a novel technique for improving the performance of turbo codes using orthogonal signalling,...

A Novel technique for Improving the Performance of Turbo Codes using Orthogonal signalling, Repetition and Puncturing

by Narushan Pillay

Supervisor: Prof. HongJun Xu

© CSIR 2006 www.csir.co.za

Breakdown of Presentation

• Structure of a new scheme Repeat-Punctured Superorthogonal Convolutional Turbo Code (RPSCTC) Encoder

• Procedure of Encoding – Parallel Concatenated Superorthogonal Recursive Convolutional Constituent Codes

• Structure of the RPSCTC Decoder• Procedure for Decoding utilizing the MAP algorithm

• Importance of Interleaving and its effects


Breakdown of Presentation

• A variation of the scheme – Dual-Repeat-Punctured Turbo Code (DRPSCTC)

• Performance Evaluation for RPSCTC using transfer function bounding techniques

• Simulation Results and Bounds for AWGN and Rayleigh Fading Channels – RPSCTC, DRPSCTC versus an existing scheme Superorthogonal Convolutional Turbo Codes (SCTC)


Encoding Structure of RPSCTC

• Structure that of a Parallel Concatenated Code (PCC)• Two constituent codes, two parity sequences• Separated by repeat and interleaver structures• Puncturing mechanism to control the code rate

S R C C 1

S R C C 2

kd

)(2,11,10,1 1...,Nmyyy

)(2,22,20,2 1...,LNmyyy

C o n s t itu e n t C o de 1


N )(2 1 Nm

)(2 1 LNmLN

N

LN

R ep ea t 'L '

LN

P u n c tu r e

Figure 1. Low-level Encoding structure of RPSCTC


Procedure of Encoding for RPSCTC

0a 1a 2a 3a

W als h - Had ar m ar d G en er a to r

F ir s t p ar itys eq u en c e

0a 1a 2a 3a

W als h - Had ar m ar d G en er a to r

R ep ea t 'L '

P u n c tu r e

S ec o n d p u n c tu r edp ar ity s eq u en c e

C o n s titu en t C o d e 1

C o n s titu en t C o d e 2

Nm 12

LNm 12

N

LN

LN

Nm 12

I n f o r m atio nS o u r c e

• More detailed structure of the RPSCTC encoder

• Identical constituent Superorthogonal Recursive Convolutional encoders

Figure 2. Detailed structure of encoder


Structure of RPSCTC Decoder

D ec o d er 1C o r r u p tedp ar ity s eq u en c e 1

ydL k |1

-Ad d d u m m y

b its

channelL

R ep eat 'L '

D ec o d er 2

ydL k |2

12eL

Ad d d u m m yb itsC o r r u p ted

p ar ity s eq u en c e 2

Av er ag e

-channelL

21eL

1

1 Av er ag e

LNea LL 12

21eL21ea LL

21ea LL

Nddd~

,...,~

,~

21

ydL k |2

ydL k |2

E s tim ate o fm es s ag e s eq u en c e

C o m p ar a to r

1

2

12

11 1,...,, p

N

ppmyyy

1

2

12

11 1,...,, p

LN

ppmyyy

LNeL 12

LNea LL 12

• Two constituent Maximum a-posteriori (MAP) decoders

• Mutual exchange of soft extrinsic

information• Cooperative network

Figure 3. Structure of the Decoder


Procedure for Decoding for RPSCTC

• The MAP algorithm• Somewhat like the Viterbi

algorithm but trellis traversed in two directions.

mk),0(

1mb

k

),1(1

mbk

),0(,01

mbk

),1(,11

mbk

1k k

mk

),0(1

mfk

),1(1

mfk

mk

,0

mk,1

k 1k

),1(,11

),1(1

),0(,01

),0(1

mbk

mbk

mbk

mbk

mk

),1(1

,1),0(1

,0 mfk

mk

mfk

mk

mk

mikk

ikk

ik

mik vyux ,, exp

Figure 4. The MAP algorithm



Starting with the log-likelihood ratio (LLR)

m

mk

m

mkkdL ,0,1 loglog)ˆ(

With the likelihood ratios defined by

m

mfk

mk

mk

m

mk

),1(1

,1,1

m

mfk

mk

mk

m

mk

),0(1

,0,0

…………….........……………..(1)

…………….........……………..(2)

…………….........……………..(3)



Extrinsic information from decoder 1 given by

)()()()( 12221 kekckdecoderke dLxLdLdL

Extrinsic information from decoder 2 given by

)()()()( 112 kakckdecoderke dLxLdLdL

……………...………..(4)

……………..….…….…..(5)


Interleaving

B = S m all

B = L ar g e

B =

Mul

tiplic

ity

D is tan c e

• Conventional turbo coding frame length set equal to interleaver size.

• Increase in frame length – performance increase

• Limit on frame length:• Transmission delay• Decoding delay• Hardware delay

• Use repeat block prior to interleaving

• Frame length constant – larger interleaver size – better performance

Figure 5. Effect of Interleaving


Interleaving

S R C C 1

S R C C 2

kd

)(2,11,10,1 1...,Nmyyy

)(2,22,20,2 1...,LNmyyy



N )(2 1 Nm

)(2 1 LNmLN

N

LN

R ep ea t 'L '

LN

P u n c tu r e

Figure 6. Effect of Interleaving

• Repeat ‘L’ block allows for the use of a larger interleaver.

• Spectral thinning• Weight one sequence – weight two sequence –

high weight turbo codeword.


Interleaving

• Weight two sequence – weight four sequence – still yields a high weight turbo codeword.


Performance Evaluation of RPSCTC

Obtain the state transition matrix from state diagram using equation (6)

jz

dilz

dil

jdildil

DILDIL

DILDIL

DILA

,0,

,00,0

),,(

Transfer function can be expressed as equation (7):

……………..…..(6)

…………….........……..(7)

0 0 0

),,(),,(l i d

dil diltDILDILT



Since the transfer function is defined by equation (8),

and,

132 )(... AIAAAI

…………….........………..(8)

……………...........…………..(9)

mmDILAIDILT0,0

1])),,([(),,(



Then the probability of producing a codeword fragment of weight d given a random input sequence of weight i is given by equation (10) for component encoder 1 and equation (11) for component encoder 2.

.),,(

),,(

),,()|( 1

1

11

1

i

N

diNt

diNt

diNtidp

d

.),,(

)|( 22

Li

LN

dLiLNtidp

………..........(10)

……………..............……..(11)



Equation (12) is used to achieve the union bound

n

dd i d dbit dpidpidp

i

N

N

ip

min 1 2221 )()|()|(

where,

)/2()(2 ob NdREQdp

for an AWGN channel.

……..(12)

……………...........……..(13)



and p2(d) is given by (14) for side-information (SI) or equation (15) for no side-information (NSI) for a Rayleigh fading channel,

d

os

SI

NEdp

/1

1

2

1)(2

d

NSI edp

1/21

1/21.

2

1)( /

2

12 ob NRE /

where

and

……………………..….….(14)

……………....…....(15)

…….……...…...(16)


Another scheme

I n f o r m atio nS o u r c e

R ep eat 'L ' S R C C 1

S R C C 2

P u n c tu r e

P u n c tu r e

1

2

N

LN

LNm 12

LNm 12

Nm 12

Nm 12

Figure 7. DRPSCTC Encoder

• Structure of the Dual-Repeat-Punctured Turbo Code (DRPSCTC) encoder

• Dual repetition prior to encoding• Puncturing at both output branches


Another scheme

D ec o d er 1

D ec o d er 2

C o r r u p tedp ar ity s eq u en c e o n ew ith d u m m y b itsa t p u n c tu r e p o s it io n s

11 2

C o r r u p tedp ar ity s eq u en c e tw ow ith d u m m y b itsa t p u n c tu r e p o s it io n s 1

Av er ag e

12

12

Av er ag e

R ep ea t 'L '

Av er ag e

R ep ea t 'L '

S o f t D ec is io n s

LNm 12

LNm 12

N

-

channelL

-channelL

C o m p ar a to r

ydL kLN |1 LN

eL 12 LNeL 12

LNeL 12

LNeL 12

LNea LL 12

LNea LL 12

12eL ydL k

LN |2

LNeL 21

LNeL 21

LNeL 21

21eL

LNeL 21

LNea LL 21

LNea LL 21

ydL kLN |2 ydL k

LN |2 ydL k |2

• Decoder for DRPSCTC

•Slightly greater complexity

•Better performance

Figure 8. DRPSCTC Decoder


Simulation Results and Bounds

-1 0 1 2 3 4 510

-12

10-10

10-8

10-6

10-4

10-2

100

SNR

BE

R

SCTC and RPSCTC Simulation for N=200

N=200 SCTC m=4

bound N=200 SCTC m=4

N=200 RPSCTC m=4

bound N=200 RPSCTC m=4

-1 0 1 2 3 4 510

-12

10-10

10-8

10-6

10-4

10-2

100

SNR

BE

R


N=200 SCTC m=2

bound N=200 SCTC m=2

N=200 RPSCTC m=2

bound N=200 RPSCTC m=2

Figure 9. SCTC and RPSCTC simulation m=2, m=4 for the AWGN channel



-1 0 1 2 3 4 510

-12

10-10

10-8

10-6

10-4

10-2

100

SNR

BE

R


N=200 RPSCTC m=4

bound RPSCTC m=4

N=200 RPSCTC m=2

bound RPSCTC m=2

Figure 10. RPSCTC simulation m=2, m=4 in the AWGN channel



-1 0 1 2 3 4 510

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

SNR

BE

R

RPSCTC N=200 R=1/15 m=4

DRPSCTC N=200 R=1/15 m=4

SCTC N=200 R=1/15 m=4SCTC bound m=4 N=200

RPSCTC bound m=4 N=200

DRPSCTC bound

Figure 11. SCTC, RPSCTC vs DRPSCTC m=4 AWGN channel

Li

LN

dLiLNtidp q

q

),,()|(

q =1,2 for component encoders for DRPSCTC



0 5 10 15 20 25 30 35 40 45 5010

-8

10-6

10-4

10-2

100

102

104

106

Codeword distance

Mul

tiplic

ity

SCTC distance spectrum N=100 R=1/15

RPSCTC distance spectrum N=100 R=1/15

DRPSCTC distance spectrum N=100 R=1/15

Figure 12. SCTC,RPSCTC, DRPSCTC distance spectrum N=100, R=1/15



2 3 4 5 6 7 8 9 10 1110

-5

10-4

10-3

10-2

10-1

SNR

BE

R

N=200 SCTC m=2

bound SCTC m=2

N=200 RPSCTC m=2

bound RPSCTC m=2

2 3 4 5 6 7 8 9 10 1110

-6

10-5

10-4

10-3

10-2

10-1

SNR

BE

R

Rayleigh fading simulation for N=200

N=200 SCTC m=4

bound SCTC m=4

N=200 RPSCTC m=4

bound RPSCTC m=4

Figure 13. SCTC and RPSCTC simulation m=2, m=4 flat Rayleigh fading channel

a novel technique for improving the performance of turbo codes using orthogonal signalling,...

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