First, Let us consider a transmission of code word C (n, k, d) of binary symbols {0, 1} using a BPSK modulation over AWGN channel, the decoder soft output for the j^th bit position of a given soft input R (r₁, r₂ ・・・ r_n) is defined as:

where J is the number of orthogonal parity check equations in code C, n_i represents the number of elements in the i^th orthogonal equation, ik represents the k^th element of the i^th

parity check equation orthogonal to the j^th bit and.

The computation of extrinsic information can be approximated by replacing the formula (2) by:

Which is a good approximation since it requires only signal adders and comparators and can be most easily implemented without undue deterioration in error performance.

The algorithmic structure of the SISO threshold decoding can be summarized as follows:

For each bit

Compute the extrinsic information Le_j as (3)

Compute the Soft-input

Compute de Soft-output: LLR_j = (4Es/N0) r_j + Le_j

Correct de bit r_j

Table 1 shows some examples of OSMLD codes that can be decoded by theses codes are also known as self orthogonal codes. In this table we used the abbreviations DSC for Difference Set Cyclic codes, EG for Euclidean Geometry codes and BCH for Bose Chaudhuri and Hocquenghem codes. For an extensive description of projective geometry codes and Euclidean geometry can be found in [9] .

The Iterative decoding process (see Figure 1) can be described as follows: In the first iteration, the decoder only uses the channel output as input, and generates extrinsic information for each symbol. In subsequent iterations, a combination of extrinsic information and channel output is used as input.

The soft input respectively the soft output of the q^th iteration is given by:

where R represent the received data, L_e(q) is the extrinsic information computed by the previous iteration. In our procedure we use a fixed value 1/J for the pparameter and this for all iterations. The value chosen for reacts as if one took an average of all J estimators which contribute in the computation of Le.

Our iterative decoding process can also be applied to the concatenated codes; in this case R is a matrix whose rows are constructed from the C¹ code and columns from C²

code. Extrinsic information computed by the SISO decoder of code C¹ is given as a priori information to the second SISO decoder of code C². Table 2 shows some code built from the concatenation of OSMLD codes.

We wish to emphasize that our algorithm can be applied to simple majority decoding codes, product codes (PC) or parallel (PCB) [2] , serial concatenated codes [4] and quasi-cyclic codes [3] . Indeed in [10] we showed that our algorithm has a good compromise between complexity and performance compared to the belief propagation algorithm (BP). For example, for the code DSC (273, 191) our algorithm is 5 times faster than BP. Figure 2 depicts the performance of iterative decoding of (1057, 813) DSC code with rate 0.69 on AWGN channel. We can see that the performance improves with each iteration and no improvement after 15^th iteration, at this stage we can said that the decoding process has converged. The number of iterations for convergence as well as the SNR at which begins the enhancement (Point of separation of the different curves) will be predicted by the Exit chart.

EXIT chart [5] [6] [7] [8] has emerged as a useful tool for analyzing convergence behavior of iterative decoding of both Turbo codes and LDPC codes. Theses diagrams propose a graphic visualization of the evolution of the quality of the extrinsic information exchanged between two SISO decoders (Turbo codes) or between variable nodes and check nodes in case of LDPC codes. We succeed to apply EXIT charts technique to our

iterative decoding process while only one SISO decoder is used. The evolution of extrinsic information at the output of our SISO decoder is visualized during iterations. Figure 3 represents the simulation model to be implemented for calculation of EXIT chart. Let X denote the binary input to an additive white Gaussian noise AWGN channel, X = {±1}, and L_CH its corresponding output expressed as a log-likelihood ratio LLR. The soft-input/soft-output (SISO) decoder receives as its input both L_CH and a priori information L_A, which is also expressed as a LLR.

For purposes of the analysis, it is assumed that the sequences {L_A} and {L_CH} at the input to the decoder are infinitely long. One can easily show that { L_CH } Gaussian (conditioned on X) with parameters (2/s², 4/s²) where s² = N₀/2Es. Under the assumption that the a priori LLR {L_A} follows a Gaussian distribution we have where and h are a Gaussian variable of zero mean and variance. The mutual information I_A = I (X; A) between X and A which can be expressed as:

We put

where and. The function J is monotonically increasing and can be calculated numerically only once because J and J⁻¹ are the same for all codes.

On the other hand the mutual information between the extrinsic LLR of the I_E and X, I_E = I (X; E) can be expressed as:

On the other hand the mutual information between the extrinsic LLR of the I_E and X, I_E = I (X; E) can be expressed as:

It is impossible to express this relation in analytical form; the distributions p_E is most conveniently determined by means of Monte Carlo simulation. Viewing I_E as a function of I_A and the signal to noise ratio SNR value, the extrinsic information transfer characteristics is defined as:

For a concatenated constructed from C¹ (n1, k1) and C² (n2, k2) and a fixed SNR, the different steps of calculation can be summarized as follow:

Initialize a word w of size (k1 × k2) with bit 0

Initialize the maximum of iteration at Max

Modulate w with Binary phase shift key (BPSK)

Transmit de code word over an AWGN channel

Do

Generate L_A for columns (AWGN/sA) and compute I_A

Decode Lines of the received word and compute I_E

sA = sA + (sA/Max)

Iteration = Iteration + 1

While (iteration! = Max)

In order to guarantee a good approximation of the graphical output, it is convenient to work with blocks of size of the order of 10⁵. However, the approximation of the distribution of L_E by a Gaussian variable is checked in the case of finite length codes (Figure 4). This assumption will allow us to validate curves Exits reports rated to the BER curves.

Figure 4 shows the evolution of the distribution of the extrinsic information at the output of the SISO threshold decoder of PC code (74529, 36481). We simulated the transmission of a frame without any approximation.

Transfer characteristics for different SNR are shown in Figure 5. The a priori input I_A is on the abscissa, the extrinsic output IE is on the ordinate. The SISO threshold decoding algorithm is applied to (819, 447) OSMLD code of rate 0.54. The numbers on different curves are the SNR values in dB. The obtained diagrams look like those of Turbo code.

To view the convergence behavior of the iterative decoding, we draw in the same figure (see Figure 6) the exit charts of the two decoders. In the case of a symmetrical concatenated code the curve of the second decoder is obtained by inverting the symmetry axis of the I_E and I_A.

Figure 6 depicts the Exit chart of the concatenated code PC (74529, 36481) constructed from DSC (273, 191) for SNR 2.5, 3, 3.15 and 3.5 dB. The expected decoding trajectory is the “stairs”-like and can be explained as follows: Let q be the iteration index. For q = 0 the decoding process starts at the origin with zero a priori knowledge I_A,0 = 0. At iteration q, the extrinsic output I_E,q = T (I_A,q) is forwarded as a priori knowledge for the next decoder I_A,q+1 = I_E,q, and so on, until I_A = I_E = 1 after about 16 iterations. Each step of the staircase curve represents one iteration of the decoding process.

Figure 7 shows the Exit chart curve for iterative decoding of asymmetric concatenated code constructed from DSC (273, 191) and DSC (73, 45) at SNR 2.8 dB. In this case and contrary to the result obtained in the figure 6 we calculated and plotted the transfer curves for each Component elementary decoder. As we can see both transfer curves enters a tunnel at SNR = 2 dB away from the x = y − axis.

BER chart vs. Exit chart

Two main regions can be defined from the curve of BER (see Figure 8). The first is called “pinch-off Region” (for SNR < 2.9 dB) whose BER remains constant and iteration has no effect. The second is called “waterfall Region” (SNR > 2.9 dB) here, the BER decreases excessively when the rate increases SNR. These three regions can be illustrated using the exit chart (see Figure 6).

“Pinch-off Region” the two decoders transfer curves cross at small values of mutual information (no convergence).

“Waterfall Region”: the decoding trajectory enters a tunnel area [5] near the bisector with a slower convergence rate.

For SNR = 3.15 dB we can observe that the decoding trajectory enters an area of tunnel close to the x = y − axis with a deceleration characteristic of convergence’s speed. The bit error rate (BER) versus SNR performance on AWGN channel is displayed in Figure 8. As we can observe, the waterfall region appears to take shape at SNR > 2.9 dB, which is 0.25 dB away from Exit chart threshold. This uncertainty of estimate is due mainly to the fact that we use a code with finite length.