a note on performance of generalized tail biting …no. 1 a note on performance of generalized tail...

No. 1

A Note on Performance of

Generalized Tail Biting Trellis Codes

Shigeich Hirasawa

Department of Industrial and Management Systems Engineering

School of Creative Science and EngineeringWaseda University, Tokyo, JAPAN

hira@waseda.jp

Masao Kasahara

Faculty of Informatics, Osaka Gakuin University, Osaka, JAPAN

kasahara@ogu.ac.jp

Pre-ICM International Convention on Mathematical Sciences (ICMS2008)Delhi, India, Dec. 18-20, 2008

No. 2

Coding Theorem … random coding arguments

1. Introduction 1. Introduction

• existence of a code

Coding theorem aspects → practical coding problem

• essential behavior of the code

• quantitative evaluation

: probability of decoding error

: rate

: decoding complexity

R

G

No. 3

• Generalized tail biting (GTB) trellis codes

c.f. Ordinary block codes/Terminated trellis codes

・Direct truncated (DT) trellis codes [3]

・Partial tail biting (PTB) trellis codes

・Full tail biting (FTB) trellis codes [4,5]

1. Introduction

• Generalized version of concatenated codes with generalized tail biting trellis inner codes ---Codes [6]

• inner codes --- GTB trellis codes

• outer codes --- Reed Solomon (RS) codes

No. 4

2. Preliminaries 2. Preliminaries

2.1 Block codes

code length

number of information symbols

rate

(N, K) block code

block code exponent

(1)

(2)

(3)

No. 5

2. Preliminaries2.2 Trellis codes

(u, v, b) trellis code

branch length

branch constraint length

number of channel symbols / branch

rate

trellis code exponent

parameter

(4)

No. 6

2. Preliminaries

Fig.0: Block code exponent and trellis code exponent for very noisy channel

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1

R/C, r/C

E(R

)/C

, e(r

)/C

E(R)

e(r)

(N, K) terminated trellis code [3]

Block codes converted from trellis codes

(6)

(7)(8)

No. 7

3. Generalized tail biting trellis codes(N, K) GTB trellis code

(i) DT (direct truncated) [3]:

(ii) PTB (partial tail biting):

(iii) FTB (full tail biting) [4, 5]:

uvv

v’ v’

parameter (9)

3. GTB trellis codes

No. 8

Figure1: Examples of FTB and PTB trellis codes

Starting states Ending states Starting states Ending states

(a) FTB trellis code for v = v’ = 2 (b) PTB trellis code for v = 2, and v’ = 1

3. GTB trellis codes

No. 9

3. GTB trellis codes3.1 Exponential error bounds for GTB Trellis codes

[Theorem 1] GTB trellis code

For

where

(10)

(11)

No. 10

3. GTB trellis codes

: starts at and ends at

: starts at and ends at

: starts at and ends at

true path : 0u

No. 11

3. GTB trellis codes

(13)

(14)

true path

true path

No. 12

3. GTB trellis codes

(16)

(15)

true path

No. 13

3. GTB trellis codes

[3]

[Example 1] GTB trellis codes for Very Noisy Channel

(a) PTB Trellis codes, (1) (2)

/ C

/ C

/ C / C

No. 14

[3]

[Example 1] GTB trellis codes for Very Noisy Channel

(b) FTB Trellis codes, (1) (2)

3. GTB trellis codes

No. 15

3. GTB trellis codes

(a)

(b)

r

eG(r)

= 0.5

= 0.4

= 0.3

= 0.2

= 0.0

PTB trellis codes ( = 0.5)

FTB trellis codes

/ C

/ C

No. 16

3. GTB trellis codes

[Corollary 1] FTB trellis code [4]:

(17)

(18)

[Corollary 2] upper bounds on

Ordinary block codes

Terminated trellis codesDT trellis codes>

Terminated trellis codes [3]:

DT trellis codes:

error exponent

decoding complexity

upper bound on Pr(・)

No. 17

3. GTB trellis codes

3.2 Decoding complexity for GTB trellis codes

[Theorem 2]

(19)

No. 18

3. GTB trellis codes3.3 Upper bounds on probability error for same decoding complexity

[Corollary 3] GTB trellis codes:

(22)

where

(N, K) block code

(3)

No. 19

3. GTB trellis codes

(a)

(b)

r

gG(r)

= 0.5

= 0.4

= 0.3

= 0.2

= 0.0

PTB trellis codes ( = 0.5 except for low rates)

FTB trellis codes

[Example 2] GTB trellis codes for Very Noisy Channel

/ C

/ C

No. 20

3. GTB trellis codes

No. 21

4. Generalized version of concatenated codes with GTB trellis inner codes

Fig.2: Error exponents for code and code for very noisy channel

[4]

4. Concatenated codes

No. 22

[Lemma 1] [4, 5]:

where

Generalized version of concatenated codes with GTB trellis codes

(J = 1)

4. Concatenated codes

No. 23

[Theorem 3] Generalized version of concatenated codes [6]:

With the DT trellis inner codes

where

Terminated trellis codes [6]:

DT trellis codes:

Length of outer code

Decoding complexity of inner code

error exponent

decoding complexity

upper bound on Pr(・)

4. Concatenated codes

No. 24

Figure 6: (a) Code with DT trellis inner codes over a very noisy channel

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

J = 1

J = 2

J = 8

4. Concatenated codes

No. 25

[Theorem 4] Code

With the GTB trellis inner codes

where

Decoding complexity

(29)

(30)

(31)

4. Concatenated codes

No. 26

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Figure 6: (b) Code with PTB trellis inner codes over a very noisy channel

J = 1

J = 2

J = 8

4. Concatenated codes

No. 27

Figure 7: Optimum parameters and over a

very noisy channel

0.5 1.0

0.5

1.0

4. Concatenated codes

PTB

FTB

0

No. 28

5. Concluding remarks

(1) GTB trellis codes over Very Noisy Channel

(2) Generalized version of concatenated codes

・ Exponent + Decoding complexity

・Terminated trellis codes = DT trellis codes

Inner codes:

Outer codes: GMD

E-O

Over-all decoding complexity is dominated by that of inner codes!

PTB

FTB

5. Remarks

No. 29

5. Concluding remarks

(3) Generalized version of concatenated codes

GTB trellis inner codes --- PTB trellis codes

・ Linear error exponent + large decoding complexity

Mail to:

hira@wadeda.jp

5. Remarks