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Analysis and design of turbo like codes

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) CIIE/ 4 @A , ,

) ( )

+ 8 , ) ) ,

.? / ,

7 +! J/I%/ - L$( )

IHB 48 LI( J 4/

1, @.+A (

, L$ ! $% LI ! I% , $/D/

1 + IBJ J$KI/

) , !10% !40%/ 1

$/D - 5 ( L $

4, ) IHB4/

"& , +' +

(

, /

- ) ) )

(1;2 .N /

,,) -

(42)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/No (dB)

Word Error Probability

q=4, n=16384

q=4, n=4096

q=3, n=16384

q=3, n=4096

$/DF " ) ( (1;2

/

)

,-.

+ , " /

L ( / ,

)

!Æ%L

-)Æ Æ

!1Æ%R ,!1%

!$/ %

,

!1Æ%

L ,!1%R! Æ%,! 1

! Æ%

%RÆ,! 1

Æ

(43)

,

!Æ% LI/

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.1

0

0.1

0.2

δ

=h/N

r(

δ

)

Weight Spectrum of 4−RA code

$/JF

!Æ%/

$/ ,

.

!%

L

%Æ% Æ

Æ

Æ

!$/ I%

!

L.

!% !$/ D%

$

& !

) -.+,

(44)

,/

!

!4% " !4%

$$KI KI

$ $ HB IBD

I I D HBI

D D IK BJ$

J J H$I H

H H B D

K K D

/

/

/

/

/

/

/

/

/

/

/

/

! DB% DB

- L $

& HB 4

, $ (

?/

G

IBD 4 ! "

$% ,

$ /

$/H ,

? " L DB 4/ ,+ ,

) DB 4 "

(1;2 / ,) (

DB 4/ )

: ) , /

# 1 $

!

L ## %

$& #

1 ,) $/ " $/I/ / 4 ,

) F

$ %! !! %

&'

! !! (

!!

(45)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−2

−1

0

1

2

3

4

5

6

7

8

Code Rate R

E

b

/N

o

(dB)

Binary Shannon Limit

RA code threshold (GD bound)

$/HF ( /

# 1 $ !

!

!

#

'$2 !$/ $%, ) !1 %L

! %L

-)

,!1%L,! %L

+ ÆL !$/ I% .

!%, ) ,

.

!%

! %

! %

L

,

!

L.

!%

(46)

, , )

!$/ J% /

, $// 1 ,

,

!Æ%/

3 1 '!1Æ%!()#*)+ ## !1Æ%

C EC E !1Æ%# !1Æ%L ÆL 1L#

'$2 ' 3 ,!1% )-

ÆC E , 1

1

C E, )

,!Æ1

R! Æ%1

%Æ,!1

%R! Æ%,!1

%

, 1

L 1

P Æ L / * 1

L 1!Æ%

1

L1!! Æ%%, /

" 1 $ ÆC E

!Æ%

, Æ L #

'$2 *1!Æ% ? ) 1

!Æ%//

1!Æ%L

-)Æ Æ

!1%R ,!1%

!$/ H%

!Æ%, )

(47)

4 * $/ !1Æ%, )

!Æ%,!1!Æ%%

!$/ K%

!$/ K% , ) 1 L 1!Æ% !1Æ% L

1 L / 4 * $/ !1Æ% L 1 L Æ L /

, Æ L /

3 1 1!Æ% ! ()#*-+

1!Æ%! 1!Æ%%

Æ! Æ%

!$/ B%

'$2 + @-A !$/ %

!1L ? )) 1L !Æ! Æ%%

)% 1 L 1!Æ% ))

!$/ % , 1 ?/

,+

! %! 1% ! %1R! %! Æ 1%R! %!Æ 1%L

-

! Æ 1%!Æ 1%

! 1% 1

L !$/%

4 1 !Æ! Æ%% !$/%

! Æ 1%!Æ 1%

Æ! Æ%/ -? 1 L

1!Æ%, )

Æ! Æ%

! 1% 1

(48)

" 1 !Æ

% , C E

Æ

L

Æ

L

1!Æ

%L#

'$2 Æ

L

Æ

! Æ

% . !$/ B%

? / !$/ B% 1!Æ% ? /

4 1 !Æ

Æ

% 1!Æ

%/

" 1 '

Æ

&

ÆÆ

1!Æ%Æ

#

'$2 *Æ / 1 !Æ Æ%LÆ !$/ B%

, 1! Æ%

/ !$/ B% ,

1!Æ%

Æ! Æ%

Æ

!$/%

!$/% , Æ I

. / - J, )

1!Æ%

Æ! Æ%

Æ

!$/$%

, Æ

, Æ

Æ/

.

!% !$/ I% Æ!% ? ) Æ//

Æ!%

L

%Æ% Æ

Æ

Æ

!$/I%

, + $//

'" 1 ', Æ!% !#

'$2 1 , , Æ!% ) C E

- / 1 ) / !$/J%,/ 4

!

L .

!%

L

Æ!%

Æ

(49)

4 , )

! % !$/D%

,

!

Æ!%

Æ!%

!Æ!%%

4 !$/ K%

!Æ% ,!1!Æ%%

!

Æ!%

Æ!%

,!1!Æ!%%% !$/J%

2, , 3 , Æ!%3

Æ

L / , Æ

L Æ

L/

, ) $/1!Æ!%%/ Æ

L , !$/J%

, )

!

Æ

Æ

,!%L

, $/$/

G Æ

L ,) $/$

1!Æ!%%Æ!%

/ !$/J% , )

!

! Æ!%%

,!Æ!%

%

Æ!%

4 ,!

% , )

!

,!Æ!%

%

Æ!%

L

$/$/

(50)

1 ,) F

'$ $ # 11 1 ) )

!

L.

!%L

Æ!%

Æ!%

Æ

( !$/D%,

!

Æ!%

Æ!%

!Æ!%%

2, $/ , + ,

!Æ%!% , )

!

Æ!%

Æ!%

4 $/

Æ!%L

!

4 , , $/$

!

/

!

(51)

! "#

& +

$ , ? ./

,/ )/ (

.

, ) !- + %

" 3 / " + )

/ G

)

- - + /

, ) ) ( /

, -,

) ! #* %/

1

, , ) ,

-+ / ( , )

, )

) /

) F %

8 % 8 % )/

G ( 5

) , ! R(R(

%/ 1

.. !%/ (

, ) / , #*

)

(52)

) ,) )

( /

! % (

) / , ? CD E/ 1

) .(!(% / 1 #*

, ,

, ) /

- /

,

)/ 1 .(.

(!((%/ ,, )#*

? ,

, ) /

& ,

, ? ..

!%/ ( /

&,) ) ? .

1 / " ,

) )

/

, I/ / ( +

)

) , ! R(R(

%/

(53)

rep. q

P

D

D

w

qw

qw

h

N

qN

qN

qN

I/ F 0 !% / .

+

+/

G ) ;0 : /

" /, ., !G10%

!% !% ) / 1

+ , G10

L

L

L/

!I/ %

G10 ) ,) ( -4/

L(

+

7

7

!I/%

, L! %R7 ( , H/

, !/ I% !% .

I/ G10F

L

L (

+

7

7

!I/$%

, L! %R7/ 4 7 , ) 7 /

L 7

(54)

7L/ 2, !I/$% # !% !/ J%

# !%L

R L R !I/I%

& ' !/ K%

' L

# !%

L !I/D%

, ;0 : .

, ) $ (

/

, ) ;0 : . .

/ G (/ ( - (/ ,

(/ ) /

# 1 (

"

8

(

L

.

L

%Æ% Æ

Æ

Æ

.&.

%

9 !(

%R ,

(55)

, , )

!I/J%/ ) ( - 4//

'" 1 &'((

9 !(

%L$! % !I/H%

%&

!I/$% !Æ% !/D% , !( - 4/$%

!Æ% L

-%#)-% !

,!%R!Æ 1%,! 1

Æ 1

%R! Æ 1%,!

1 ÆR:

Æ 1 %

R 1,!:1%R1,!

Æ R:

1

%% !I/K%

, ,!%L ! %! % / (

!Æ%!Æ%R$! ( - 4/$%/

+ (

L

, I/ I/ )

, F

# 1 & '(( )

&.

L$! !

% !I/B%

%& ' L

.

L

%Æ%

Æ

Æ

Æ

I/ "2) I/,

(56)

-/ - I #* JKH4

, I ( D 4/

$ I D J H K

P$ PI PD PJ PH PK

( /HB ./D ./IK ./H$I ./B . / D

./ KB ./JKH ./B . /D . / I . /H

./ID$ ./HHI ./BD . /JJ . / ID . /$

4 " * ./IBD ./HBI ./BJ$ . /H . / D . /

I/ F 2

/

) ,

, #* ( / ,

, ) / )

.1 ? CIK DE / .

1 - / ,

@A ) + /

S

S

Yn

Xn

n+1

n

(57)

,

-

.1 ? ,

I//

2, , + , , ) .

? . O ,

, , 8 .

4' , CD $E/

Æ .1 /

I/$ , .1 ? L$ !J% /

S0

U1

U2

S5

S4

S3

S1

S2

S6

X5

X6

X3

X4

X2

X1

State node

Variable node

Constraint node

y2

y3

y4

y5

y6

y1

I/$F .1 /

) , (1;2 / I/I

, $ / )

*# )/ 4+ I IBJ / I/I

, 40 , 10 /

(58)

( J4 ,H 4) /

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

SNR (dB)

Error Rate

RDD Simulation, LCM Interleaver, Repetition 3

Length 1026 WER

Length 1026 BER

Length 4096 WER

Length 4096 BER

I/IF " ) (1;2

/

&" '++

, CD E/ .

)

)/ "

, ) CD E , ) , .

+ IBJ )

(59)

, ) + / 1

) - ,

, , ) (1;2 I/ /

!

, $KI 4 / 4

. " /

I/D , ( /

) , ! R(

R(R(

%

/

D

D

M

U

X

P

Acc.

I/DF ) .( /

+C

E +C

E

L

L

L

L

L

R

L

R

R

/

/

/

L R

L R R

!I/ %

!"#$

,) , /

), , ) ,

(60)

, ) , ,

F

L

, - // , ),

/ & G10

!I/$% , ),/

G10 ) / ( +

)

) - -

/ , !

% !

%

!

% !

%/

,

L

!I/ %

,

) / !$/$%/

#

;) , (

! %F

!Æ% L

-#)-. 1,

*

I1

R1,

:

1

R! 1%,

1 :

1

R! 1%,

!* %I

1

R! Æ%,

*

Æ

RÆ,

*

Æ

,!*%

(61)

#* / "2

(/ $KI 4 ) KI4/

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.4

−0.2

0

0.2

0.4

δ

=h/N

r(

δ

)

Random Codes

RA Codes

CA Codes

I/JF 1 ) .( /

1 ( 4' !,.+,% .

*.+ . ( /

I/H, (, + I

J$KI/ ) 40 ( J$KI

D 4 4 , " 3 /

&& , ++++

(62)

0

0.5

1

1.5

2

2.5

3

3.5

4

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/No (dB)

Word Error Probability

Block Length=1024

Block Length=16384

I/HF ) ) .(

(1;2 /

) /

, ) ) ,

/ ", I/K ,

) + ) ..!((%

/ , ) CBE CIE/

acc.

P

rep. q

acc.

P

(63)

,

L

!I/ %

,

G10

)/ ) !$/% !$/$%/ ,

F

!Æ%L

-#Æ Æ

-# -#

,!% +!% +!Æ% !I/ $%

, +!%

+!%L,!% ,!

% ! %,!

%

1 ) (( I/ /

G , (( I//

- " 3 / -

I (( ) "2 I$H 4 , " IBD

4/

$ I D J H K

P P$ PI PD PJ PH PK

(( /IIDD ./I$H ./HH ./BD . /JJ . / ID . /$

/$K ./ID$ ./HHI ./BD . /JJ . / ID . /$

4 " * / KI ./IBD ./HBI ./BJ$ . /H . / D . /

I/F 2

(( /

( , #*

(( 3 , , ) / .

(64)

, #*

( / , , C BE )

$(( I 4 B 4

( /

0

0.5

1

1.5

2

2.5

3

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

Eb/No (dB)

Word Error Probability

RAA codes

RDD cods

RA codes

I/BF $ (( ( ,

(65)

$

. +

) " . / 4 @" .

A , )

- + / 1

- -.+

) .

- #*/ + ,#*

/

, , + ,,

CIEF

*+ ,

-. */

/

4 / CIE , , )

, /

- !:

% 3 4 Æ

, , , ) -.+

? !

, ' ! . % @ .

) A - 4 / BBJ CIE/

(66)

, , / " D/I D/D

) , .

)/ " D/J )/ " D/H

, ) -/ ( " D/K

/ " ) ( - /

.

, D/ /

)! %

) )

/ ( + )

! %

,

L / , / )

!% + , L

/

L

) L!

% /

4 S )

, , D/ / 1 C

E

/ !( ) " //%

G !D/$% C

E

@A , ? /

( , D// (

+

,

, )

, /

)

) ) / )

!% + , L

,

/ 4

$

$ ! !

(67)

P

E

k

k

k

k

k

k

w

w

w

w

w

w

1

1

P

P

E

E

2

2

n

h

k

w

J

J

N

N

N

h

h

h

J

2

1

.

.

.

D/ F 0 , / )

. + ,

& , +/

) S , , D// 1

C

E /

G !D/I%

C

E @A

, ? /

, Æ ) !.

D/$ D/I% , !Æ% C

E

C

E / , ) !Æ% !

(68)

P

k

E

1

E

2

N

N

n

d

d

h

w

D/F 0 , / )

. + ,

& , +/

."

, #*,

/

,

- .

- # - .

, / 1 ,.,

, , .

/

, / (

-

(

%& !D/ %

(

!D/%

- (

L

, / " ,

F

.

L

,%Æ

!Æ%Æ !D/$%

(69)

!,

!Æ% " / %

.

L

.

!D/I%

,F

9 !(%

L ,

, ( , ( / , 9 !(% )

, , (

/ !

(/%

# 1 % .

! !D/I% !

# !#&.

# '

"

% &

9 !(

%R ,

!D/D%

'$2 " - # ) , )

!/ % !/ % " /

,

R

/,

L 9 !(

%R

/,

Æ Æ

!D/J%

(70)

# .

&%/ & &

&(

,)

#

!Æ%

Æ

# .

&%

Æ Æ

!D/H%

/,

Æ Æ

/,

L ,

!D/K%

, L ! %/ " !D/K% !D/J%, ) !D/D%/

" 1 9 !(

%L$! !

% ' & #&.

L$! !

% !D/B%

'$2 2 !

L !

/ ,, !D/D% !D/%/

, .

, , ,

/

# 1 .

!

, %

%

&(

%

.

L

,%Æ%

!Æ%Æ !D/ %

!#

'$2

, %Æ%

!Æ%

Æ

, %Æ

!Æ%

(71)

.

!D/$% . % / , , , . . ) . )/ .

) Æ

Æ ( Æ , !Æ % Æ

L !D/ %

Æ & 0Æ !Æ%L ,!Æ%/ !Æ % Æ ,!Æ % Æ

, !D/ %/ Æ

L/ & , %

L !Æ

%, ) , %Æ% !Æ% Æ !Æ % Æ

!D/ % ) , , .

/

" 1 !Æ% !

L$!( %

!Æ% !

R!Æ% Æ Æ Æ

. ! Æ !Æ% Æ !D/ %

'$2 1 D// %

(72)

D/ .

/

.& / 0

C

E ) G10

G10 @ )A CIE !

" /$ .+%F

L !D/ $% , G10

! D/ %/

! " / %

L L ! % L

2- , /$( - !D/ I%/ , L

/ ; L

(73)

; / 1

- 3

3 L/ !) ;% ,F 8

!R % ; !8 % ; !D/ D%

, R

/

!D/ I% !D/ D%,

8 1

8 & / Æ

,F 2 2 8 1

! !D/ J% < / %/ 0.

!D/ J% , 9 !(

%

D/ /

!Æ% Æ , !D/ J% ,

- / " 1 3 1 =

< , )

(74)

0# 0#

!R %

0# 0# !D/ H%

, L ! ) % L/

!Æ% !/I% , !D/ J% !D/ H%, )

!Æ% L

R)ÆR

%#2Æ ,! % ! %,!

% !D/ K%

, ) / 0 !D/ K% , , D/ ) .

C

E /

.. / - C

E )G10 , .

G10

CKE!D/%F

L !D/ B% & L !D/% "

L < / !, < L <!

%%

(75)

> , L 2 2 8 4 > 3 ; 3 !D/ % , / ! 5 % L ! 5 % , L

, )>

L > L L > L L# L L' > L L! !D/ % 2 8 ! ! 3 ; 3 !D/%

Æ L , );L;Æ

! 3 ; 3 ! ; !D/$%

3 / /

!D/% , R !D/$%, )

2 8 !

(76)

! R

/% !D/I% ,

9 !(

%/

!Æ% , !D/I%/ 7

1 3 1

!D/I% -

, ) 8 2 1 -2 # ! ' !D/D%

7 +!D/ J%!D/ K%

, - , L F

!Æ%

R)ÆR

%#2Æ # ,!

#

%R',! ' % !,! ! % !D/J%

, ) / 0 !D/J% , , D/ ) .

C

E /

.1 0 ,

, ) / .

: CIE CKE - C KE/ D/$

? C

E , .

) - R )

- R / D/I ? C

E , ) - ) - /

# 1 C

E .

! #&.

%&

L $! %

(77)

'$2 ;) D/ D/ , Æ ) .

, , /

3 1 C

E .

!#

'$2 1 D/ , !D/ K%

F

!

L

!Æ% L )ÆR

%#2Æ

!,!% ! %,! %%

, !Æ%Æ , , , F

Æ

Æ

%#%2Æ !,!

% ! %,!

%%

4 /$ ! % L R

/

3 1 C

E

9 !(

%L$! %

%#

'$2 7 !D/ J%

,)

9 !(

% L

,

L

,

2,

8

1,

2,

T

(

(78)

!% ,) , ! !/I% %F 8 1, !;( % !;( % 8 ! % ! % !<( %

/ & T

, / !% !D/H% <(

, / L / 2

(

L-!

% ) % !D/ %/

2- , ) /

# 1 C

E ! $ .

! #&.

L $!   % L $! % %&

( D/ D/ 3Æ ) ,

, /

3 1 C

E ! . !#

'$2 D/+ Æ ,F

Æ Æ %#%2Æ !,!

%R,!

% ,!%%

4 /$

R

/

3 1 C

E $ 9 !(

(79)

'$2 1 !D/I% ,) 9 !( % L , 2, 8 ! ;( 2, T ( L $!   %

!% , ) , ! !/I%%F !# % 8 ! !'% !'% !<( % 8 8 1, !;( % /

!% <(

, / / L / , ( L-! % )%/ .2 )

4 / CDE .

, L ) ) ,

L

L

) L $/ 0- ) - ) ,

(1;2 )

D

4 - )

+? , ?/ ,

D/$ , )

, ) , )

L $! %

L $! % , - /

,) F ! % - , ) .

, D/$ ,-.+ 8 !%D/$

(80)

- , , ) , D/$

) ) )/

.) !(% , .

,

L .

L )

) , ! R(% /

L / & D/I .

(), ?$

? / ,

) ,

! " $/I%

!Æ%L

-#

,!%R! Æ%,!

! Æ%

%RÆ,!

Æ %

, ;.

) ) ,/

!

" *

$ P$ /B / $$

I PI / $ / B

D PD / J$ /K

J PJ / KH /DI

H PH /H /HI

D/ F ( 4" /

!

!4% " *!4%

$ P$ / ./D

I PI /B$ ./K

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References

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