Turbo-like codes based on RSC code decomposition Hong Sun

Didier Le Ruyet

Department of Communication Engineering Wuhan University 430072 Wuhan, China e-mail: [email protected]

Abstract |

Han Vu Thien

Laboratoire Signaux et Systemes Laboratoire Signaux et Systemes Conservatoire National des Arts et Conservatoire National des Arts et Metiers Metiers 75141 Paris Cedex 03, France 75141 Paris Cedex 03, France e-mail: [email protected] e-mail: [email protected]

In this paper, we show that recursive

systematic convolutional code with primitive denominator polynomial can be decomposed .

As a conse-

quence, a RSC code can be view as a kind of parallel concatenation of one memory RSC code. This decomposition gives an original relation between RSC codes and turbo codes.

I.

Decomposition of RSC Codes

A recursive convolutional code is de ned with a generator ( ) = Q(D)=P (D), where D is the unit-delay operator, Q(D ) and P (D ) are polynomials of degree m over the nite eld GF (2) : G D

( )=1+

Q D

X

m

1

i=1

i

q D

i + Dm ; P (D) = 1 +

X

m

1

i=1

i

p D

i + Dm

The input-output equation of the encoder with input U (D) and output X (D) is written as X (D) = G(D)U (D). It is well known that this graph is not cycle free. In order to generate maximum length sequences of period 2m 1 the polynomial P (D) is chosen as a primitive polynomial. As a m consequence, there exists a polynomial m M (D) of degree 2 1 m so that M (D)P (D) = 1 + D2 1 . From the input-output relation, we have : [2]

m

( ) ( ) + (1 + D2

H D U D

where

1

)X (D) = 0

( ) = Q(D)M (D) = 1 +

H D

u2

u3

all k: k=

s

m X

2

(1)

2

i=1

i

h D

i + D 2m

u4

u5

u6

m X

2

2

i=1

) + sk 2m +1

i (

h u k

i

As a consequence, the sequence of parity check bits X is split into 2m 1 dierent sequences X (L) = fx(jL) g; L = 1; 2; : : : ; 2m 1 as shown in Fig. 2 where g (D) = (L) X (D)=U (D)D L . Each parity bit is described with the following state and output equations :

8 > < jL > : jL

Pm

( )

= s(jL) +

( )

= sLj + u(jL)

s +1

x

L i j ()

( ) 2 2 h u i =1

i

where u(jL) (i) = u[L 1+(2m 1)j

x4 x(1) 2

x5 x(2) 2

x6 x(3) 2

] and u(jL) = u(jL) (i = 0)

i

u7

II. Turbo-like codes based on RSC code decomposition

We propose to replace each delay element D 1 by dierent interleavers. This structure is a multiple parallel concatenated convolutional code or turbo-like code. Compared to classical turbo codes, the constituent codes are one memory RSC codes and consequently the decoder is much simpler. In order to avoid low weight codewords, the interleavers are build according to the constraint given by the decomposition of RSC codes. Using this principle we can derive a new class of eÆcient codes as recently proposed independently in [3]. References

x1 x2 x3 (3) (1) (2) x x1 x1 1

(2)

1

A RSC code can be represented using a conventional Tanner graph with binary variable and parity check nodes [1]. The Tanner graph of (1) is shown in Fig.1 for the (7,5) RSC code.

u1

Figure 2: Decomposition of the (7,5) RSC codes

x7 x(1) 3

Figure 1: Tanner graph of the (7,5) RSC code from (1)

Considering the binary hidden variable sk = uk + xk , then (1) is corresponding to the following parity-check equation for

[1] R. M. Tanner, \A recursive approach to low complexity codes", IEEE Trans. Inform. Theory vol. 27, no. 5, pp. 533{547, Sept. 1981. [2] H. Sun, Z. J. Li, D. Le Ruyet, \Dierent representations of recursive systematic convolutional codes and their associated decoding for turbo codes", Proc. of Int. Symp. on Turbo Codes and Related Topics, Brest, France, pp. 323{326, Sept.2000. [3] L. Ping, K. Y. Wu, \Concatenated tree codes", Proc. of Int. Symp. on Turbo Codes and Related Topics, Brest, France, pp. 161{164, Sept.2000.