linear-feedback mac-bc duality for correlated bc …...linear-feedback mac-bc duality for correlated...

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Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr} Based on joint work with Michèle Wigger {michele.wigger@telecom-paristech.fr} 53rd Annual Allerton Conference on Communication, Control, and Computing October 2, 2015 Contributions BC-noises correlation factor λ 2 [-1, 1] MAC with “non-standard” sum-power constraint depending on λ MAC-BC Duality with Correlated Noises at BC-Receivers C linfb BC (h 1 , h 2 , λ; P )= C linfb BC (h 1 , h 2 ; λ, P ). Transfer MAC results to BC I (Ozarow’84) = ) Achievable region for BC with correlated noises Constructive Sum-Rate Optimal BC-Scheme (λ = 0) Ozarow’s MAC-encoders and MAC-decoders “rearranged” ) Constructive sum-rate optimal BC-scheme. 1 / 18

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Page 1: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Linear-Feedback MAC-BC Duality for CorrelatedBC-Noises, and Iterative Coding

Selma Belhadj Amor{[email protected]}

Based on joint work with Michèle Wigger{[email protected]}

53rd Annual Allerton Conference on Communication, Control, and Computing

October 2, 2015

Contributions

BC-noises correlation factor � 2 [�1, 1]MAC with “non-standard” sum-power constraint depending on �

MAC-BC Duality with Correlated Noises at BC-Receivers

C linfbBC (h1, h2,�;P) = C linfb

BC (h1, h2;�,P).

Transfer MAC results to BCI (Ozarow’84) =) Achievable region for BC with correlated noises

Constructive Sum-Rate Optimal BC-Scheme (� = 0)Ozarow’s MAC-encoders and MAC-decoders “rearranged” ) Constructivesum-rate optimal BC-scheme.

1 / 18

Page 2: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Two-User Gaussian BC with Perfect Feedback

�⌦

⌦h2

Transmitter(M1,M2)

h1

xt

Z1,t

Z2,t

Receiver 1M1

Receiver 2M2

Y1,t

Y2,t

Y1,t

Y2,t

Transmitter sends M1 to Receiver 1 and M2 to Receiver 2.

Messages M1 and M2 independent ; Mi ⇠ U{1, . . . , 2nRi}

Pr(error) = Pr⇢(M1 6= M1) or (M2 6= M2)

2 / 18

Two-User Gaussian BC with Perfect Feedback

�⌦

⌦h2

Transmitter(M1,M2)

h1

xt

Z1,t

Z2,t

Receiver 1M1

Receiver 2M2

Y1,t

Y2,t

Y1,t

Y2,t

h1, h2 2 R deterministic, non-fading; Xt ,Y1,t ,Y2,t 2 R

Yi,t = hixt + Zi,t ,0

@Z1,t

Z2,t

1

A i.i.d. ⇠ N

0

@0,

0

@1 �

� 1

1

A

1

A where � 2 [�1, 1]

BC is physically-degraded if: � = h1h2

or � = h2h1

.

2 / 18

Page 3: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Two-User Gaussian BC with Perfect Feedback

�⌦

⌦h2

Transmitter(M1,M2)

h1

xt

Z1,t

Z2,t

Receiver 1M1

Receiver 2M2

Y1,t

Y2,t

Y1,t

Y2,t

Power constraint: E⇥kXnk2⇤ nP ; Xn , (X1, . . . ,Xn)

T

Perfect output feedback:

Xt = '(n)t (M1,M2,Y1,1, . . . ,Y1,t�1,Y2,1, . . . ,Y2,t�1), t 2 {1, . . . , n}.

2 / 18

Capacity of Gaussian BC with Feedback

Capacity region CfbBC unknown

Sum-capacity C

fbBC,⌃ at high SNR: (limP!1 C

fbBC,⌃ � CHigh = 0)

(Gastpar et al.’14)

CHigh =

8>>>>><

>>>>>:

12

log✓

1 +h

21 + h

22 � 2h1h2�

1 � �2 P

◆if �1 < � < 1,

12

log�1 + h

21P

�+

12

log�1 + h

22P

�if � 2 {�1, 1} and h1 6= �h2

12

log�1 + h

21P

�if � 2 {�1, 1} and h1 = �h2.

3 / 18

Page 4: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Achievable Regions with Feedback for Gaussian BC

'

&

$

%

Ozarow/Leung’84, Kramer’02: LMMSE-based Schalwijk-Kailath-type

schemes

Elia’04, Wu et al.’05, Ardestanizadeh et al.’12: Control-theory based schemes

Gastpar et al.’11,

Linear-Feedback Schemes

◆✓

⇣⌘

Venkataramanan/Pradhan’11, Shayevitz/Wigger’13, Wu/Wigger’14

Non-Linear-Feedback Schemes

4 / 18

Achievable Regions with Feedback for Gaussian BC

'

&

$

%

Ozarow/Leung’84, Kramer’02: LMMSE-based Schalwijk-Kailath-type

schemes

Elia’04, Wu et al.’05, Ardestanizadeh et al.’12: Control-theory based schemes

Gastpar et al.’11,

Linear-Feedback Schemes! best regions!

◆✓

⇣⌘

Venkataramanan/Pradhan’11, Shayevitz/Wigger’13, Wu/Wigger’14

Non-Linear-Feedback Schemes

4 / 18

Page 5: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Linear-Feedback Schemes (LFSs) for BC

Feedback used linearly:

Xn = Wn + A1,BCYn1 + A2,BCYn

2 ,

A1,BC,A2,BC: strictly lower-triangular matrices,

Xn , (X1 . . .Xn)T, Yni , (Yi,1 . . .Yi,n)T, Wn , ⇠(n)(M1,M2) 2 Rn.

Decoding can be arbitrary

C linfbBC (h1, h2,�;P) = all rates achievable with LFCS over MIMO BC

/ Optimization step to determine C linfbBC is very difficult.

! Tricky part: Identify optimal feedback matrices A?1,BC and A?

2,BC.

5 / 18

Two-User Memoryless Gaussian MAC with Feedback

Transmitter 1M1h1x1,t

h2x2,tTransmitter 2M2

Zt

(M1, M2)ReceiverYt

Yt

Yt

Transmitters 1 and 2 send independent messages M1 and M2 to ReceiverMi ⇠ U{1, . . . , 2nRi},

Pr(error) = Pr⇢(M1, M2) 6= (M1,M2)

6 / 18

Page 6: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Two-User Memoryless Gaussian MAC with Feedback

Transmitter 1M1h1x1,t

h2x2,tTransmitter 2M2

Zt

(M1, M2)ReceiverYt

Yt

Yt

hi 2 R: deterministic non-fading,X1,t ,X2,t ,Yt 2 R,Yt = h1X1,t + h2X2,t + Zt ,

{Zt} i.i.d.⇠ N (0, 1).

6 / 18

Two-User Memoryless Gaussian MAC with Feedback

Transmitter 1M1h1x1,t

h2x2,tTransmitter 2M2

Zt

(M1, M2)ReceiverYt

Yt

Yt

Xni , (Xi,1 . . .Xi,n)T, i 2 {1, 2}.

Sum-power constraint:

E⇥kXn

1k2⇤+ E⇥kXn

2k2⇤ nP .

Perfect output feedback:

Xi,t = '(n)i,t (Mi ,Y1, . . . ,Yt�1), i 2 {1, 2}, t 2 {1, . . . , n}.

6 / 18

Page 7: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Linear-Feedback Schemes (LFSs) for MAC

Feedback used linearly:

Xni = Vn

i + Ai,MACYn, i 2 {1, 2},

A1,MAC,A2,MAC: strictly lower-triangular matrices,Xn

i , (Xi,1 . . .Xi,n)T, Yn , (Y1 . . .Yn)T, Vni , '(n)

i (Mi ) 2 Rn.

C linfbMAC(h1, h2;P) = all rate pairs achievable with LFSs over MAC.

7 / 18

Linear-Feedback Capacity for MAC

Ozarow’84: feedback capacity region under individual power constraintssum-power constraint P �! union over all P1 + P2 = P

LMMSE-based LFS that achieves (linear-) feedback capacity:

CfbMAC(h1, h2;P) = C linfb

MAC(h1, h2;P) =

[

P1,P2�0:P1+P2=P

[

⇢2[0,1]

8>>>>><

>>>>>:

(R1,R2) :

R1 12

log�1 + h

21P1(1 � ⇢2)

�,

R2 12

log�1 + h

22P2(1 � ⇢2)

�,

R1 + R2 12

log�1 + h

21P1 + h

22P2 + 2

qh

21h

22P1P2⇢

�.

9>>>>>=

>>>>>;

8 / 18

Page 8: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Previous MAC-BC Duality: Independent BC-Noises (� = 0)

MAC-BC Duality with Linear-Feedback Schemes—Independent BC-noisesBelhadj Amor/Steinberg/Wigger’14

C linfbBC (h1, h2, 0;P) = C linfb

MAC(h1, h2;P) = CfbMAC(h1, h2;P).

Explicit expression of C linfbBC (h1, h2, 0;P).

Feeback always increases capacity of Gaussian BC with independent noises.

9 / 18

Previous MAC-BC Duality: Independent BC-Noises (� = 0)

MAC-BC Duality with Linear-Feedback Schemes—Independent BC-noisesBelhadj Amor/Steinberg/Wigger’14

C linfbBC (h1, h2, 0;P) = C linfb

MAC(h1, h2;P) = CfbMAC(h1, h2;P).

Explicit expression of C linfbBC (h1, h2, 0;P).

Feeback always increases capacity of Gaussian BC with independent noises.

What happens if the BC-noises are correlated (� 6= 0)?

9 / 18

Page 9: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Dual MAC with “Non-Standard” Power Constraint

Transmitter 1M1h1x1,t

h2x2,tTransmitter 2M2

Zt

(M1, M2)ReceiverYt

Yt

Yt

� 2 [�1, 1]: BC-noise correlation factor“Non-standard” sum-power constraint:

E⇥kXn

1k2⇤+ E⇥kXn

2k2⇤+ 2�E[hXn1,Xn

2i] nP (1)

C linfbMAC,CorrPower (h1, h2;�,P): all rates achievable with LFSs over this MAC.

10 / 18

Main Results

MAC-BC Duality for BC with Correlated Noises

C linfbBC (h1, h2,�;P) = C linfb

MAC,CorrPower (h1, h2;�,P) .

11 / 18

Page 10: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Main Results

Let |h1| � |h2|.Assume BC is physically degraded, i.e. � = h2

h1.

MAC power constraint:

E⇥kh1Xn

1 + h2Xn2k2⇤

| {z }received power

+(h21 � h

22)E

⇥kXn

2k2⇤ h

21nP

Since (h21 � h

22)E

⇥kXn

2k2⇤� 0 ) E

⇥kh1Xn

1 + h2Xn2k2⇤

| {z }received power

h

21nP .

With and without feedback

CMAC,⌃ 12

log✓

1 + (1n

)E⇥kh1X1 + h2X2k2⇤

◆ 1

2log(1 + h

21P)

Gastpar’04: Feedback does not increase CMAC,⌃ under consideration.! Feedback does not increase sum-capacity of dual physically-degraded BC.

12 / 18

Main Results

Adapt Ozarow’84’s scheme under ind. powers P1 and P2 to our dual MAC:

I Choose P1 � 0 and P2 � 0 so that

P1 + P2 + 2�⇢pP1P2 P,

for some parameter ⇢ 2 [0, ⇢?(h1, h2,P1,P2)].

⇢?(h1, h2,P1,P2) is the unique solution in (0, 1) to:

(1+h

21P1(1�x

2))(1+h

22P2(1�x

2)) = 1+h

21P1+h

22P2+2|h1||h2|x

pP1P2.

13 / 18

Page 11: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Main Results

New Achievable Region over BC with Feedback and Correlated Noises

R1 12

log(1 + h

21P1(1 � ⇢2))

R2 12

log(1 + h

21P2(1 � ⇢2))

R1 + R2 12

log(1 + h

21P1 + h

22P2 + 2|h1||h2|⇢

pP1P2)

for some P1,P2, and ⇢ 2 [0, 1] satisfying

P1 + P2 + 2�⇢pP1P2 P ,

for some parameter ⇢ 2 [0, ⇢?(h1, h2,P1,P2)].

14 / 18

Proof–Step 1: Optimal Block-Feedback Schemes

Class of MAC- and BC-block-feedback schemes is optimal

I Divide the blocklength into subblocks of length-⌘

I Inner Code (described by {A1,MAC,A2,MAC} for MAC and {A1,BC,A2,BC} for

BC):

F uses feedback linearly.

F transforms ⌘ channel uses into 1 channel use of new super Gaussian MIMO

MAC or BC

I Outer Code:

F ignores feedback

F codes to achieve nofeedback capacity of super Gaussian MIMO MAC or BC

! Multi-letter expressions for C linfbMAC,CorrPower (h1, h2;�,P) and C linfb

BC (h1, h2,�;P)

15 / 18

Page 12: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Proof–Step 2: Dual Optimal Block-Feedback Schemes

Identify pairs of MAC- and BC-block-feedback schemes that are dual! For each MAC-params, 9 BC-params achieving the same region, vice-versa.

If we chooseAi,BC = Ai,MAC,

then super Gaussian MIMO MAC and BC have same nofeedback capacity.

(C: mirror image of C along counter-diagonal)

I Proof: nofeedback MAC-BC duality & equivalence relations on capacityregions of super Gaussian MIMO channels

Ozarow’84 �! Optimal BC parameters

16 / 18

Proof–Step 2: Dual Optimal Block-Feedback Schemes

Identify pairs of MAC- and BC-block-feedback schemes that are dual! For each MAC-params, 9 BC-params achieving the same region, vice-versa.

If we chooseAi,BC = Ai,MAC,

then super Gaussian MIMO MAC and BC have same nofeedback capacity.

(C: mirror image of C along counter-diagonal)

I Proof: nofeedback MAC-BC duality & equivalence relations on capacityregions of super Gaussian MIMO channels

Ozarow’84 �! Optimal BC parameters There is more than duality ofachievable regions!

16 / 18

Page 13: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Main Results:Constructive Sum-Rate Optimal BC-Scheme with Independent Noises

Ozarow’s MAC linear-feedback-sum-rate optimal scheme:

Oz-Enc 2

Oz-Enc 1 Oz-Dec 1

Oz-Dec 2

M1

M2

M1

M2

MAC-Tx 1

h1x1,t

h2x2,t

MAC-Tx 2

Zt

MAC-RxMAC-Channel

Yt

Yt

Yt

BC-Channel

BC-Rx 1

BC-Rx 2

Y1,t

Y2,t

Y1,t

Y2,t

h2

BC-Tx

h1

xt

Z1,t

Z2,t

17 / 18

Main Results:Constructive Sum-Rate Optimal BC-Scheme with Independent Noises

We can use same encoding and decoding functions!

Oz-Enc 2

Oz-Enc 1 Oz-Dec 1

Oz-Dec 2

M1

M2

M1

M2

MAC-Tx 1

h1x1,t

h2x2,t

MAC-Tx 2

Zt

MAC-RxMAC-Channel

Yt

Yt

Yt

BC-Channel

BC-Rx 1

BC-Rx 2

Y1,t

Y2,t

Y1,t

Y2,t

h2

BC-Tx

h1

xt

Z1,t

Z2,t

17 / 18

Page 14: Linear-Feedback MAC-BC Duality for Correlated BC …...Linear-Feedback MAC-BC Duality for Correlated BC-Noises, and Iterative Coding Selma Belhadj Amor {selma.belhadj-amor@inria.fr}

Main Results:Constructive Sum-Rate Optimal BC-Scheme with Independent Noises

Our new linear-feedback-sum-optimal BC-scheme:

Oz-Enc 2

Oz-Enc 1 Oz-Dec 1

Oz-Dec 2

M1

M2

M1

M2

MAC-Tx 1

h1x1,t

h2x2,t

MAC-Tx 2

Zt

MAC-RxMAC-Channel

Yt

Yt

Yt

BC-Channel

BC-Rx 1

BC-Rx 2

Y1,t

Y2,t

Y1,t

Y2,t

h2

BC-Tx

h1

xt

Z1,t

Z2,t

17 / 18

Summary

BC-noises correlation factor � 2 [�1, 1]MAC with “non-standard” sum-power constraint depending on �

MAC-BC Duality with Correlated Noises at BC-Receivers

C linfbBC (h1, h2,�;P) = C linfb

BC (h1, h2;�,P).

Transfer MAC results to BCI (Ozarow’84) =) Achievable region for BC with correlated noises

Constructive Sum-Rate Optimal BC-Scheme (� = 0)Simple rearrangement Ozarow’s MAC-encoders and MAC-decoders achievesC

linfbBC,⌃ for non-symmetric BC.

18 / 18