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Department of Electrical Engineering École Polytechnique de Montréal. David Haccoun, Eng., Ph.D. Professor of Electrical Engineering Life Fellow of IEEE Fellow , Engineering Institute of Canada. Engineering training in Canada. 36 schools/faculties. 3. 1. 2. 1. Vancouver. 2. 11. 13.

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Department of electrical engineering cole polytechnique de montr al

Department of Electrical Engineering

École Polytechnique de Montréal

David Haccoun, Eng., Ph.D.

Professor of Electrical Engineering

Life Fellow of IEEE

Fellow , Engineering Institute of Canada


Department of electrical engineering cole polytechnique de montr al

Engineering training in Canada

36 schools/faculties

3

1

2

1

Vancouver

2

11

13

1

2

Montréal

Undergraduate students

Canada: 55,000

Québec: 14,600

Toronto


Cole polytechnique cradle of engineering in qu bec

ÉcolePolytechnique, cradle of engineering in Québec

The oldest engineering school in Canada .

The third-largest in Canada for teaching and research.

The first in Québec for the student body size.

Operating budget $85 million Canadian Dollars (C$).

Annual research budget $60.5 million C$.

Annual grants and research contracts $38 million C$.

15 Industrial Research Chairs.

24 Canada Research Chairs.

7863 scientific publications over the last decade.

220 professors, and 1,100 employees.

1,000 graduates per year, and 30,000 since 1873.


Department of electrical engineering cole polytechnique de montr al

11 engineering programs

  • Biomedical

  • Civil

  • Chemical

  • Electrical

  • Geological

  • Industrial

  • Computer

  • Software

  • Mechanical

  • Mining

  • Engineering physics


Our campus

Our campus

Polytechnique


Novel iterative decoding using convolutional doubly orthogonal codes

Novel Iterative Decoding Using Convolutional Doubly Orthogonal Codes

A simple approach to capacity

David Haccoun

Éric Roy, Christian Cardinal


Department of electrical engineering cole polytechnique de montr al

Modern Error Control Coding Techniques

Based on Differences Families

  • A new classof threshold decodablecodes leading to simple and efficient error control schemes.

  • No interleaver, at neither encoding nor decoding

  • Far less complex to implement than turbo coding schemes, attractive alternatives to turbo coding at moderate Eb/N0 values

  • High rate codes readily obtained by puncturing technique

  • Low complexity and high speed FPGA-based prototypes at bit rate >100 Mbps.

  • Extensions to recursive codes  Capacity

  • Rate adaptive schemes Punctured Codes

  • Reduced latency Simplified Codes

  • Reduced complexity

7


Department of electrical engineering cole polytechnique de montr al

8


Department of electrical engineering cole polytechnique de montr al

D2

D

Dm

A

J

m

– Set of connection positions

– Number of connection positions

– Memory length

– Coding span

One-Dimensional NCDO Codes

  • Nonrecursive systematic convolutional (NSC) encoder ( R = 1/2 )

Information sequence

Shift register of length m

0 1 2 m-1 m

...

...

D1

AWGN

Channel

=0

=m

... ...

... ...

J

Parity sequence

9


Department of electrical engineering cole polytechnique de montr al

+

  • Simpleorthogonal properties : CSOC

Differencesaredistinct

Example of Convolutional Self-Orthogonal Code

CSOC, R=1/2, J=4, m=15,

10


Department of electrical engineering cole polytechnique de montr al

Example of CSOC, J=4,

DistinctSimple Differences

  • All the simple differences are distinct

  • CSOC codes are suitable for threshold decoding

11


Department of electrical engineering cole polytechnique de montr al

Threshold (TH) Decoding of CSOC

  • CSOC are Non iterative, systematic and non recursive

  • Well known symbol decoding technique that exploits the simply-orthogonal properties of CSOC

  • Either hard or soft-input soft-output (SISO) decoding

  • Very simple implementation of majority logic procedure

12


Example of one step threshold decoder j 3 a 0 1 3 d min 4

D

D

D

S

>

<

D

D

D

Example of One-Step Threshold DecoderJ = 3, A= {0, 1, 3}, dmin= 4

Soft outputs

in LLR

(2 -1)=1

(3 -0)=3

(3 -1)=2

0

0

1

ûi

Decoded

bits

1 =0

3 =3

2 =1

= tanh/tanh-1(sum-product)or add-min (min-sum) operator

, are LLRs values representing the received symbols ,

13


Department of electrical engineering cole polytechnique de montr al

Novel Iterative Error Control Coding Schemes

  • Extension to Iterative Threshold Decoding

  • Convolutional Self-Doubly-Orthogonal Codes : CSO2C

  • All the differences (j -k )are distinct;

  • The differences of differences (j -k )–(l -n ),j  k, k  n, n  l, l  j, must be distinctfrom all the differences (r -s ),r s ;

  • The above differences of differences are distinctexcept for the unavoidable repetitions

  • Decoder exploits thedoubly-orthogonal properties of CSO2C

  • Asymptotic error performance (dmin=J+1 ) at moderate Eb/N0

Issues : Search and determination of new CSO2Cs

Extention of Golomb rulers problem (unsolved)

14


Department of electrical engineering cole polytechnique de montr al

Example of CSO2C, J=4,

Differences of Differences

(1,3,1,3)=((-12)-(12))= -24

(2,0,1,0)=((13) -( -3))= 16

(2,0,2,0)=((13)-(-13))= 26

(2,1,0,1)=((10)-( 3))= 7

(2,1,2,0)=((10)-(-13))= 23

(2,1,2,1)=((10)-(-10))= 20

(2,3,0,1)=(( -2)-( 3))= -5

(2,3,0,3)=(( -2)-( 15))= -17

(2,3,1,0)=(( -2)-( -3))= 1

(2,3,1,3)=(( -2)-( 12))= -14

(2,3,2,0)=(( -2)-(-13))= 11

(2,3,2,1)=(( -2)-(-10))= 8

(2,3,2,3)=(( -2)-( 2))= -4

(3,0,1,0)=((15)-( -3))= 18

(0,1,0,1)=(( -3)-( 3))= -6

(0,2,0,1)=((-13)-( 3))= -16

(0,2,0,2)=((-13)-(13))= -26

(0,3,0,1)=((-15)-( 3))= -18

(0,3,0,2)=((-15)-(13))= -28

(0,3,0,3)=((-15)-(15))= -30

(1,0,1,0)=(( 3)-( -3))= 6

(1,2,0,2)=((-10)-(13))= -23

(1,2,1,0)=((-10)-( -3))= -7

(1,2,1,2)=((-10)-(10))= -20

(1,3,0,2)=((-12)-(13))= -25

(1,3,0,3)=((-12)-(15))= -27

(1,3,1,0)=((-12)-( -3))= -9

(1,3,1,2)=((-12)-(10))= -22

(3,0,2,0)=((15)-(-13))= 28

(3,0,3,0)=((15)-(-15))= 30

(3,1,0,1)=((12)-( 3))= 9

(3,1,2,0)=((12)-(-13))= 25

(3,1,2,1)=((12)-(-10))= 22

(3,1,3,0)=((12)-(-15))= 27

(3,1,3,1)=((12)-(-12))= 24

(3,2,0,1)=(( 2)-( 3))= -1

(3,2,0,2)=(( 2)-( 13))= -11

(3,2,1,0)=(( 2)-( -3))= 5

(3,2,1,2)=(( 2)-( 10))= -8

(3,2,3,0)=(( 2)-(-15))= 17

(3,2,3,1)=(( 2)-(-12))= 14

(3,2,3,2)=(( 2)-( -2))= 4

  • All the differences of differences are distinct

  • These codes are suitable for iterativethreshold or belief propagation decoding

15


Department of electrical engineering cole polytechnique de montr al

Spans of some best known CSO2C encoders

  • Issue : minimization of memory length (span) m of encoders

  • Lower bound on span

16


Department of electrical engineering cole polytechnique de montr al

Non-Iterative Threshold Decoding for CSOCs

Approximate MAP valueli:

Extrinsic

Information

Received Inform. Symb.

=

+

: Addmin operator;

where

Decision rule :

, otherwiseûi= 0

ûi=1 if and only ifli

0

CSOC i is an equation of independent variables

17


Department of electrical engineering cole polytechnique de montr al

Depends on the simple differences

Estimation of at Iteration 

Feedforward for future symbols

Feedback for past symbols

depends on the simple differences

and on

the differences of differences

Iterative Threshold Decoding for CSO2Cs

General Expression:

Iterative Expressions:

  • 1 Iteration:DistinctDifferences

  • 2 Iterations: Distinct Differences of differences

  • Distinct Differences of differences from Differences

18


Department of electrical engineering cole polytechnique de montr al

Iterative Threshold Decoder Structure for CSO2Cs

Delay m

Delay m

Delay m

Delay m

Delay m

Delay m

Forward-Only Iterative Decoder

Last Iteration

...

...

Soft

output

Soft output

Soft

output

Soft

output

Information

symbols

threshold

threshold

threshold

...

threshold

...

decoder

decoder

decoder

decoder

Iteration

Iteration

Iteration

Iteration

 =1

 =M

 =2

 =I

Hard Decision

From channel

Parity-check

symbols

...

...

Decoded

Information

symbols

  • No interleaver

  • One ( identical ) decoder per iteration

  • Forward-only operation

Features:

19


Department of electrical engineering cole polytechnique de montr al

Block Diagram of Iterative Threshold Decoder (CSO2Cs)

  • One-step TH decoding per iteration

  • Iterative TH decoder ( M iterationsM one-step decoders)

  • Each one-step decoder for a distinct bit

Latency m bits

Latency m bits

Latency m bits

Input

For

Output

For

Total Latency M mbits

20


Department of electrical engineering cole polytechnique de montr al

Iterative Belief Propagation (BP) Decoder of CSO2C

p

p

w

w

-

t

Mm

t

u

u

w

w

-

t

Mm

DEC 1

DEC 2

DEC

M

t

0

l

l

(

2

)

(

M

)

ˆ

l

(

1

)

u

-

-

-

-

2

t

Mm

t

m

t

Mm

t

m

1

Threshold Decoder

(TH)

Latency

m bits

Latency

m bits

Latency

m bits

BP Decoder

p

p

w

w

-

t

Mm

t

u

w

u

w

-

t

Mm

DEC 1

DEC

2

DEC

M

t

(BP)

(

)

M

v

{

}

Latency m bits

Latency m bits

-

,

t

Mm

j

Latency m bits

0

l

l

(

2

)

(

M

)

ˆ

l

(

1

)

u

-

-

-

-

2

t

Mm

t

m

t

Mm

t

m

1

M(BP)~ ½M(TH)

BP Latency ~ ½ TH Latency

1-step BP complexity ~ J X 1-step TH complexity

21


Department of electrical engineering cole polytechnique de montr al

Error Performance Behaviors of CSO2Cs

J=9, A={0, 9, 21, 395, 584, 767, 871, 899, 912}

BP

Waterfall region

TH

Waterfall region

BP

Error floor region

TH

Error floor region

TH, 8 iterations

Both BP and TH decoding approach the asymptotic error performance in error floor region

BP, 8 iterations

BP, 4 iterations

22


Department of electrical engineering cole polytechnique de montr al

Analysis Results of CSO2Cs

  • Effects of Code Structure on Error Performance

  • With iterative decoding, error performance depends essentially on the number of connections , rather than on memory lengths (spans) .

  • Shortcomings of CSO2Cs

  • Best known codes: rapid increase of encoding spans with J:

  • Optimal codes unknown (Minimum span m )

  • Improvements : Span Reduction

  • Reduce span by relaxing conditionson the double orthogonality at small degradation of the error performance Simplified S-CSO2C

  • Search and determination of new S-CSO2Cs with minimal spans

23


Department of electrical engineering cole polytechnique de montr al

Definition of S-CSO2Cs

Normalized simplification factor

,

  • The set of connection positions A satisfies :

  • All the differences (j -k ) are distinct ;

  • The differences of differences (j -k)-(l -n ), j  k, k  n, n  l, l  j, are distinct from all the differences (r -s ), r  s ;

  • The differences of differences are distinct except for the unavoidable repetitions and a number of avoidable repetitions

Maximal number of distinct differences of differences

(excluding the unavoidable repetitions)

Number of repeated differences of differences

(excluding the unavoidable repetitions)

  • Search and determination of newshort spanS-CSO2Cs

    yielding value

24


Department of electrical engineering cole polytechnique de montr al

Comparison of Spans of CSO2Cs and S-CSO2Cs

25


Department of electrical engineering cole polytechnique de montr al

Uncoded BPSK

coding gain

asymptotic coding gain

Performance Comparison for J=10 S-CSO2C

26


Department of electrical engineering cole polytechnique de montr al

Performance Comparison for J=8 Codes (BP Decoding)

CSO2C: A = { 0, 43, 139, 322, 422, 430, 441, 459 }S-CSO2C: A = { 0, 9, 22, 55, 95, 124, 127, 129 }

27


Department of electrical engineering cole polytechnique de montr al

Eb/No = 3.5 dB

8th iteration

CSO2C

BER

S-CSO2C

14000

3000

Latency (x 104 bits)

Performance Comparison CSO2Cs / S-CSO2Cs (TH Decoding)

28


Department of electrical engineering cole polytechnique de montr al

Small Span

Analysis of Orthogonality Properties (span)

Convolutional Self-Orthogonal Codes

(CSOC)

Simple

Orthogonality

Extension

Orthogonalproperties of set A

Convolutional Self-Doubly-Orthogonal Codes (CSO2C)

Double

Orthogonality

Large Span

Relaxed Conditions

Relaxed Double

Orthogonality

Simplified CSO2C

(S-CSO2C)

Substantial

Span Reduction

29


Department of electrical engineering cole polytechnique de montr al

Analysis of Orthogonality Properties (computational tree)

Decoded symbol

  • The computational tree represents the symbols used by the decoder to estimate each information symbol in the iterative decoding process.

  • Error performances function of

    Independency VS Short cycles

  • Analysis shows that the parity symbols are limiting the decoding performancesof the iterative decoder because of their degree 1in the computational tree

    (no descendant nodes).

  • Impact : The decoder does not update these values over the iterative decoding process : limiting error performances.

LLR for final hard decision

Iter (-1)

Iter (-2)

  • Simple orthogonality  Independence of inputs overONE iteration

  • Double orthogonality  Independence of inputs overTWO iterations

30


Department of electrical engineering cole polytechnique de montr al

Analysis of Orthogonality Properties (cycles)

Conditions

on associated sets

Cycles on Graphs

Codes

No 4-cycles

Distinct differences

CSOC

Minimization of

Number of

6-cycles

Uniformly Distributed

Distinct differences

from difference of differences

CSO2C

S-CSO2C

Minimization of

Number of

8-cycles

Distinct differences of differences

Uniformly Distributed

A Number of Additional

8-cycles

A number of repetitions of differences of differences

Approximately Uniformly Distributed

31


Department of electrical engineering cole polytechnique de montr al

Summary of Single Register CSO2Cs

  • Structure of Tanner Graphs for Iterative Decoding

  • No 4–cycles

  • A minimal number of 6–cycles which are due to the unavoidable repetitions

  • A minimal number of 8–cycles

  • Uniform distribution of the 6 and 8–cycles

Relaxing doubly orthogonal conditions of CSO2C adds some 8-cycles leading to codes with substantially reduced coding spans S-CSO2C

  • Error performance

  • Asymptotic coding gain

  • Correspond to the minimum Hamming distance at moderate

    Eb/N0 values.

32


Department of electrical engineering cole polytechnique de montr al

In order toimprovetheerror performancesof the iterative decoding algorithmthe degree of the parity symbolsmust be increased

Extension : Recursive Convolutional Doubly-OrthogonalCodes (RCDO)

Solution : Use Recursive

Convolutional Encoders

(RCDO)

33


Rcdo codes

3rd register

2nd register

1st register

RCDO codes

  • RCDO are systematic recursive convolutional encoder

  • RCDO can be represented by their sparse parity-check matrixHT(D)

Forward connections

Feedback connections

  • RCDO encoder example : R=3/6, 3 inputs 6 outputs

34


Rcdo protograph structure

The parity-check matrix HT(D) completely defined the RCDO codes.

The memory of the RCDO encoder mis defined by the largest shift register of the encoder

Each line of HT(D)represents one output symbol of the encoder.

Each column of HT(D) represents one constraint equation.

Protograph representation of a RCDO codes is defined by HT(D).

The degree distributions of the nodes in the protograph become important in the convergence behavior of the decoding algorithm.

Regular RCDO (dv, dc) : dv = degree of variable (rows)

dc = degree of constraint (col.) (same numbers of nonzero elements of HT(D) )

RCDO protograph structure

Irregular RCDO protograph

35


Rcdo doubly orthogonal conditions

RCDO doubly-orthogonal conditions

  • The analysis of the computational tree of RCDO codes shows that, as for the CSO2C, three conditions based on the differences must be respected by the connection positions of the encoder.

  • For RCDO the decoding equations are completely independent over 2 decoding iterations.

  • Estimation of parity symbols are now improved from iteration to iteration

  • Resulting in improving the error performances

36


Rcdo codes error performances

50thiter

LDPC

n=1008

decoder limit’

RCDO (3,6)

1.10 dB

Increasing

number of shift registers

RCDO codes error performances

  • Error performances of RCDO (3,6) codes, R=1/2, 25th iteration

  • Characteristics :

    • Small shift registers

    • Error performances

      VS

      number of shift registers

    • Low number of iterations compared to LDPC

  • The complexity per decoded symbol of all the decoders associated with the RCDOs ( in this figure ) is smaller than the one offered by the LDPC decoder of block length 1008. Attractive for VLSI implementation.


Rcdo codes error performances1

Characteristics :

Coding rate-15/30

15 registers

m = 149

Regular HT(D) (3,6)

40th Iteration

Close to optimal convergence behavior of the iterative decoder.

After 40 iterations

 0.4 dB

Low error floor

RCDO codes error performances

  • Asymptotic error performances of RCDO close to BP decoder limit

38


Comparisons

Comparisons

  • Error performances comparisons with other existing techniques

Pb = 10-5

CSO2C

good error performances at moderate SNR

RCDO

good error performance at low SNR

39

  • Figure from : C. Schlegel and L. Perez,Trellis and Turbo coding, Wiley, 2004.


Comparison of the techniques

Comparison of the techniques

Block lengthN, IterationsM

40


Conclusion

Conclusion

  • New iterative decoding technique based on systematic doubly orthogonal convolutional codes : CSO2C, RCDO.

  • CSO2C : good error performances at moderate Eb/No :

    • Single shift register encoder; J dominant

  • Recursive doubly orthogonal convolutional codes RCDO.

    • Error performances improvement at low Eb/No.

    • Multiple shift registers encoder ; mdominant

    • Error performances comparable to those of LDPC block codes.

  • Simpler encoding and decoding processes.

  • Attractive for VLSI high speed implementations

  • Searching for optimal CSO2C & RCDO codes : open problem

41


Department of electrical engineering cole polytechnique de montr al

Merci

THANK YOU

42


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