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