On the Duality of Gaussian Multiple-Access and Broadcast Channels ...
On the Duality of Gaussian Multiple-Access and Broadcast Channels Nihar Jindal, Sriram Vishwanath, and Andrea Goldsmith Ruiyuan Hu Lehigh University
April 5, 2005
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 1 / 14
Purpose Capacity Region of the Gaussian BC is equal to the union of capacity regions of the dual MAC, where the union is taken over all individual power constraint that sum up to the BC power constraint. [ CMAC (P1 , . . . , PK ; h). CBC = {Pi }K 1 :
P
K i=1
Pi =P
MAC capacity region is equal to the intersection of dual BC capacity regions. ! K \ X Pi CMAC (P 1 , . . . , P K ; h) = CBC ; α1 h1 , . . . , αK hK αi K {αi }1 : αi >0
Ruiyuan Hu (Lehigh)
i=1
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 2 / 14
Outline
System Model Capacity Region Analysis Capacity Region of the MAC Capacity Region of the BC
Constant MAC and BC MAC to BC BC to MAC
Fading MAC and BC MAC to BC BC to MAC
Conclusion
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 3 / 14
System Model
BC: Yj [i] =
q
hj [i]X [i] + nj [i]
MAC: Y [i] =
K q X hj [i]Xj [i] + n[i] j=1
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 4 / 14
Capacity Region Analysis (MAC)
The capacity region of a Gaussian MAC with channel gains h = (h1 , . . . , hK ) and power constraints P = (P1 , . . . , PK ) is X 1 1 X R: Rj ≤ log(1 + 2 hj P j ) ∀S ⊆ {1, . . . , K } 2 σ j∈S
j∈S
With success decoding with interference cancellation, the corner points are ! hπ(j) P π(j) 1 Rπ(j) = log 1 + , j = 1, . . . , K . P 2 σ2 + K i=j+1 hπ(i) P π(i)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 5 / 14
Capacity Region Analysis (BC)
The capacity region of a Gaussian BC with channel gains h = (h1 , . . . , hK ) and power constraints P is (
B
hj P j 1 P R : Rj ≤ log 1 + 2 2 σ + hj k=1 KPkB 1[hk > hj ]
Ruiyuan Hu (Lehigh)
!
) , j = 1, . . . , K
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 6 / 14
MAC to BC (Constant) Theorem The capacity region of a constant Gaussian BC with power constraint P is equal to the union of capacity regions P of the dual MAC with power constraint (P1 , . . . , PK ) such that K j=1 Pj = P CBC =
[
CMAC (P; h).
P:1·P=P
Corollary The capacity region of a constant Gaussian MAC with power constraints P = (P1 , · · · , PK ) is a subset of the capacity region of the dual BC with power constraint P = 1 · P CMAC (P; h) ⊆ CBC (1 · P; h) Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 7 / 14
MAC to BC (Constant)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 8 / 14
MAC to BC (Constant)
Channel Scaling: from (P; h) to ( P α ; αh)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 9 / 14
BC to MAC (Constant)
Theorem The capacity region of a constant Gaussian MAC is equal to the intersection of the capacity regions of the scaled dual BC over all possible channel scalings CMAC (P; h) =
\ α>0
Ruiyuan Hu (Lehigh)
CBC (1 ·
P ; αh) α
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels10 / 14
BC to MAC (Constant)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels11 / 14
Fading MAC and BC Theorem MAC to BC: The capacity region of a fading Gaussian BC with power constraint P is equal to the union of ergodic capacity Pregions of the dual MAC with power constraint (P1 , . . . , PK ) such that K j=1 Pj = P CBC =
[
CMAC (P; h).
P:1·P=P
Theorem BC to MAC: The capacity region of a fading Gaussian MAC is equal to the intersection of the ergodic capacity regions of the dual BC over all scalings CMAC (P; h) =
\ α>0
Ruiyuan Hu (Lehigh)
CBC (1 ·
P ; αh) α
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels12 / 14
Fading MAC and BC
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels13 / 14
Conclusion
Duality between the Gaussian MAC and BC is defined by establish fundamental relationships between the capacity regions of the MAC and BC with the same channel gains and the same noise power at all receivers.
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels14 / 14
April 5, 2005
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 1 / 14
Purpose Capacity Region of the Gaussian BC is equal to the union of capacity regions of the dual MAC, where the union is taken over all individual power constraint that sum up to the BC power constraint. [ CMAC (P1 , . . . , PK ; h). CBC = {Pi }K 1 :
P
K i=1
Pi =P
MAC capacity region is equal to the intersection of dual BC capacity regions. ! K \ X Pi CMAC (P 1 , . . . , P K ; h) = CBC ; α1 h1 , . . . , αK hK αi K {αi }1 : αi >0
Ruiyuan Hu (Lehigh)
i=1
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 2 / 14
Outline
System Model Capacity Region Analysis Capacity Region of the MAC Capacity Region of the BC
Constant MAC and BC MAC to BC BC to MAC
Fading MAC and BC MAC to BC BC to MAC
Conclusion
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 3 / 14
System Model
BC: Yj [i] =
q
hj [i]X [i] + nj [i]
MAC: Y [i] =
K q X hj [i]Xj [i] + n[i] j=1
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 4 / 14
Capacity Region Analysis (MAC)
The capacity region of a Gaussian MAC with channel gains h = (h1 , . . . , hK ) and power constraints P = (P1 , . . . , PK ) is X 1 1 X R: Rj ≤ log(1 + 2 hj P j ) ∀S ⊆ {1, . . . , K } 2 σ j∈S
j∈S
With success decoding with interference cancellation, the corner points are ! hπ(j) P π(j) 1 Rπ(j) = log 1 + , j = 1, . . . , K . P 2 σ2 + K i=j+1 hπ(i) P π(i)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 5 / 14
Capacity Region Analysis (BC)
The capacity region of a Gaussian BC with channel gains h = (h1 , . . . , hK ) and power constraints P is (
B
hj P j 1 P R : Rj ≤ log 1 + 2 2 σ + hj k=1 KPkB 1[hk > hj ]
Ruiyuan Hu (Lehigh)
!
) , j = 1, . . . , K
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 6 / 14
MAC to BC (Constant) Theorem The capacity region of a constant Gaussian BC with power constraint P is equal to the union of capacity regions P of the dual MAC with power constraint (P1 , . . . , PK ) such that K j=1 Pj = P CBC =
[
CMAC (P; h).
P:1·P=P
Corollary The capacity region of a constant Gaussian MAC with power constraints P = (P1 , · · · , PK ) is a subset of the capacity region of the dual BC with power constraint P = 1 · P CMAC (P; h) ⊆ CBC (1 · P; h) Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 7 / 14
MAC to BC (Constant)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 8 / 14
MAC to BC (Constant)
Channel Scaling: from (P; h) to ( P α ; αh)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, Channels 2005 9 / 14
BC to MAC (Constant)
Theorem The capacity region of a constant Gaussian MAC is equal to the intersection of the capacity regions of the scaled dual BC over all possible channel scalings CMAC (P; h) =
\ α>0
Ruiyuan Hu (Lehigh)
CBC (1 ·
P ; αh) α
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels10 / 14
BC to MAC (Constant)
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels11 / 14
Fading MAC and BC Theorem MAC to BC: The capacity region of a fading Gaussian BC with power constraint P is equal to the union of ergodic capacity Pregions of the dual MAC with power constraint (P1 , . . . , PK ) such that K j=1 Pj = P CBC =
[
CMAC (P; h).
P:1·P=P
Theorem BC to MAC: The capacity region of a fading Gaussian MAC is equal to the intersection of the ergodic capacity regions of the dual BC over all scalings CMAC (P; h) =
\ α>0
Ruiyuan Hu (Lehigh)
CBC (1 ·
P ; αh) α
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels12 / 14
Fading MAC and BC
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels13 / 14
Conclusion
Duality between the Gaussian MAC and BC is defined by establish fundamental relationships between the capacity regions of the MAC and BC with the same channel gains and the same noise power at all receivers.
Ruiyuan Hu (Lehigh)
On the Duality of Gaussian Multiple-Access and Broadcast April 5, 2005 Channels14 / 14
