Design of Short, High-Rate DVB-S2-Like Semi-Regular LDPC Codes

Abstract

This work focuses on high-rate (𝑅>4/5) moderate-length (1000≤𝑛<5000) low-density parity-check codes. High-rate codes allow to maintain good quality of the preliminary decisions that are used in carrier recovery, while a moderate code length allows to keep the latency low. The interleaver of the LDPC matrix that we consider is inspired to the DVB-S2 standard one. A novel approach for avoiding short cycles is analyzed. A modified BP decoding algorithm is applied in order to deal with longer cycles. Simulations and results for the AWGN channel are presented, both for BPSK signalling and for coded modulation based on the partition ℤ²/4ℤ².

1. Introduction

An (𝑛,𝑘) LDPC code is a binary block code defined as the null space of the sparse “parity check” matrix 𝐇_{(𝑛−𝑘)×𝑛}, where 𝐇 can be viewed as the adjacency matrix of the bipartite graph 𝐺({𝐶,𝑉},𝐸), the Tanner graph, with ℎ_𝑖𝑗=1 meaning that the 𝑖th c-node is connected to the 𝑗th v-node. Richardson et al. [1] studied a method named Density Evolution (DE) for finding the pair of distributions (𝜆,𝜌), as 𝑛→∞, that allows to reach the fastest BER convergence at some probability error target 𝜖, under BP algorithm. For short and moderate-length LDPC codes, that is, for 10³≤𝑛<10⁴, the design of 𝐇 via DE shows relatively high error floors (@𝑃_𝑏≈10⁻⁶), therefore a different approach is required. The performance can be improved by conditioning the construction of 𝐇 in such a way that short cycles are avoided. A technique for the elimination of short cycles in regular LDPC codes has been recently presented in [2]. However, regular codes are outperformed by irregular or semiregular codes.

Eroz et al. [3] proposed a particular structure of 𝐇 that leads to irregular codes. This structure has been adopted for standardization of DVB-S2 applications (the DVB-S2 standard includes 2 versions with 𝑛=64800 and 𝑛=16200) and belongs to the family of IRA codes [4].

This paper addresses the problem of designing short DVB-S2-like codes, introducing a new technique for the elimination of cycles of length 4 and 6.

The remainder of the paper is outlined as follows. Section 2 introduces the system models taken under consideration and the matrix structure of the DVB-S2 standard. Section 3 explains the features adopted in designing the short-length DVB-S2-like LDPC codes. Section 4 shows simulation results and some comparisons with short codes designed via DE. Finally, some conclusions are drawn in Section 5.

2. DVB-S2-like Parity-Check Matrix

Here follow some recalls of the parity-check matrix structure described in [3].

The LDPC matrix 𝐇 is composed of two submatrices 𝐇^{1(𝑛−𝑘)×𝑘} and 𝐇^{2(𝑛−𝑘)×(𝑛−𝑘)}, 𝐇=[𝐇₁,𝐇₂]; the rightmost one is a lower bidiagonal matrix which offers the advantage of simple systematic encoding. Specifically, taking modulo-2 additions of consecutive rows of 𝐇₂, such that 𝐡^𝑖=𝐡^𝑖−1⊕𝐡_𝑖 for 𝑖=1,…,𝑛−𝑘 and 𝐡^0=𝟎, one gets the identity matrix 𝐈_{(𝑛−𝑘)}. Another advantage is to avoid short cycles in Tanner graph involving low-degree v-nodes, leading to improved performance as stated in [1].

Submatrix 𝐇₁ is built as follows: chosen an opportune value 𝑀 that divides both 𝑛−𝑘 and 𝑘, randomly generate the first column of 𝐇₁, 𝐡₁, and get the next 𝑀−1 columns by cyclic shifting of 𝑗((𝑛−𝑘)/𝑀) positions, respectively, for 𝑗=2,…,𝑀. The second group of 𝑀 columns is created in the same way, just after the first columns of the new group have been generated. This procedure allows to fill the entire submatrix 𝐇₁.

3. Code Design

3.1. Short Cycle Elimination

A cycle of length 𝑙 in the Tanner graph is a set of 𝑙/2v-nodes and 𝑙/2c-nodes such that a path of consecutive 𝑙 edges starts from a v-node (c-node) and ends at the same v-node (c-node).

Given two columns 𝐡_𝑖 and 𝐡_𝑗 of 𝐇, a length-4 cycle is formed when the inner product between the 2 columns is greater than 1:𝐡^𝑇𝑖⋅𝐡_𝑗≥2(1)for some 𝑖,𝑗∈{1,…,𝑛∶𝑖≠𝑗}. This condition suggests a practical way to check the presence of length-4 cycle in a parity-check matrix.

Focusing on a column of 𝐇, 𝐡_𝑖, let us define the set of the “cyclic-distances” between the nonzero entries of the 𝑖th column𝐷_𝑖=𝑑^{(𝑖)𝑝,𝑞}∶𝐡_𝑝,𝑖=1,𝐡_𝑞,𝑖=1,(2)where 𝑝,𝑞∈{1,…,𝑛−𝑘∶𝑝≠𝑞} and𝑑^{(𝑖)𝑝,𝑞}=⎧⎪⎨⎪⎩||𝑝−𝑞||if0<||𝑝−𝑞||≤𝑛−𝑘𝑛−𝑘−||𝑝−𝑞||if𝑛−𝑘2,2<||𝑝−𝑞||<𝑛−𝑘.(3)For example, if the nonzero entries of 𝐡_𝑖 are located at the positions {12,23,83} with 𝑛−𝑘=100, then 𝐷_𝑖={𝑑^(𝑖)12,23=11,𝑑^(𝑖)12,83=29,𝑑^(𝑖)23,83=40}. Also we have 𝐷_𝑖={1} for all the columns of 𝐇₂, briefly denoted with 𝐷_𝐇2={1}.

Given three columns 𝐡_𝑖, 𝐡_𝑗, and 𝐡_𝑘, a length-6 cycle is formed when the sum of two cyclic-distances of two distinct columns 𝑖,𝑗 equals a cyclic-distance of the third column 𝑘. An example is shown below (4) with 𝐷₂=2, 𝐷₄=1, and 𝐷₅=1 a length-6 cycle is formed: 𝑑^(4)2,3+𝑑^(5)1,2=𝑑^(2)1,3.

A complete search for the sets {𝐷_𝑖∶𝑖=1,…,𝑛} that avoid the presence of length-6 cycles is computationally expensive as 𝑛 and the average weight of the columns grow.

The problem of finding the longest (𝑛,𝑘) LDPC code that is length-4 and length-6 cycle free is described as follows: let 𝐷₀=𝐷_𝐇2 be the initial set of admissible cyclic-distances (i.e., length-4 and length-6 cycle free); when a new column 𝐡_𝑐 is added in the design of 𝐇₁ a new set of cyclic-distances 𝐷_𝑐 is added to the current set 𝐷_𝑗 according to the following update rule:𝐷_𝑗+1=𝐷_𝑗∪𝐷_𝑐∪(𝐷_𝑗⊕𝐷_𝑐),(5)as 𝑗 runs in ℕ₀={0,1,2,…}; the objective is to find the maximum 𝑗 s.t. the set 𝐷_𝑗 is cycle free. The binary operation 𝐴⊕𝐵 between two sets 𝐴 and 𝐵 is defined as𝐴⊕𝐵=min|||𝑎_𝑖−𝑏_𝑗|||,𝑛−𝑘−|||𝑎_𝑖−𝑏_𝑗|||,∀𝑖≠𝑗.(6)A new column 𝐷_𝑐 is an eligible set if𝐷_𝑗∩𝐷_𝑐∪𝐷_𝑗⊕𝐷_𝑐=∅.(7)A smart strategy is to choose the cyclic distances from a class of rest modulo a small integer 𝑚: in this way all cyclic distances will be eligible. For example, choosing the class of rest 1 modulo 𝑚=4, namely [1]₄, the sum of two cyclic-distances drawn from two distinct columns is in the class [1]₄+[1]₄=[2]₄≠[1]₄, so a length-6 cycle involving 3 columns cannot be created. When the sum of two distances is greater than (𝑛−𝑘)/2, the sum is complemented to 𝑛−𝑘 as illustrated in (3); usually 𝑛−𝑘∈[0]₄ so 𝑛−𝑘−([1]₄+[1]₄)=[0]₄−[2]₄=[2]₄≠[1]₄. This strategy automatically avoid the creation of length-4 cycles.

For some particular choices of 𝑘, 𝑀, 𝑚, and the average column weight, it is possible that the total number of eligible cyclic distances is not sufficient to build all the 𝑘/𝑀 columns. Actually the total number of original columns can be reduced to 𝑘/𝑀⋅𝑀/(𝑛−𝑘)=𝑘/(𝑛−𝑘), where (𝑛−𝑘)/𝑀 is the cyclic offset between two consecutive columns in the same c-node 𝑀-subset. Without loss of generality, consider the first c-node 𝑀-subset 𝐇_𝑀={𝐡_𝑖,𝑖=1,…,𝑀} and its shifted replica by 1 position denoted with 𝐇^𝑀≫1. Obviously, no cycles are created between the two subsets 𝐇_𝑀 and 𝐇^𝑀≫1; not only, but also the matrix [𝐇_𝑀,𝐇^𝑀≫𝑖] for 𝑖∈{1,2,…,(𝑛−𝑘)/𝑀−1} does not contain length-4 and length-6 cycles, so an original column can generate ((𝑛−𝑘)/𝑀)⋅𝑀−1=𝑛−𝑘−1 columns without any cycle.

3.2. v-Node Degree Distribution

In the semiregular construction the v-node degree distribution has the form𝜆(𝑥)=2(𝑛−𝑘)2(𝑛−𝑘)+𝑤_𝑣𝑘𝑥+𝑤^𝑣𝑘2(𝑛−𝑘)+𝑤_𝑣𝑘𝑥^𝑤𝑣−1,(8)that is, the submatrix 𝐇₁ has a constant column degree 𝑤_𝑣. The parameter 𝑤_𝑣 can assume only small values to keep the low-density nature of 𝐇 and hence few cycles in the factor graph.

It is known that there exists a tradeoff between the performance in the cliff region and the performance in the floor region. In [5] it is shown that the floor improves when 𝑤_𝑣 increases. However the weight 𝑤_𝑣 cannot be increased above a maximum value that depends on the code length and the rate, in order to maintain the low-density nature of 𝐇. This makes it difficult to obtain good performance in the floor region with codes of moderate length and high rate.

3.3. Modified BP Algorithm

The BP decoding algorithm is optimal with tree-like factor graph, a condition practically reached when 𝑛 is very large. When the block length is finite, the BP becomes a suboptimal algorithm in the sense that it no longer performs MAP decoding [6, 7]. The main idea in order to outperform the BP decoder is to include a procedure of hard decisions on the v-nodes' posteriors, called decimation. The modified algorithm resembles that in [8]: in the generic iteration the mutual information from the v-nodes to c-nodes 𝐼_𝑣𝑐 is estimated and some hard decisions are taken on the most reliable v-nodes to reach the desired value of 𝐼_𝑣𝑐 for the current iteration. The increment Δ𝐼_𝑣𝑐 between two consecutive iterations is assumed to be Δ𝐼_𝑣𝑐=1/𝑙_max, such that the expected maximum number of iteration is 𝑙_max.

4. Results

4.1. BPSK Signalling

In the BPSK system, the receiver computes the a priori metrics for the LDPC decoder𝐋_𝟎=2𝜎²𝐲,(9)where 𝐲 is the received vector and 𝜎²=𝑁₀/2 is the variance per dimension of the AWG noise. Several LDPC codes are compared, that are different for code rate (𝑅=9/10 and 𝑅=4/5), code length (𝑛=1000 and 𝑛=5000). Also, we considered LDPC codes designed via DE. The results obtained with DE are not reported here due to the poor performance in the floor region.

In Figure 1, there is a comparison between the original BP decoding algorithm and the modified version, working on the same semiregular code with 𝑛=5000, 𝑅=0.8, and 𝑤_𝑣=8. The modified version of the BP algorithm exploits the original version with a constant gain in the cliff region.

Figure 1 

Proposed watermarking system: (a) encoder and (b) decoder.

In Figure 2 are plotted 8 BER curves versus 𝐸_𝑏/𝑁₀=1/8𝑅𝜎² in dB. The solid curves represent the performance of the codes designed with the short cycle elimination procedure, while the dashed curves represent the performance of the codes designed as described in Section 2. The code with 𝑛=5000 and 𝑅=9/10 is 1 dB from capacity and the code with 𝑅=4/5 is about 1.6 dB from capacity (@𝑃_𝑏=10⁻⁵); the floor does not appear up to 𝑃_𝑏=10⁻⁷.

Figure 2 

Synchronization code detection.

4.2. Coded Modulation

Let us consider the coded modulation based on the 16-way partition ℤ²/4ℤ² where log₂16=4 coded bits 𝐜=[𝑐₁,𝑐₂,𝑐₃,𝑐₄] address one coset leader in [ℤ²/4ℤ²], and log₂(128/16)=3 uncoded bits 𝐮=[𝑢₁,𝑢₂,𝑢₃] address one out of 8 subconstellations. The coded vector 𝐜 is drawn from an LDPC encoder of rate 𝑅=𝑘/𝑛; the normalized redundancy is 𝜌=4((𝑛−𝑘)/𝑛)[b/2D] and the volume to noise ratio (VNR) is𝛼²=2^𝜌2𝜋𝑒𝜎².(10)

As suggested in [9] it has been adopted a partial Gray mapping rule.

At the receiver metrics for the 𝑗th received symbol and each of the 4 Gray-mapped bits are computed according to𝐿^(𝑖)0,𝑗=12𝜎²|||𝑦^𝑗−𝑧^0(𝑖)|||²−|||𝑦_𝑗−𝑧^1(𝑖)|||²(11)for 𝑖=1,…,4 and 𝑗=1,…,𝑛/4, such that 4⋅𝑛/4=𝑛 metrics are provided to the LDPC decoder.

Four different LDPC codes are compared; they differ for code rate (𝑅=9/10 and 𝑅=4/5 lead to overall redundancies of 𝜌=0.4[b/2D] and 𝜌=0.8[b/2D], resp.), code length (𝑛=2000 and 𝑛=6000 mean 𝑁=500 and 𝑁=1500 2D symbols, resp.). The LDPC decoder computes 𝑙_max=30 iterations of the modified BP algorithm. In Figure 3 are plotted 4 BER performance curves versus SNR in dB. The longer semiregular code shows good performance and error floor below 𝑃_𝑏=10⁻⁸. Figure 4 shows some WER curves versus the volume to noise ratio 𝛼², which is recalled in (10), and the sphere lower bound on the WER for the finite dimensionality of the lattice [10].

Figure 3 

Damaged synchronization code.

Figure 4 

Word error rate (WER) curves versus the volume to noise ratio (𝛼2) of some semiregular LDPC codes. Coded modulation based on the partition ℤ2/4ℤ2.

5. Conclusion

The design of the moderate length instance of the DVB-S2-like parity-check matrix has been considered. Several codes designed with a semiregular approach are worked out and their performance is obtained by running few iterations of the decoding algorithm. The proposed edge assignment and the modified BP algorithm, with a decimation procedure, allow to lower the error floor compared to the same codes designed via density evolution.