Near Shannon Limit Error - Correcting Coding and Decoding :

This paper deals with a new class of convolutional codes called Turbo-codes, whose performances in terms of Bit Error Rate (BER) are close to the SHANNON limit. The Turbo-Code encoder is built using a parallel concatenation of two Recursive Systematic Convolutional codes and the associated decoder, using a feedback decoding rule, is implemented as P pipelined identical elementary decoders. I INTRODUCTION Consider a binary rate R=1/2 convolutional encoder with constraint length K and memory M=K-1. The input to the encoder at time k is a bit dk and the corresponding codeword Ck is the binary couple ( X k , Y k ) with K -I i = O x k = z g I i d k ; m d . 2 g l i =0,1 (la) K-1 Yk = zg2idk-i m d . 2 82i =0,1 ( I b ) i = O where GI: {gl i ) , G2: ( g 2 i } are the two encoder generators, generally expressed in octal form. It is well known, that the BER of a classical Non Systematic Convolutional (NSC) code is lower than that of a classical Systematic code with the same memory M at large SNR. At low SNR, it is in general the other way round. The new class of Recursive Systematic Convolutional (RSC) codes, proposed in this paper, can be better than the best NSC code at any SNR for high code rates. A binary rate R=1/2 RSC code is obtained from a NSC code by using a feedback loop and setting one of the two outputs Xk or Yk equal to the input bit dk. For an RSC code, the shift register (memory) input is no longer the bit dk but is a new binary variable ak. If Xk=dk (respectively Yk=dk), the output Y k (resp. X k ) is equal to equation (lb) (resp. la) by substituting ak for dk and the variable ak is recursively calculated as K 1 ak = dk + 1 ria,-; m d . 2 ( 2 ) i = l where yi is respectively equal to gli if Xk=dk and to g 2 i if Yk=dk. Equation (2) can be rewritten as K -1 i = O dk = Z 7 ; U k i tmd.2. (3) One RSC encoder with memory M=4 obtained from an NSC encoder defined by generators G1=37, G2=21 is depicted in Fig. 1. Generally, we assume that the input bit dk takes values 0 or 1 with the same probability. From equation (2), we can show that variable ak exhibits the same statistical property 0-7803-0950-2/93/$3.00Q1993IEEE 106.1 Pr{ak =El,..ak-l =Ek-l)=Pr(dk = E ) = 1 / 2 (4) with E is equal to K -1 I=1 E = C y i e i m d . 2 E =0,1. ( 5 ) Thus the trellis structure is identical for the RSC code and the NSC code and these two codes have the same free distance df However, the two output sequences (Xk} and { Yk ) do not correspond to the same input sequence (dk) for RSC and NSC codes. This is the main difference between the two codes. When punctured code is considered, some output bits X k or Y k are deleted according to a chosen puncturing pattern defined by a matrix P . For instance, starting from a rate R=1/2 code, the matrix P of rate 2/3 punctured code is