Investigations on Binary and Non-Binary LDPC Codes

I would also like to thank the other faculty members of the Department for their continuous support and encouragement throughout this course of study. I am grateful to all the members of the research and technical staff of the Department.

Abstract

Moreover, a technique for rate-compatible puncturing of the non-binary LDPC codes is proposed. Simulation studies are conducted to assess the performance of the proposed techniques on standard binary and non-binary LDPC codes.

List of Acronyms

List of Symbols

Lchj LLR channel for xj for NB LDPC codes M Total number of CN in Tanner graph M(i) Set of adjacent VN ofci. Ndi,bp Total number of stages di VN having bp punched bits N(T) Array of CNs associated with VN in TS T.

Introduction

Background
Motivation and Objectives

Trapping Sets and Short Cycles in Binary LDPC Codes
Puncturing of Non-binary LDPC Codes

Thesis Contribution
Organization of the Thesis

The behavior of the iterative decoder is different in the waterfall and fault floor regions. This technique is based on the lengths and EMD values of the short NB cycles found in the NB-LDPC codes.

Figure 1.1: Typical BER or FER curve for the LDPC codes

Low-density Parity-check Codes

Introduction

This chapter reviews some of the popular construction and decoding algorithms for the LDPC codes. Two popular algorithms for constructing short-length LDPC codes are presented in Section 2.5.

Graphical Representation of LDPC Codes

The structure of the Tanner graph determines the performance of an LDPC code under iterative decoding. The rate r of an LDPC code can be lower bound in terms of the degree distributions.

Figure 2.1: Tanner graph for the parity-check matrix shown in Example 2.1

Decoding of LDPC Codes

Sum-Product Algorithm

Pr (xj = 1|y, all checks involving xj have been satisfied). 2.6) In the expression above, Pr (xj =u|y, all checks involving xj are fulfilled) is the conditional probability that xj = u, u ∈ {0,1} given that (i) all the CNs ci, i ∈ N(j) is satisfied (i.e. all such components of the syndrome vector are zero) and (ii) the received sequence is y. In the above expression, Pr (check ci is satisfied|xj =u,y) is the conditional probability that the ith CN is satisfied (i.e. the ith element of the syndrome vector is zero) given that (i) the value of xj isu and ( ii) the received row is y.

Figure 2.3: Diagrammatic representation of the message passing in the SPA

Decoding Threshold for an Ensemble of LDPC Codes

Under the assumption that all associated checks tovj are conditionally independent given the value of xj, Vji can be calculated as. Based on the sign of LLR a posteriori, the hard decision is made according to (2.7).

Finite-length Construction of LDPC Codes

Progressive Edge-Growth Algorithm
Approximate Cycle Extrinsic Message Degree Algorithm

The GOS detection procedure during the construction of the jth column is presented in Algorithm 2.4. At the beginning of the algorithm, p(µt) for all nodes (VNs and CNs) except vj is initialized to ∞ (unvisited).

Error Floor of LDPC Codes

Using ηexpACE, two different Tanner plots with the same ηACE can be distinguished. Useful information about the degree of involvement of the VNs in the formation of the cycles can be obtained from ηexpACE.

Non-binary LDPC Codes

Decoding by FFT-based Q-ary Sum-product Algorithm

The NB-LDPC codes can be represented in terms of the parity check matrices as in the case of the binary codes. The complexity of the CN processing for the QSPA in the rudimentary form is Oqdc for a degree-dc CN. The iterative decoding of the NB-LDPC codes is usually described in the probability domain due to the feasibility of accomplishing the CN processing via the FFT.

Conclusions

To reduce the complexity of the MS algorithm, Declercq and Fossorier proposed the extended MS (EMS) algorithm [ 20 ]. The EMS decoding algorithm considers only a few number (nm) elements from the q-elements of a vector message occurrence at a CN. Since a limited number of elements are considered, the accuracy of the EMS decoding may be less than that of the FFT-QSPA.

Finding Dominant Trapping Sets of Irregular LDPC codes

Introduction

To address the error floor problem of the LDPC codes, the most crucial task is to find a list of the dominant TSs. The list of dominant TSs is useful for estimating the high SNR performance of the LDPC codes. This chapter proposes a method for identifying the dominant TSs for the irregular LDPC codes.

Definitions

The number of VNs present in a TS is commonly referred to as the size of the TS. In that case, the decoder will never be able to get out of the wrong state, even if the number of iterations is increased. In the remainder of the chapter, definition 3.2 will be used for a TS unless otherwise stated.

Figure 3.2: A different representation of the (5,3) TS in Figure 3.1

A Brief Literature Review on the Methods to Find Trapping Sets in LDPC Codes

Hierarchical Approach to Find the Trapping Sets

The adequacy of the two types of extensions, shown in Figure 3.4, is verified below. This specific type of expansion can be achieved by performing the expansion shown in Figure 3.4(a) sequentially for each of the two VNs to be added. Therefore, by considering only the two types of extensions in Figure 3.4, Nguyen et al.

Figure 3.3: Expansion of a (4,4) TS to a (5,3) TS

Proposed Method of Finding the Dominant Trapping Sets

According to Definition 3.2, the smallest possible TS for an irregular class code is (2,2). Construct the set L1 of classes of the form (a−1,∗) needed to find the TS of each class (a, b) in Ta. Construct the set L2 of classes of the form (a−2,∗) needed to find the TS of each class (a, b) in Ta.

Figure 3.9: An (a, b) TS from an (a − 1, b) TS

Numerical Results for Enumeration of Trapping Sets

The number of different classes of dominant TSs, ASs, and FASs discovered by the proposed technique, along with the results of the KB algorithm, are shown in Table 3.1. The numbers of dominant TSs of different classes found with the proposed technique, together with the results of theADDR method [3], are shown in Table 3.3. An interesting observation from Table 3.3 is that all numbers of different TS classes found by the proposed algorithm are multiples of 27.

Table 3.3: The numbers of dominant trapping sets of N = 648, M = 324 IEEE 802.11n standard code obtained by the proposed algorithm and by the ADDR method [3]

Conclusions

It is clear that the proposed method can find significantly more number of TSs than the ADDR method. Due to the cyclic nature of the submatrices of the parity check matrix, a TS with a specific subgraph has at least 26 other isomorphic copies. This fact also suggests that all TSs of each class have been discovered by the proposed algorithm.

On the Equality of the ACE and the EMD of a Cycle for ACE Spectrum

Introduction
Inequality of the ACE and the EMD in the Presence of Sub- cyclescycles
Sufficient Conditions for the Equality of the ACE and the EMD of a Cycle for the ACE Spectrum Constrained Codes
Simulation Results and Discussions
Conclusions

Therefore, the sufficient condition for equality between ACE and EMD is for C2i. Using (4.10) in (4.3), the sufficient condition for equality between ACE and EMD can be written as. We have derived three sufficient conditions for the equality of ACE and EMD for one cycle for the ACE spectrum-limited LDPC codes.

Figure 4.1: A cycle of length 6 with EMD=2

EXIT Chart Analysis of Puncturing for Non-binary LDPC Codes

Introduction

The authors in [43] noted that the grouping algorithm can also be applied to the NB-LDPC codes. We also describe the EXIT card models for the binary and the NB-LDPC codes. In Section 5.3, we propose the EXIT card model for puncturing for the NB-LDPC codes.

Background

Rate-compatible Puncturing Scheme for Binary LDPC Codes
Rate-compatible Puncturing Scheme for Non-binary LDPC Codes [43]
EXIT Chart for Binary LDPC Codes [68]
EXIT Chart for NB-LDPC Codes [10]

Formulation of the Decoding Steps in the LLR Domain
Formulation of the Mutual Information for an LLR-vector Random Variable
Modeling of the Message Vectors
Steps of the EXIT Chart for the NB-LDPC Codes

The EXIT diagram is essentially a superposition of the input/output mutual information transfer curves for the VN decoder (VND) and the CN decoder (CND). The non-Gaussianity of the channel LLR makes modeling the EXIT diagram difficult for the NB-LDPC codes. The steps of the EXIT scheme for the NB-LDPC codes can now be summarized as follows:

Proposed EXIT Chart for Punctured NB-LDPC Codes

Random Selection of the VNs and Regular Puncturing

Study on Quasi-regular NB-LDPC Codes

Selection of the VNs According to the Grouping Algorithm and Irregular PuncturingPuncturing

The effects of the puncture patterns with different values of bp can be examined with the help of the EXIT chart. Suppose IE,CND(IA,CND, j, τ) is the EXIT function for the outputU of the CN of degree j when the number of pierced VNs is in N τ. The number of Ndi,bp of the VNs to be pinked of a particular gradedi withbp number of pierced bits must not exceed the maximum numberNdg.

Figure 5.6(a) shows the thresholds for punctured codes over GF(2 6 ) at r l = 0.6. The threshold values are plotted against b p for different values of t

Simulation Results

Rate thresholds [30, 31] Optimized models rl ENb. of a given degree from GR can provide, then we select the remaining VNs of the same degree randomly from G0.

Figure 5.11: SER of the mother code over GF 2 4 with N = 142, M = 71 and the degree distribu- distribu-tions as in [30, 31]

Conclusions

A Cycle-based Rate-compatible Puncturing Technique for Non-binary

Introduction

In this chapter, we investigate the performance of puncturing schemes for NB-LDPC codes from a finite-length perspective. The order of piercing within each group is determined using a sorting algorithm. In [6], the ACE values of short cycles were taken into account when selecting punched bits.

Cycles in NB-LDPC Codes

If all CNs in the cycle path are not satisfied, that cycle is detrimental to the iterative decoding of the NB-LDPC codes. We derive below a theorem regarding the probability of a cycle becoming a legitimate NB cycle. Suppose the number of cycles of length 2l in binary sense present in the Tanner graph of the code is Nb,2l.

Figure 6.1: A cycle of length 6 along with the edge labels

Proposed Rate-compatible Puncturing Scheme

Let the set of VN candidates for puncture be denoted by Γ. VNs of the lowest recovery rate not yet selected are included in Γ. Search space. Γ is gradually reduced by taking into account the involvement of VN Γ in the formation of NB cycles. The two selection criteria for the sorting algorithm in [32] are:. a) number of surviving neighboring CNs associated with VN and (b) degree of VN.

Simulation Results

The selection of the VNs in a particular group is performed according to the sorting algorithm [32]. The selection of the VNs in a particular group is performed according to the proposed cycle-based sorting algorithm. This is due to the selection of the puncture VNs according to the cycle-based sorting technique in scheme (3).

Table 6.1: Numbers of VNs in different groups for the (4,8)-regular mother code over GF 2 4 with N = 250, M = 125

Conclusions

The advantages of the NB-LDPC codes over the equivalent binary counterparts in the context of the error floor are confirmed. For such kind of codes, we proposed an improved version of RC puncture algorithm by calculating the lengths and the EMD values of the short NB cycles. Observations in different contexts have established that the selection of the pierced VNs according to the cycle-based criterion can significantly improve the performance.

Conclusions

Summary of Contributions

Finding a list of dominant trapping sets of an LDPC code is an important task. Based on the hierarchical approach, we proposed a technique to find the dominant trapping sets of the irregular LDPC codes. The proposed technique has been able to find the dominant trapping sets of several commonly considered irregular LDPC codes.

Possible Future Work

The improved RC puncturing scheme for the NB LDPC codes discussed in Chapter 6 takes into account the EMD values of the short NB cycles. By comparing this scheme with the proposed scheme, the benefit of considering the EMD instead of the ACE can be analyzed. The cycle-based puncturing scheme proposed in Chapter 6 for the NB LDPC codes can be explored in the case of the binary LDPC codes.

Bibliography

Whang, “Eﬃcient Puncturing Method for Rate-Compatible Low-Density Parity-Check Codes,” IEEE Transactions on Wireless Communications, vol. Urbanke, "Design of capacity-approximation irregular low-density parity-check codes," IEEE Transactions on Information Theory, vol. Ashikhmin, "Design of low-density parity-check codes for modulation and detection," IEEE Transactions on Communications, vol.

List of Publications