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CONSTELLATION MAPPING FOR EPoC LDPC CODING

Presenters: Rich Prodan, BZ Shen and Avi Kliger

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This presentation defines Algebraic formulaic mapping for all possible QAM constellations Recursive procedure starting with BPSK Can be implemented either in hardware or a look-up-table

Gray code mapping for even constellations QPSK, 16-QAM,64-QAM,256-QAM,1024-QAM and 4096-QAM

The least Gray code penalty mapping for odd constellations 8-QAM,32-QAM,128-QAM,512-QAM and 2048-QAM

The defined constellation mappings demonstrate consistently good performance on EPoC LDPC codes Long size (16200,14400) code for both downstream and upstream Medium size (5940,5040) code for upstream Short size (1120,840) code for upstream

CONSTELLATION MAPPING FOR LDPC CODES

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Gray code for binary numbers Listing all n-bit numbers so that successive numbers differ in exactly one bit position There are many Gray code mappings

Proposed Gray code mapping for 2k integers, 2i-(2k-1) (k=0,1,…,2k-1)

k=1:

k>1: Example : k=2:

ONE DIMENSIONAL GRAY CODE MAPPING

G1(0)= 1, G1(1)=-1 -1 1

−=+−=−=+−=

=+==+=

1))1(2( )11(3))0(2( )10(

1))1(2()01(3))0(2()00(

12

12

12

12

GGGG

GGGG

-3 3 -1 1

( )( ) 1,,1,0for 1or 0b ,)(221)( i1111

0110 −==+−= −−−

− kibbGbbbbG kkk

kk

10 11 01 00

1 0

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2n+m Rectangular constellation points {(2i-(2n-1), 2j-(2m-1)) | i=0,1,…,2n-1, j=0,1,…,2m-1}

Mapping 2-D symbols with (n+m) binary bits using 1-D Gray code mapping I and Q are mapped independently (or orthogonal)

Example n=m=1(QPSK)

Special case: square constellation 22n -QAM points {(2i-(2n-1), 2j-(2n-1)) | i=0,1,…,2n-1, j=0,1,…,2n-1}

Example: 16-QAM (see figure on the right)

GRAY CODE MAPPING ON RECTANGULAR CONSTELLATION

( )),(),())(),((Let

110110

110110

−−

−−

==

mmnnrctrct

mn

bbbGaaaGQIbbbaaa

ddd

( ) ( )( ) ( ) )1,1())1(),1(()11(),11( ),1,1())0(),1(()10(),10(

)1,1())1(),0(()01(),01( ),1,1())0(),0(()00(),00(

1111

1111

−−==−==−====

GGQIGGQIGGQIGGQI

rctrctrctrct

rctrctrctrct

-1 1

-1

1

11 01

10 00

16-QAM

QPSK

Qrct(b)

Irct(b)

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Defined by Joel G. Smith “Odd-bit quadrature amplitude shift keying,” IEEE Trans. Commun., vol. COMM-23, no. 3, pp. 385–

389, Mar. 1975 All one-dimensional Gray codes and two-dimensional Gray codes of orthogonal components are pure.

All other Gray codes are “impure,” and suffer a “Gray code penalty”

Mathematic definition

If l provides a Gray code mapping then GP(l)=1, i.e. no penalty The proposed mapping on rectangular constellation has GP=1

GRAY CODE PENALTY

{ }

( )( )

∑∑−

=

∈=

=−=−

12

0

)(

)(

)(),(

21)( : PenaltyCodeGray

)( and )( between distance hamming :)(),(mapping by the given labelingbinary :)(

of neighbors distance) Euclidean(innearest theis |)(12,,1,0, symbols,2 are thereQAM,2

n

ij

i i

SNSij

nP

jiji

i

ijji

ni

nn

SN

SlSlwtlG

SlSlSlSlwtSl

SSSSNiS

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MAPPING ON CROSS CONSTELLATION CASE I: 8-QAM

10/1 11/1 01/1 00/1

10/0 11/0 01/0 00/0

10/1 11/1 01/1 00/1

10/0 11/0 01/0 00/0

00/0

00/1

( ))(),())(),(( 01102

010

bGaaGQIbaa

rctrct ==

ddd

-3 3 -1 1

-1

1

( )( )

=+=−=−=321)000(134)000(

cr

cr

QI ( )

( )

=+−==−−=

321)001(134)001(

cr

cr

QI

( )( )( )

+=−−=

<

==

otherwise2)()()()(4)(()(

3)()()()()(

dddddd

ddddd

rctrctcr

rctrctcr

rectrctcr

rctcr

QQsignQIIsignI

IQQII

Starts with rectangular constellation mapping

Transform the right side wing to cross constellation mapping

-3 -1 1

3 1 -1 -3

3125.11621)21

4511

4512(

81

==+++++++=PG

transform to cross constellation

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Starts with an n x (n+1) rectangular constellation Transform two vertical wings to two horizontal wings Pseudo Gray code mapping on two horizontal wings (with the least Gray Code Penalty) Algebraic formula

See an example in the next page

MAPPING ON CROSS CONSTELLATION CASE II: 22n+1-QAM (n>1)

( )

( )( )( )( )

( )( )( )( )

≤

+=−=

>

−=−=

<

==

=

==

−

−−+

−

otherwise

)(Q2)()()()(4)()(

)(Q)(4)()(

2)()()(

3)( if)()()()(

2Let

),(),())(),((

rct

rct

11101101

11010

ssQQsignQ

IsIsignI

sQsQsignQ

sIIsignI

sIQQII

s

bbbGaaaGQIdbbaaad

rctrctcr

rctrctcr

rctrctcr

rctrctcr

rctrctcr

rctcr

nnnnnrctrct

nn

dddd

ddd

dddd

ddd

ddddd

dd

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MAPPING ON 32 CROSS CONSTELLATION QAM

-7 -5 -3 -1 1 3 5 7

3 1 -1 -3

100/00 101/00 111/00 110/00 010/00 011/00 001/00 000/00

100/01 101/01 111/01 110/01 010/01 011/01 001/01 000/01

100/11 101/11 111/11 110/11 010/11 011/11 001/11 000/11

100/10 101/10 111/10 110/10 010/10 011/10 001/10 000/10

-7 -5 -3 -1 1 3 5 7

5 3 1 -1 -3 -5

100/00 101/00 111/00 110/00 010/00 011/00 001/00 000/00

100/01 101/01 111/01 110/01 010/01 011/01 001/01 000/01

100/11 101/11 111/11 110/11 010/11 011/11 001/11 000/11

100/10 101/10 111/10 110/10 010/10 011/10 001/10 000/10

( )( )

=−==−=538)0000(347)00000(

cr

cr

QI ( )

( )

−=−−==−=

538)00010(347)00010(

cr

cr

QI( )

( )

=+==−=

541)00001(178)00001(

cr

cr

QI ( )

( )

−=+−==−=

541)00011(178)00011(

cr

cr

QI

100/00 000/01

000/10 000/11

( )( )

=−=−=−−=

538)10000(347)10000(

cr

cr

QI ( )

( )

=+=−=−−=

541)10001(178)10001(

cr

cr

QI ( )

( )

−=+−=−=−−=

541)10011(178)10011(

cr

cr

QI ( )

( )

−=−−=−=−−=

538)10010(347)10010(

cr

cr

QI

100/01 000/00

100/11 100/10

Mapping on rectangular constellation

Transform two wings to cross constellation mapping

( )

( )( )( )( )( )( )( )( )

≤

+=−=

>

−=−=

<

==

==

=

otherwise2)(

4)()()()(8)()(

2)()(8)()(

4)()()(

6)( if)()()()(

2Let )(),())(),(( 1022103

10210

dddd

ddd

dddd

ddd

ddddd

dd

rctrctrctcr

rctrctcr

rctrctrctcr

rctrctcr

rctrctcr

rctcr

rctrct

QQQsignQ

IIsignI

QQQsignQ

IIsignI

IQQII

sbbGaaaGQI

bbaaad

( ) 1667.16/7614/54/54/54/512/33/43/42/3322

==++++++++++=PG

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ON LONG SIZE LDPC CODE (W/ 15 ITERATIONS) (DOWNSTREAM AND UPSTREAM)

10 15 20 25 30 35

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SNR (dB)

BE

R

8-QAM16-QAM32-QAM64-QAM128-QAM256-QAM512-QAM1024-QAM2048-QAM4096-QAMAWGN, 15 its.LDPC(16200,14400)

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ON UPSTREAM MEDIUM SIZE LDPC CODE (W/ 15 ITERATIONS)

10 12 14 16 18 20 22 24 26 28 30

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SNR (dB)

BE

R

8-QAM16-QAM32-QAM64-QAM128-QAM256-QAM512-QAM1024-QAMAWGN, 15 its.LDPC (5940,5040)

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ON UPSTREAM SHORT SIZE LDPC CODE (W/ 15 ITERATIONS)

8 10 12 14 16 18 20 22 24 26 2810

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SNR (dB)

BE

R

8-QAM16-QAM32-QAM64-QAM128-QAM256-QAM512-QAM1024-QAMAWGN, 15 its.LDPC(1120,840)

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A recursive constellation mapping procedure for both downstream and upstream LDPC codes has been presented Procedure can be implemented either by hardware logic or by look-up-tables On even-bit QAM constellations, i.e. 22n-QAM Gray code mapping

On odd-bit QAM constellations, i.e. 22n+1-QAM Cross constellation defined Transform from the Gray code mapping of same size rectangular constellation Mapping with the least Gray code penalty (pseudo-Gray code)

Performance Proposed constellation mappings to all LDPC codes were thoroughly simulated Performance across all codes is good and comparable SNR for threshold Bit Error Ratio is approximately 3 dB apart for two consecutive constellations

Proposal to adopt this constellation mapping procedure for EPoC LDPC coding

CONCLUSION

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Move to: Adopt the constellation mapping procedure in prodan_3bn_02_1113.pdf for EPoC. Moved: BZ Shen Second:

PROPOSED MOTION

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Thank You