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