For practical applications on channels other than the BEC the belief propagation algo-rithm is rather complicated, and often leads to a decrease in the speed of the decoder.
Therefore, often a discretized version of the belief propagation algorithm is used. The lowest level of discretization is achieved when the messages passed are binary. In this case one often speaks of a hard decision decoder or the bit-flipping decoded, as opposed to a soft decision decoder as introduced in the last section. Gallager himself described two hard decision decoding algorithms on the BSC alongwith the soft-decoder, when he initially proposed LDPC codes. Rather than the decoder complexity, he was interested in the analytical analysis of the decoder and conjectured that though the mathematical analysis of probabilisic decoding is difficult, a very weak bound on the probability of error (pe) of probabilistic decoding can be found by the much easier analytical analysis of these hard-decoders. Below is the basic structure of the commonly cited Gallager A and Gallager B hard-decoders alongwith their pe analysis.
4.3.1 Gallager A and Gallager B Hard-Decoding Algorithms
Gallager A Algorithm :
• Messages are from the set 0,1 and represent the current estimate of the decoder of a particular bit.
• At Check Nodes : The outgoing message from a check-node results from the computation of the XOR sum of the incoming (extrinsic) messages
• At Variable Nodes: The outgoing message equals the originally received message, except if all other incoming messages agree, in which case this common value is
4.3 Majority-Based (MB) Hard-Decoding Algorithm for Irregular LDPC Codes 93
sent
To derive the pe analysis for the above algorithm, we assume a Binary Symmetric Channel (BSC) with crossover probability p0e and a regular (n,j,k) code with n as the code-length, j as the variable-degree and k as the check-degree. If a variable node is received in error ( an event of probability p0 ) and pi is the probability of error of a variable node after iteration i, then it was showed [34] that pi+1 is given by
pi+1 =p0−perr−corr(pi) +perr−add(pi) (4.3) where perr−corr(pi) is the probability of correcting an error and perr−add(pi) is the probability of adding an error during iteration i+ 1. In the case of Gallager A algorithm perr−corr(pi) is given by Gallager proposed that stronger results for decoding will be achieved if for some integer b, a variable-node is changed whenever b or more of the parity-check constraints rising from the variable-node are violated, unlike the case of the Gallager A algorithm where all the parity-check constraints of the incoming messages need to be violated, for a variable node to change its value . This modified decoding is commonly termed as Gallager B decoding algorithm. The value of perr−corr(pi) in equation 4.4 this case is given by where Clj−1 indicates the binomial coefficient.
94
Chapter 4 : Simplistic Algorithm for Irregular LDPC Codes Optimization Based on Wave Quantification
4.3.2 Concept and Algorithm for MB Hard Decoding
The basic underlying concept of MB-decoding is evident from its name i.e. the decision on whether the flipping of a variable-node’s value at each iteration is taken based upon the consensus over themajorityof incoming messages. First detailed works on the MB-based algorithms were performed by Zarrinkhat et al. [49] where he defined MB hard-decoding algorithms of varying order (w). The order w defines the strength of the majority i.e.
how many more than the at least half the incoming messages are in agreement with each other where 0 ≤ w≤ ⌊(j−1)/2⌋. The basic per iteration computations for an order w MB hard-decoder can be summarised as:
• Messages are from the set 0,1 and represent the current estimate of the decoder of a particular bit.
• At Check Nodes : The outgoing message from a check-node results from the computation of the XOR sum of the incoming (extrinsic) messages
• At Variable Nodes: The outgoing message equals the originally received message, except if a ⌈(j−1)/2⌉+w or more of the incoming messages disagree, in which case this common value is sent
The value of w determines the order and hence the strength of the disagreement required for the change. Extreme cases are standard majority-decoding algorithm (order 0) i.e. w = 0 , and the case where all the incoming messages unanimously agree on a value (maximum order and same as Gallager A algorithm) i.e.w = ⌊(j−1)/2⌋. The pi
evolution as represented by 4.4 in this case is given by the modification in the value of perr−corr(pi) as
4.3 Majority-Based (MB) Hard-Decoding Algorithm for Irregular LDPC Codes 95 Figure 4.6 Convergence of Probability of Error for Ensemble (3,6) for p0 = 0.0350 and p0 = 0.0450 Regular LDPC Code decoded by MB algorithm of order 0
4.3.2.1 Important Characteristics of MB Hard-Decoding Algorithms
Majority-based algorithms are especially attractive for their remarkably simple imple-mentation (per iteration) because of the exchange of only hard (0,1) messages alongwith their superieur performance to the classical Gallager A algorithm. Many important char-acteristics of the Majority-Based algorithms were studied in [49] that has led to renewed research interest in them in the recent years.
Classical graphical approach is generally employed to visualize the behavior of these iterative algorithms. Important characteristics like that of threshold values and speed of convergence can be easily depicted by means of such figures. The graphical approach, introduced originally by Gallager, has been used in similar contexts by others, including ten Brink [40]. It involves plotting down the values of pi as a function of pi−1. A use of this graphical interpretation of the convergence of pi (denoted by h(x) function) to 0 for the ensemble of (3,6) regular LDPC codes, decoded by MB algorithm of order 0 (w= 0) and for two varying initial probabilities of error (p0 = 0.0350) and (p0.0450 = 0) is depicted in the figure 4.6 [49]. It can be observed from the figure that depending on the channel parameter, there is either a decreasing (p0 = 0.0350) or an increasing (p0.0450 = 0) trend of average error probability with iterations. For a starting point on the line x =y , the algorithm is stuck right from the beginning, and there will be no decreasing or increasing trend.
Threshold Value of MB Algorithms :
96
Chapter 4 : Simplistic Algorithm for Irregular LDPC Codes Optimization Based on Wave Quantification
As clearly observed from the above figure that pi can follow both an increasing or a decreasing trend for a p0 and w. An important theorem with respect to the threshold value states [49] that for a given order w and a given p0 ∈ [0,0.5], pi is a decreasing sequence which converges to zero, if and only if the curve y=pi is below the liney=x,
∀x∈(0, p0). Threshold is conventionally defined as (p∗0), such that for all values p0 < p∗0, liml→∞pi = 0. Since it can be shown that pi is an increasing function of p0 such that p0 ∈ [0,0.5] and p0 ∈[0,0.5], then using the above mentioned theorem as well, it can be seen that if for a given p0 and an MB algorithm of order w (= M Bw) pi is decreasing and converges to zero, then for every pi ∈ [0, p0], pi is decreasing and converges to zero.
Hence, an alternate definition for the threshold p∗0 of an M Bw algorithm can be defined as the supremum of all p0 ∈ [0,0.5], such that the curve pi is below the line y = x for every pi−1 ∈(0, p0].
Convergence Speed of MB Algorithms :
Using the same graphical interpretation of the evolution of pi with respect to itera-tions as mentioned above, it was observed [49] that for a given p0, the convergence speed of an M Bw at any iteration depends on how far the curve pi is from the line y = x at that iteration. The farther the curve, the faster the trend. This explains the slow con-vergence of Gallager A algorithm in decoding the ensemble (dv, dc) regular LDPC codes over a channel withp0 < p∗0 and close top∗0 as inn this case, there is only a narrow tunnel between the curve and the line in the vicinity of x= 0 taking it longer to converge. Sim-ilarly it was concluded that between two majority-based algorithms used to decode the same ensemble of regular LDPC codes over the same channel, the one with smaller order eventually converges faster at the later stages of decoding. Another important result re-garding the convergence of the MB algorithms was that in comparison with the Gallager A algorithm, whose convergence to zero is exponential with respect to the number of iterations, the convergence of MB algorithms is super-exponential.