Weighted-Bit-Flipping-Based Sequential Scheduling Decoding Algorithms for LDPC Codes

Abstract

Low-density parity-check (LDPC) codes can be applied in a lot of different scenarios such as video broadcasting and satellite communications. LDPC codes are commonly decoded by an iterative algorithm called belief propagation (BP) over the corresponding Tanner graph. The original BP updates all the variable-nodes simultaneously, followed by all the check-nodes simultaneously as well. We propose a sequential scheduling algorithm based on weighted bit-flipping (WBF) algorithm for the sake of improving the convergence speed. Notoriously, WBF is a low-complexity and simple algorithm. We combine it with BP to obtain advantages of these two algorithms. Flipping function used in WBF is borrowed to determine the priority of scheduling. Simulation results show that it can provide a good tradeoff between FER performance and computation complexity for short-length LDPC codes.

1. Introduction

Low-density parity-check (LDPC) codes were first invented by Gallager [1] but had been neglected for decades until Mackay brought them back to light in 1996 [2]. Since then, much attention had been attracted for their excellent Shannon limit approaching error-correcting performance through belief propagation (BP) [3] decoding algorithm. This iterative decoding algorithm, sometimes also called sum-product algorithm (SPA) [4], is a powerful algorithm to approximately solve many NP hard problems such as statistical inference in physics [5–7], hypothesis testing, cooperative localization, and channel coding.

There are lots of researches with various decoding algorithms. Among existing LDPC decoding algorithms, bit-flipping (BF) algorithms are the simplest. The operations of check-nodes in BF are modulo-two additions while the variable-nodes only need simple comparison operations. BF decoding algorithms are easy to implement, but they usually perform not so well when compared to BP decoding algorithms, so various weighted BF (WBF) [8–12] decoding algorithms were proposed.

To improve the standard BP decoding performance, several sequential scheduling strategies in BP have been invented. In sequential scheduling strategies, the messages are computed in a serial manner using the newest updated information. Sequential strategies were introduced as a sequence updates based on check-node (CSBP) [13–15] or variable-node (VSBP) [16, 17]. Simulations demonstrate that sequential strategies converge about twice as fast as the standard parallel BP decoding algorithms (Flood) without any extra computing burden. The kernel steps of sequential updating algorithms focus on finding the order of message updating which converges fastest. To our knowledge, the best decoding algorithms in the sense of performance is informed dynamic scheduling (IDS) [18–24] algorithms, which update messages dynamically. A metric called residual [18] is used in IDS which decides the updating order of propagated messages. Metric computing and selecting operations can cause significant increase in computational complexity in IDS. In order to achieve the tradeoff between decoding performance and computational complexity, a low-complexity sequential WBF-based scheduling algorithm is proposed, in which the priority used in WBF determines the order of scheduling. Simulation results show that it can provide a good tradeoff between block errors performance and complexity for short-length LDPC codes.

This paper is organized as follows. Section 2 reviews BP (Flood and CSBP) and WBF. Section 3 introduces our WBF-based serial BP (WBFSBP) strategies. Section 4 analyzes the computing complexity of WBFSBP. Section 5 reveals the simulation results. Section 6 draws the conclusions.

2. WBF and BP

Let 𝑁 and 𝐾 be the block length and the information length of a binary LDPC code; thus the rate of the code is 𝑟=𝐾/𝑁. Let 𝑀=𝑁−𝐾; an LDPC code is described by an 𝑀×𝑁 parity-check matrix 𝐇. 𝐻_𝑚𝑛  presents the entry of row 𝑚 and column 𝑛 in 𝐇. The set 𝒩(𝑚) denotes the nodes that adjoin 𝑚 by 𝒩(𝑚)={𝑛∶𝐻_𝑚𝑛=1}, similarly for the set 𝒩(𝑛)={𝑚∶𝐻_𝑚𝑛=1}. 𝒩(𝑚)/𝑛 denotes the set 𝒩(𝑚) excluding 𝑛. Binary phase shift keying (BPSK) modulation over an additive white Gauss noise (AWGN) channel is assumed for information transmission, which maps a codeword vector 𝐜=(𝑐₁,𝑐₂,…,𝑐_𝑛) into a vector 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛), where 𝑐_𝑖=2𝑥_𝑖−1, 𝑖∈[1,𝑁]. The received vector is 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑛), where 𝑦_𝑖=𝑥_𝑖+𝑛_𝑖, 𝑖∈[1,𝑁], and 𝑛_𝑖 is the AWGN. 𝐙=(𝑧₁,𝑧₂,…,𝑧_𝑛) denotes the hard-decision vector acquired from 𝐲∶𝑧_𝑖=1 if 𝑦_𝑖>0 and 𝑧_𝑖=0 if 𝑦_𝑖≤0. 𝐒=(𝑠₁,𝑠₂,…,𝑠_𝑀)=𝐳⋅𝐻^𝑇 denotes the syndrome of hard-decision vector 𝐳.

2.1. WBF

Firstly, a metric called “reliability” and a function-associated flipping probability of each check-node are calculated via [9] 𝑤_𝑛,𝑚=min_{(𝑚)𝑖∈𝒩⧵𝑛}𝑦_𝑖[](𝑚),𝑒,𝑚∈1,𝑀,𝑛∈𝒩_WBF,𝑛=_{(𝑛)𝑚∈𝒩}2𝑠_𝑚−1𝑤_𝑛,𝑚−𝛼𝑦_𝑛. (1) The WBF algorithm is described as follows [9]:(1)set iteration number 𝑘=0, take 𝑘_max as the maximum number of iterations, and pick up 𝑤_𝑛,𝑚,(2)calculate 𝑠^𝑘=𝑧^𝑘𝐻^𝑇; if 𝑠^𝑘=0, stop decoding and output 𝑧^𝑘,(3)for 𝑛=1,2,…,𝑁, calculate 𝑒^𝑘_WBF,𝑛 as follows: 𝑒^𝑘_WBF,𝑛=_{(𝑛)𝑚∈𝒩}2𝑠^𝑘_𝑚−1𝑤_𝑛,𝑚−𝛼𝑦_𝑛, (2)(4)𝑧^𝑛∗=𝑧^𝑛∗+1 based on 𝑛^∗=argmax_{[]𝑛∈1,𝑁}𝑒^𝑘_WBF,𝑛, (3)(5)𝑘=𝑘+1 and go to step (3), until stopping rules are satisfied.

2.2. Standard BP (Flood) Decoding Algorithm

Commonly, the communication of the nodes in the corresponding Tanner graph comprises the Flood algorithm. For every variable-node 𝑣_𝑗 and check-node 𝑐_𝑖, the corresponding message generation functions 𝑚_{𝑐𝑖→𝑣𝑗}=𝑓_{𝑐𝑖→𝑣𝑗}(𝑚) and 𝑚_{𝑣𝑗→𝑐𝑖}=𝑓_{𝑣𝑗→𝑐𝑖}(𝑚) are defined as follows [2]: 𝑚_{𝑣𝑗→𝑐𝑖}=_{𝑐𝑎𝑣∈𝒩𝑗⧵𝑐𝑖}𝑚_{𝑐𝑎→𝑣𝑗}+𝐶_𝑣𝑗,𝑚_{𝑐𝑖→𝑣𝑗}=2arctanh_{𝑣𝑏𝑐∈𝒩𝑖⧵𝑣𝑗}𝑚tanh^{𝑣𝑏→𝑐𝑖}2,𝐿𝑣_𝑗=_{𝑐𝑎𝑣∈𝒩𝑗}𝑚_{𝑐𝑎→𝑣𝑗}+𝐶_𝑣𝑗, (4) where 𝐶_𝑣𝑗=log(𝑝(𝑦_𝑗∣𝑣_𝑗=0)/𝑝(𝑦_𝑗∣𝑣_𝑗=1)) is the channel information of 𝑣_𝑗. 𝐿(𝑣_𝑗) denotes the LLR of 𝑣_𝑗.

As shown in Figure 1, in time slot 1, messages are passed to all of these check-nodes by all variable-nodes that connect to them, and in time slot 2, all of the processed messages are sent back to all variable-nodes. The Flood decoding algorithm is formally described in Pseudocode 1.

1: Set every 𝑚𝑐→𝑣=0 2: Set every 𝑚𝑣𝑗→𝑐𝑖=𝐶𝑣𝑗 3:   while stop rules unsatisfied do 4:  for all 𝑐𝑎 (𝑎=1,2,…,𝑀) 5:   for each 𝑣𝑏∈𝒩(𝑐𝑎), compute and generate 𝑚𝑣𝑏→𝑐𝑎 (𝑎=1,2,…,𝑀) simultaneously 6:   for each 𝑣𝑏∈𝒩(𝑐𝑎), compute and generate 𝑚𝑐𝑎→𝑣𝑏 (𝑎=1,2,…,𝑀) simultaneously 7: end while

Pseudocode 1 

Pseudocodes of Flood.

Figure 1 

Standard BP algorithm (Flood).

2.3. CSBP

The messages calculated in CSBP [15] are the same as those in the Flood decoding algorithms. The only difference between them is the way of messages updating. In every iteration of CSBP, the messages are updated in a sequential way: update every check-node one by one in an ascending order (or descending order) arranged in the corresponding Tanner graph. The passing process of messages in CSBP is shown in Figure 2. Check-nodes are represented by (𝑐₁,𝑐₂,𝑐₃), and (𝑏₁,𝑏₂,…,𝑏₆) for variable-nodes. In time slot 1, messages are passed to 𝑐₁ by all variable-nodes that connect to 𝑐₁. In time slot 2, the processed messages are sent back to those variable-nodes. The updating rules of 𝑐₂ and 𝑐₃ are identical with those of 𝑐₁. The realization of CSBP is described in Pseudocode 2.

1: Set every 𝑚𝑐→𝑣=0 2: Set every 𝑚𝑣𝑗→𝑐𝑖=𝐶𝑣𝑗 3:   while stop rules unsatisfied do 4:    for 𝑖  do 5:   for check-node 𝑐𝑖 6:     for every 𝑣𝑏∈𝒩(𝑐𝑖)  do 7:    compute and generate 𝑚𝑣𝑏→𝑐𝑖 8:     end for 9:     for every 𝑣𝑏∈𝒩(𝑐𝑖)  do 10:    compute and generate 𝑚𝑐𝑖→𝑣𝑏 11:     end for 12:    end for 13:   end while

Pseudocode 2 

Pseudocodes of CSBP.

Figure 2 

CSBP algorithm.

3. WBF-Based Serial BP

We will introduce our WBFSBP algorithm using a new message updating schedule. In general, sequential decoding algorithms are composed of two major steps.

Step 1. Determine the order of nodes to be updated.

Step 2. Compute the messages.

In Flood and CSBP, Step 1 is skipped. In IDS, these two steps are operated alternately for every node. It means that after updating each single node, “residual” computing and sorting operation is needed, which causes much extra computing complexity. For the sake of getting the tradeoff between performance and computing complexity, a simple sequential decoding algorithm called WBF-based serial BP is proposed. First, let us review the flipping function (𝑒^𝑘_WBF,𝑛) of a variable-node in WBF decoding algorithm described in Section 2. The larger the 𝑒^𝑘_WBF,𝑛 is, the more unreliable the corresponding variable-node is. For each check-node, a new metric is defined as follows: 𝐸^𝑘_WBF,𝑚=max_{𝑛∈𝒩(𝑚)}𝑒^𝑘_WBF,𝑛−submax_{(𝑚)𝑛∈𝒩}𝑒^𝑘_WBF,𝑛, (5) where submax (⋅) denotes the second maximal value of (⋅). If there is more than one unreliable variable-node connected to a check-node, the message passing within this check-node is almost nonsense. If all neighbor variable-nodes of a check-node are reliable, the priority of updating this check-node should be very low. So we concentrate on the check-node which has only one unreliable neighbor variable-node. The check-nodes with larger value of 𝐸^𝑘_WBF,𝑚 are more likely containing just one unreliable neighbor variable-node. So updating these check-nodes firstly can correct errors in time and speed up convergence, thus reducing the iterations. In contrast to IDS, the sorting operation is not dynamically decided node by node but is simply obtained from sorting the set of 𝐸^𝑘_WBF,𝑚 in each iteration. When the sorting is finished, the rest of updating computation is the same as that of CSBP. The detailed steps of WBFSBP are shown in Pseudocode 3.

0: Set iteration number 𝑘=0, take 𝑘max as the maximum number of iteration, pick up 𝑤𝑛,𝑚,  Set every 𝑚𝑐→𝑣=0, every 𝑚𝑣𝑗→𝑐𝑖=𝐶𝑣𝑗. 1: Compute 𝑠𝑘=𝑧𝑘𝐻𝑇, if 𝑠𝑘=𝟎, stop decoding and output 𝑧𝑘. 2: For 𝑛=1,2,…𝑁, compute           𝑒𝑘WBF,𝑛=∑𝑚∈𝒩(𝑛)(2𝑠𝑘𝑚−1)𝑤𝑛,𝑚−𝛼𝑦𝑛 3: sort 𝐸𝑘WBF,𝑚 in descending order and put the corresponding check-nodes to 𝑈 4: while stop rules unsatisfied  do 5:  for 𝑖  do 6:   for 𝑖th check-node 𝑐𝑈𝑖 in 𝑈 7:   for every 𝑣𝑏∈𝒩(𝑐𝑈𝑖)  do 8:   compute and generate 𝑚𝑣𝑏→𝑐𝑈𝑖 9:   end for 10:    for every 𝑣𝑏∈𝒩(𝑐𝑈𝑖)  do 11:    compute and generate 𝑚𝑐𝑈𝑖→𝑣𝑏 12:    end for 13:  end for 14: end while

Pseudocode 3 

Pseudocodes of WBFSBP.

4. Complexity

Let 𝑑_𝑣 and 𝑑_𝑐 represent the degree of any variable-node and check-node, respectively; 𝑒 is the total number of all edges in the corresponding Tanner graph, so 𝑒=𝑑_𝑣⋅𝑁=𝑑_𝑐⋅𝑀, where 𝑁 is the number of variable-nodes and 𝑀 is the number of check-nodes. In Tables 1 and 2, we have listed the computation complexity of check-to-variable phase and variable-to-check phase, respectively. In an iteration of Flood, every edge of the Tanner graph should be updated once for each direction, so the number of updated messages via check-to-variable and variable-to-check is 𝑒, respectively. Sequential updating strategies surpass the Flood without any extra cost. We note that the check-to-variable message updating computations in one IDS iteration is the same as reverse message updating computations in one WBFSBP iteration, both equal 𝑒. In each iteration, sorting of {𝐸^𝑘_WBF,𝑚} requires 𝒪(𝑁log𝑁) operations while dynamic scheduling strategies in IDS need 𝒪(𝑁²).

Table 1 

Computation complexity from variable-nodes to check-nodes.

Table 2 

Computation complexity from check-nodes to variable-nodes.

5. Simulation Results

The decoding performance of the Flood, CSBP, and WBFSBP over AWGN channels is presented in this part. The LDPC codes are constructed based on Gallager's random method without any 4-cycle.

Figures 3 and 4 present the FER performance of the (128,256) Gallager LDPC code with BP (Flood), CSBP, and WBFSBP decoding algorithms. The 𝑘_max is set to 20 and 50, respectively, and 𝛼 is set to 1.3 in the computation of 𝐸^𝑘_WBF,𝑚. Figure 3 shows that, at the FER of 1𝑒−4, WBFSBP acquires about 0.2 dB and 0.35 dB promotions over the CSBP and BP in the case of 𝑘_max=20, while in Figure 4, the coding gains are 0.25 dB and 0.45 dB in the case of 𝑘_max=50. Figure 4 also shows that at 𝐸_𝑏/𝑁₀=4, WBFSBP-20 can achieve the same FER performance as what BP-50 and CSBP-50 do. In other words, WBFSBP algorithm can reduce more than half of the iterations of that BP algorithm needed at some 𝐸_𝑏/𝑁₀.

Figure 3 

FER versus 𝐸𝑏/𝑁0 performance of BP (Flood), CSBP, and WBFSBP decoding algorithms with maximal number of iterations of 20 with LDPC code of blocklength of 256 and rate of 0.5. BP algorithm with maximal number of iterations of 20.

Figure 4 

FER versus 𝐸𝑏/𝑁0 performance of BP (Flood), CSBP and WBFSBP decoding algorithms with maximal number of iterations of 50 with LDPC code of blocklength of 256 and rate of 0.5. As compared, performance of max iteration 20 of WBFSBP is given.

Figure 5 presents the FER performance of the (102,204) Gallager LDPC code with above-mentioned decoding algorithms. The 𝑘_max is set to 20 and 𝛼 is set to 1.3. We see that WBFSBP-20 can beat BP-50 for all  𝐸_𝑏/𝑁₀. Compared to CSBP-20, WBFSBP-20 can acquire about 0.15 dB coding gain.

Figure 5 

FER versus 𝐸𝑏/𝑁0 performance of BP (Flood), CSBP and WBFSBP decoding algorithms with maximal number of iterations of 20 with LDPC code of blocklength of 204 and rate of 0.5. BP-20 denotes BP algorithm with maximal number of iterations of 20. As compared, performance of max iteration 50 of BP is given.

6. Conclusion

In this paper, a new sequential scheduling decoding algorithm is proposed, in which the order of message passing is based on the weight factor computed in WBF algorithm. We focus on the check-nodes which have just one unreliable neighbor variable-node. Updating these check-nodes preferentially can avoid errors of propagation, thus speeding up the convergence of decoding algorithm. Compared to IDS, our decoding algorithm gives a good tradeoff between error performance and decoding complexity. As a future work, the scheduling algorithm can be generalized to LDPC codes over GF(𝑞).

Acknowledgments

The authors would like to thank all anonymous reviewers whose constructive suggestions were very helpful. The work of this paper is sponsored by the State 863 Project (2008AA01Z227), the National Natural Science Foundation of China (NSFC), under Grant 61271204.