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1
Codage avec Information Adjacante (DPC : Dirty paper coding)
et certaines de ses applications : Tatouage (Watermarking)MIMO broadcast channels
Gholam-Reza MOHAMMAD-KHANI
2
Gel’fand and Pinsker’s channel
Channel definition
Channel capacity (Gel’fand and Pinsker 1980)
Encoder
3
Gaussian case (DPC) Channel description (Dirty paper coding - Costa 1983)
Coding
4
Gaussian case (DPC) Channel description (Dirty paper coding - Costa 1983)
Coding
S
EncoderW U X
5
DPC Application for Watermarking Channel description (Dirty paper coding - Costa 1983)
Watermarking Application : X : Mark (Weak Signal) , S : Host (Strong Signal) , Z : Noise Capacity Achieving for Mark Signal
6
Problem statement in MIMO BC
:
Decoder #1
Decoder #K
:
r1 antennas
rK antennas
Y1
YK: Encoder
W1
WK
t antennas
X
p(y|x,H)
H
ZXY
ZXY iii
H
H
H1
HK
PTrXXETrH Σ.
7
Performance Criteria in BC
:
Usual Criteria (Information Theory Aspects) :• Capacity Regions• Throughput (Sum Capacity)
New Criteria (Practical Aspects) :• BER Regions• Number of Satisfied Users (of Rates or of BER)
8
Some Relateds Works
:
-Sato : Upperbound for Sum Capacity of BC
- Cover [72] : Definition of Broadcast Channels
- Weingarten & Shamai [06] : Capacity Region of Gaussian MIMO BC
- Caire & Shamai [03] + Viswanath & Tse [03] + Vishwanath & Goldsmith [03] + Yu & Cioffi [04]:
Achievable Throughput of Gaussian MIMO BC
DPC scheme : Achieve Sum Capacity and Capacity Region for MIMO BC
9
DPC and MIMO BC
:
Decoder #1
Decoder #K
:
r1 antennas
rK antennas
Y1
YK: Encoder
W1
WK
t antennas
X
p(y|x,H)
H
ZXY
ZXY iii
H
H
H1
HK
PTrXXETrH Σ.
i
i
ijjiiii Z
S
XXY HH
10
One Simple Case : Gaussian SISO BC Channel model and capacity region
Superposition coding:
11
DPC vs TDMATheorique Comparison :- Jindal & Goldsmith [05] :
Best performance of DPC on Sum Capacity
- Weingarten & Shamai [06] :
Best Performance of DPC on Capacity Region
Practical Comparison :- Belfiore [06]
- Mohammad-Khani & Lasaulce [06]
Sensibility to Channel Estimation
BER Comparison
12
Structure of DPC schemes for Gaussian MIMO BCs
Outer encoders Tomlinson Harashima precoder (THP)Scalar Costa’s scheme (SCS)Trellis coded quantization (TCQ) + turboNested lattices
Encoder structure
Inner Encoder
Outer Encoder:
W1
WK
X
H
Outer encoders : LinearPre-equalizers: MF, ZF, MMSEZF-DPC MMSE-DPC
X~
13
Structure of DPC schemes for Gaussian MIMO BCs Encoder structure
14
Comparison of outer coders
0 5 10 1510
-6
10-5
10-4
10-3
10-2
10-1
100
SNR
BE
R
THPSCSNL, A2HexagonalTCQ
15
Inner coding
CommentsInner coding space-time coding or beamformingInner + outer coding implements a good multiple access scheme
Received signal structure
PuEK
ip
i
i
1
2 ; Kizuuuy i
ijjji
ijjjiiiii ,,1 , ; ,,,
K
ii
K
iiiK
i
u11
1 , . ; . xBxBBBuBxzxHy
x
zuWHBy .
Possible approachesLinear precoding with successive coding using DPC as outer coding (the outer coder treats the interference)
Linear pre-equalizer with independent outer coder (the outer coder does not treat the interference)
ij jji
iiii
p
pR 2
,
2,
11log
ij jji
iiii
p
pR 2
,
2,
11log
16
MMSE-DPC
Main featuresOptimum in the sense of the sum-capacityTwo ways of implementing it:
Yu & Cioffi 04 (GDFE precoder)Viswanath & Tse 03 (duality BC – MAC)
Precoding filters depend on power allocation
Coding order: no effect on sum capacity (not true for the capacity region)Power allocation: we used the policy proposed by Boche & Jorswieck 04 (corresponding numerical algorithms converge)
PHHIP
†det log sup A
sumC
Kkp k
K
kjjjjk ,,1 , †
1
1
†
hhhIB
Kopt ppdiag ,, 1 P Numerical technique
17
ZF-DPC Main features
Introduced by Caire & Shamai 03 (for single-antenna receivers)
We generalized this scheme to multi-antenna receivers Simpler than MMSE-DPC but suboptimum in terms of sum-capacityQuasi-optimal in terms of sum-capacity, when H is full row rank
Number of served users limited to rank of HSensitive to coding order
†QBG.QH
)( ;
,,1 ,
2,
,,
Hrankmgd
mizugugy
iii
iij
jjiiiii
m
ii
m
ii
zfdp
Pd
dR
1
1
1
log
Waterfilling :
18
Influence of the coding order: example
ConclusionsCoding order has no effect on sum rate for MMSE-DPCSum rate of ZF-DPC strongly depends on coding order Coding order can be optimized by a greedy algorithm [Tu & Blum03]If the coding order is not well chosen: TDMA can perform better than DPC (especially for low SNRs)
-10 -5 0 5 10 15 200
2
4
6
8
10
12
SNR(dB)
Sum
Ca
pac
ity
(bit
s)
Influence of coding order on Sum Capacity
TDMAMIMOSato BoundAchieved by MMSE-DPCZF-DPC : best order, (1,3,2)ZF-DPC : worst order , (3,2,1)
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
SNR
Sum
Ra
te (
bit
s)
Influence of coding order on sum rate
TDMA
Sato Upper Bound
MIMO
order (1,2,3)
order (1,3,2)
order (2,1,3)
order (2,3,1)
order (3,1,2)
ZF-DPC : order (3,2,1)
211
023
005
H
19
Conventional pre-equalizers Definitions
ZF :
MMSE :
MF :
Comments The outer coder does not help to the interference cancellation task (separate coding)No successive coding = no coding order Most simple schemes when the CSI is known
mr
mrtI
I
si , )(
si , .1††
H
HHHHB
)(
,,1 ,
Hrankm
mizuy iii
†1† H.P.HHIB
†HB Numerical Method to compute Sum Rate
Water-Filling
20
Comparison of inner coders (1/2)
120
211
023
HSum Rate Comparison
-5 0 5 10 15 20 25 300
5
10
15
20
25
30
35
SNR(dB)
Cap
aci
ty (
bits
)
Sum Capacity Comparison of BC-MIMO
Sato : DP-OptZF-DPCMMSEMFZFTDMA
12
21
13
H
-25 -20 -15 -10 -5 0 5 100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
SNR(dB)
Sum
Ca
pac
ity
(bit
s)
Sum Capacity Comparison of BC-MIMO
ZF-DPCSato Upper BoundMMSE-DPC
21
Region of achieved Rate ComparisonComparison of inner coders (2/2)
2 trK
11
23H
14.0
4.01H2 trK
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
R1
R2
Region of Capacity
TDMA
ZF-DPC
ZF
MMSE
MF
MMSE-DPC : 1,2
MMSE-DPC : 2,1
Sum Capacity
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
R1
R2
Region of Capacity
TDMA
ZF-DP : 1,2
ZF-DP : 2,1
ZF
MF
MMSE-DP : 1,2
MMSE-DP : 2,1
MMSE
Sum Capacity
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R1
R2
Region of Capacity
TDMA
ZF-DP : order 1,2
ZF-DP : order 2,1
ZF
MF
MMSE-DP : 1,2
MMSE-DP : 2,1
MMSE
data10
P=7dB
P=20dB
P=10dB
22
Overall performance (1/2)Degraded channel (No need to inner coder)
0 0.2 0.4 0.6 0.8 110
-6
10-5
10-4
10-3
10-2
10-1
100
beta
BE
R
user1 : P=20
user1 : P=1
user2 : P=20
user2 : P=1
mean : P=20
mean : P=1
user1 : P=20, SCS
user2 : P=20, SCS
P=20
P=1
N1=0dBN2=N1+5dB
Application de TCQ pour un BC scalaire dégradé 2 utilisateurs
22122
12111
ZXXZXY
ZXXZXY
x2y2 Viterbi
Decoder
1w
2w2w
1w
y1
TCQ
TCQu1 x1
u2
xz1
z20
1
1111 NPP
ViterbiDecoder
10 ; 12 ; 1
, , 2221
21
2
.PPPP
NZENZEPXE
23
Overall performance (2/2)
10-4
10-3
10-2
10-1
100
10-4
10-3
10-2
10-1
100
Pe1
Pe2
MFMMSE-DPMMSEZFZF-DP
14.0
4.01HdBPtrK 10 ,2
0 2 4 6 8 1010
-4
10-3
10-2
10-1
100
p1
BE
R
P=10dB
ZF-DPC : user 1
ZF-DPC : user 2
ZF : user 1
ZF : user 2
MMSE : user 1
MMSE : user 2
MF : user 1
MF : user 2
MMSE-DP : user 1
MMSE-DP : user 2
THP: user1
THP :user2