→ Y1;2. M22 −→ U.
Decoding: Joint typicality decoding.
Cooperation: Bin number of V n - 2nR12 bins.
Gain: Dec. 2 reduces search space of V by R12.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
EncEncEncXn
Channel
Y n1
Y n2
Dec 1
Dec 2
v(m10, m20)
...
...
y1-codebook ∼ P nY1|V
u-codebook ∼ P nU|V
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
H(M2) − nǫn
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
H(M2) − nǫn ≤ I(M2; Y n2 |M12) + I(M2; M12)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
H(M2) − nǫn ≤ I(M2;Yn
2|M12) + I(M2; M12)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
H(M2) − nǫn ≤ I(M2;Yn
2|M12) + I(M2; M12)
=
n∑∑∑
i=1
[
I(M2;Yn
2,i|M12, Yi−11
)−I(M2;Yn
2,i+1|M12, Yi
1)]
+I(M2; M12)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
H(M2) − nǫn ≤ I(M2;Yn
2|M12) + I(M2; M12)
=
n∑∑∑
i=1
[
I(M2;Yn
2,i|M12, Yi−11
)−I(M2;Yn
2,i+1|M12, Yi
1)]
+I(M2; M12)
=
n∑∑∑
i=1
[
I(M2;Y2,i|M12, Yi−11
, Y n
2,i+1)−I(M2;Y1,i|M12, Yi−11
, Y n
2,i+1)]
+ I(M2; M12)
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12
Semi-Deterministic BC with Cooperation - Converse
Outline
Via telescoping identities:
1. Auxiliaries: Vi = (M12, Y i−11 , Y n
2,i+1) and Ui = M2.
2. Telescoping identities [Kramer, 2011], e.g.,
H(M2) − nǫn ≤ I(M2;Yn
2|M12) + I(M2; M12)
=
n∑∑∑
i=1
[
I(M2;Yn
2,i|M12, Yi−11
)−I(M2;Yn
2,i+1|M12, Yi
1)]
+I(M2; M12)
=
n∑∑∑
i=1
[
I(M2;Y2,i|M12, Yi−11
, Y n
2,i+1)−I(M2;Y1,i|M12, Yi−11
, Y n
2,i+1)]
+ I(M2; M12)
⋆ Replaces 2 uses of Csiszar Sum Identity.
Goldfeld/Permuter/Kramer BCs with Cooperation: Capacity and Duality 12 / 12