Channel Coding and Information Theory

DOI: 10.1049/cp:20061337 ISBN: 0 86341 644 6 Location: Hangzhou, China Conference date: 6-9 Nov. 2006 Format: PDF The BER performance of presented quasi-cyclic (QC) low-density parity-check (LDPC) codes is not good as that of randomly constructed LDPC codes, while the lack of structure of randomly constructed LDPC codes implies serious disadvantages in terms of storing and accessing a large parity-check matrix. To solve the problems, this paper proposes a design of good quasi-cyclic (QC) low-density parity-check (LDPC) codes, the obtained algebraically structured codes have large minimum distances and good BER performance. The proposed design is based on index matrices, which determines the shifts of the circulant matrices of the sparse parity-check matrices. Compared with the presented design of QC LDPC codes, the designed QC LDPC codes are free from girth 4, sometimes free from girth 6, and have much larger minimum distances. Based on the algebraic code structure, the conditions of the girth and minimum distance of the codes are found. The BER performance of the designed QC LDPC block codes compares with that of randomly constructed LDPC codes for any block lengths. Better BER performance is obtained by increasing the circulant size of the base QC code. Simulations verify the construction of the QC LDPC codes to be valid. (4 pages) Inspec keywords: AWGN channels; error statistics; channel coding; algebraic codes; parity check codes; sparse matrices Subjects: Algebra; Other topics in statistics; Codes

Low-density Parity check codes (LDPC) are gaining interest for high data rate application in both terrestrial and spatial communications which requires low bit error rate. This paper proposes an algorithm through which the best parity check matrix is obtained for irregular low density parity check codes, which reduces the Bit Error Rate ,Frame Error Rate and computational complexity. Tanner graph also called a bipartite graph format in an Low Density Parity Check (LDPC) code design contains many short cycles; it will produce a computational complexity in a code design process and it degrades the code performance. To overcome this effect, we proposed an new algorithm which obtains a best final parity check matrix. Throughout the paper we will give the detailed description of an efficient algorithm proposed. The complexity is studied graphically and analytically. Using proposed algorithm, cycles of length four observed in corresponding matrix are removed. Each matrix is evaluated over a noisy Additive White Gaussian Noise (AWGN) channel. Bit Error rate and Frame error rate is calculated. The results are compared with randomly generated best parity check matrix. Simulation results shows that proposed method successfully overcomes the computational complexity and meets the near Shannon limit.

This letter gives a random construction for Low Density Parity Check (LDPC) codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has great flexible choice in LDPC code’s parameters including codelength, code rate, the least girth of the graph, the weight of column and row in the parity check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Systemic codes have many applications in digital communication, so this letter proposes a construction of the generator matrix of systemic LDPC codes from the parity check matrix. Simulations show that the method performs well with iterative decoding.

In this chapter we cover some of the more common applications of LDPC codes. The breadth and depth of work reported in the literature on the application of LDPC codes to data transmission and storage would fill many volumes. Thus, this chapter has the modest goal of introducing selected applications. These applications include coded modulation, turbo equalization and code design for intersymbol-interference channels, estimation of the level LDPC error floors, and decoder design for low error floors. We also provide in this chapter a brief introduction to LDPC convolutional codes. Because of their importance, we also include a section on so-called digital fountain codes, which have application to file transfer over networks and closely resemble LDPC codes.

We propose a novel concatenated coding scheme combined Low Density Parity Check(LDPC) code and vertical Bell Labs layered space-time (VBLAST) code to improve the bit error rate (BER) performance for Free-space optical (FSO) communication links with On Off Keying(OOK) modulation. In the scheme, π-rotation encoding and belief-propagation (BP) decoding algorithm are employed in LDPC code, and minimum mean-square error (MMSE) soft-decision decoding algorithm is used for VBLAST code. Moreover, a joint iterative detection and decoding algorithm of VBLAST code and LDPC code is used to improve the overall link performance. At last, the BER of the proposed scheme over lognormal atmospheric turbulent fading channels is analyzed by using Monte Carlo method. The result shows that the BER of the proposed concatenated coding scheme is much improved than that only use VBLAST code scheme.

The queued-code based on low-density parity-check (LDPC) codes, which exploit both the near Shannon-limit performance LDPC codes and the instantaneous channel state information (CSI), can provide excellent performance in the block fading channels. The existing work utilized the conventional random puncturing and repetition operation, which may lead to considerable performance loss due to the changed intrinsic connections of parity-check matrix (PCM) for the receiver. In this paper, we present a reversed sequential puncturing (RSP) scheme and a construction method with modified dual-diagonal structure for the LDPC coded queued-code. The consecutive puncturing pattern in reversed direction can ensure the uniform row-weight distribution of the punctured PCM. The modified dual-diagonal form and the RSP scheme can ensure no more than one punctured symbol involved in each check-sum equation. The simulation results demonstrate that our proposed scheme can achieve noticeable performance gains.

In this paper we propose an innovative channel coding scheme called 'Accumulate Repeat Accumulate codes' (ARA). This class of codes can be viewed as serial turbo-like codes, or as a subclass of Low Density Parity Check (LDPC) codes, thus belief propagation can be used for iterative decoding of ARA codes on a graph. The structure of encoder for this class can be viewed as precoded Repeat Accumulate (RA) code or as precoded Irregular Repeat Accumulate (IRA) code, where simply an accumulator is chosen as a precoder. Thus ARA codes have simple, and very fast encoder structure when they representing LDPC codes. Based on density evolution for LDPC codes through some examples for ARA codes, we show that for maximum variable node degree 5 a minimum bit SNR as low as 0.08 dB from channel capacity for rate 1/2 can be achieved as the block size goes to infinity. Thus based on fixed low maximum variable node degree, its threshold outperforms not only the RA and IRA codes but also the best known LDPC codes with the dame maximum node degree. Furthermore by puncturing the accumulators any desired high rate codes close to code rate 1 can be obtained with thresholds that stay close to the channel capacity thresholds uniformly. Iterative decoding simulation results are provided. The ARA codes also have projected graph or protograph representation that allows for high speed decoder implementation.

Approximate calculation of channel log-likelihood ratio (LLR) for wireless channels using Padé approximation is presented. LLR is used as an input of iterative decoding for powerful error-correcting codes such as low-density parity-check (LDPC) codes or turbo codes. Due to the lack of knowledge of the channel state information of a wireless fading channel, such as uncorrelated fiat Rayleigh fading channels, calculations of exact LLR for these channels are quite complicated for a practical implementation. The previous work, an LLR calculation using the Taylor approximation, quickly becomes inaccurate as the channel output leaves some derivative point. This becomes a big problem when higher order modulation scheme is employed. To overcome this problem, a new LLR approximation using Padé approximation, which expresses the original function by a rational form of two polynomials with the same total number of coefficients of the Taylor series and can accelerate the Taylor approximation, is devised. By applying the proposed approximation to the iterative decoding and the LDPC codes with some modulation schemes, we show the effectiveness of the proposed methods by simulation results and analysis based on the density evolution.

The increasing demand for high-density and high-speed digital optical recording systems has made the development of advanced coding and signal processing techniques for fast and reliable data recovery increasingly important. In recent years, the parity-check (PC)-code-based reception technique has been widely studied for magnetic recording systems, and it is projected to be highly promising for high-density optical recording. The PC code is an inner error correction code (ECC), which can detect dominant short error events of the system, using only a few parity bits. This reduces the loss in error correction capability of the outer ECC due to random short errors and results in a simple and efficient solution to improve the overall performance. This thesis is dedicated to the design and analysis of PCcode- based recording systems to achieve higher recording capacity with low implementation complexity, for high-density blue laser disc systems. In particular, most of the key components of the PC-code-based optical recording system have been designed and optimized for different recording densities, and different proportions of white noise and media noise. During the development of advanced coding and detection techniques, it becomes necessary to investigate the system’s performance with different coding schemes and recording densities. In Chapter 2 of the thesis, we propose a generalized Braat-Hopkins model for optical recording channels, which provides a fair basis for the performance comparison of detection schemes over different coding schemes and recording densities, for channels with additive, white Gaussian noise (AWGN) and media noise. Various basic issues associated with the PC-code-based systems are investigated in Chapter 3. These include bounds for bit error rates and error event probabilities, the dominant error events at channel detector output, different PC codes, simple and efficient post-processors, the impact of error events that are split across data block boundaries and the corresponding remedy, as well as the effects of code rates and recording densities. Simulation results show that a 4-bit PC code achieves the best performance. The corresponding bit error rates (BERs) are very close to the performance bounds, at both nominal and high densities. Constrained codes, which serve as a key component in the read-write channel of data storage systems, are desired to have a high code rate with low-complexity encoder/decoder. In Chapter 4, we investigate the design of certain capacity-approaching constrained codes for optical recording systems. In particular, we derive analytically the relationship between the number of encoder states and the maximum size of these codes. We identify the minimum number of encoder states that maximizes the code rate, for any desired codeword length. The design of constrained PC codes is key to the development of PCcode- based systems, and the systematic design of efficient constrained PC codes remains a challenging problem. In Chapter 5, we propose a general and systematic code design technique for constructing capacity-approaching constrained PC codes, which can detect any type of error events of the system, with the minimum code rate loss. Compared to the rate 2/3 code without parity, the newly designed constrained 4-bit PC code achieves a performance gain of 2 dB at nominal density, and 1.5 dB at high density, at BER = 10-5. As the dominant noise for high-density optical recording systems, media noise severely degrades the performance of channel detectors and postprocessors that are designed for AWGN. In Chapter 6, we propose two novel modifications to the bit detector to combat media noise. We further develop a data-dependent post-processing scheme for error correction. Compared to the system designed without considering media noise and without PC codes, the overall performance gain of the developed scheme can be more than 11 dB at high media noise levels. In data storage systems, the ECC failure rate (EFR) serves as the ultimate measure of the data recovery performance. In Chapter 7, we develop semi-analytical approaches for estimating the EFR, and analyze EFRs for the developed PC-code-based systems, for the cases without and with interleaving. Our analysis shows that with the optimum interleaving degrees, compared to the rate 9/13 code without parity, the 2-bit PC code achieves a gain of around 0.5 dB, and the 4-bit PC code gains 0.6 dB, at high record ing density and EFR = 10-16. The thesis concludes in Chapter 8 with some suggestions for further work.

Nonbinary LDPC coding is a class of excellent error correcting codes in wireless communication systems. The hardware realization of this decoder architecture put forth a challenge at making a better tradeoff between the computation complexities and decoding performance. This letter proposes modifications in the structure of H-matrix which characterizes the Low Density Parity Check (LDPC) codes. The modification is aimed towards the reduction of computation complexity by retaining the properties of H-matrix. Two masking matrices are proposed with reduced number of non- zero elements, forming the diagonals of the parity check matrix. Simulation results indicate improved decoding performance and reduced computational complexity with the proposed H-matrix structures.

In this paper a coded modulation method for the additive white Gaussian noise channel is presented which incorporates signal shaping. The scheme uses a modulation map to map bits to channel inputs. A modulation map is considered that has the advantage it is can easily be combined with binary error-correcting codes. The decoding strategy is a combination of bit-interleaved coded modulation and multistage decoding. The number of required decoding stages is only small compared to other multistage decoding schemes for coded modulation. It is shown that modulation maps can be designed with a constrained capacity limit very close to the capacity of the AWGN channel. When combined with low-density parity-check codes, simulation results show that a near-capacity performance is achieved for high spectral efficiencies.

The principle of coding in the signal space follows directly from Shannon’s analysis of waveform Gaussian channels subject to an input constraint. The early design of communication systems focused separately on modulation, namely signal design and detection, and error correcting codes, which deal with errors introduced at the demodulator of the underlying waveform channel. The correct perspective of signal-space coding, although never out of sight of information theorists, was brought back into the focus of coding theorists and system designers by Imai’s and Ungerböck’s pioneering works on coded modulation. More recently, powerful families of binary codes with a good tradeoff between performance and decoding complexity have been (re-)discovered. Bit-Interleaved Coded Modulation (BICM) is a pragmatic approach combining the best out of both worlds: it takes advantage of the signal-space coding perspective, whilst allowing for the use of powerful families of binary codes with virtually any modulation format. BICM avoids the need for the complicated and somewhat less flexible design typical of coded modulation. As a matter of fact, most of today’s systems that achieve high spectral efficiency such as DSL, Wireless LANs, WiMax and evolutions thereof, as well as systems based on low spectral efficiency orthogonal modulation, feature BICM, making BICM the de-facto general coding technique for waveform channels. The theoretical characterization of BICM is at the basis of efficient coding design techniques and also of improved BICM decoders, e.g., those based on the belief propagation iterative algorithm and approximations thereof. In this text, we review the theoretical foundations of BICM under the unified framework of error exponents for mismatched decoding. This framework allows an accurate analysis without any particular assumptions on the length of the interleaver or independence between the multiple bits in a symbol. We further consider the sensitivity of the BICM capacity with respect to the signal-to-noise ratio (SNR), and obtain a wideband regime (or low-SNR regime) characterization. We review efficient tools for the error probability analysis of BICM that go beyond the standard approach of considering infinite interleaving and take into consideration the dependency of the coded bit observations introduced by the modulation. We also present bounds that improve upon the union bound in the region beyond the cutoff rate, and are essential to characterize the performance of modern randomlike codes used in concatenation with BICM. Finally, we turn our attention to BICM with iterative decoding, we review extrinsic information transfer charts, the area theorem and code design via curve fitting. We conclude with an overview of some applications of BICM beyond the classical coherent Gaussian channel.

Low Density Parity Check (LDPC) codes are Shannon-Limit approaching codes. Shannon stated the theoretical bounds for digital data transmission in a given channel capacity with error correction coding. Spread Spectrum also called as CDMA (Code Division Multiple Access) is an efficient way of radio wave communication deployed in commercial broadcasts, ham radios, military communications, police radios, etc. CDMA is a means of signal transmission in which the signal frequency is spread over a very wide bandwidth. Spreading of data is achieved prior to transmission with the help of an independent code called ‘Pseudo Noise Code’ (PN sequences), which is subsequently used in the receiver end to de-spread the signal received in order to recover the signal data. PN sequences based on prime numbers like Legendre & Weil sequences are used in the present work. An impressive part of LDPC lies in its decoding algorithm. Sum-Product Algorithm (SPA) decoding offers the best error performance amongst all the types of decoding techniques of LDPC codes and thus applied here. This paper demonstrates an idea of introducing LDPC codes along with DS-Spread Spectrum technique using Legendre & Weil sequences. In the present work, the encoding and decoding algorithms are applied after spreading of the signal. SPA decoding technique is successfully employed and simulated results are plotted taking an example design to illustrate the concepts discussed above.

Low-density parity-check (LDPC) codes are very powerful error correction codes for bit-interleaved coded modulation (BICM) schemes. In BICM schemes the bits of one symbol have unequal error protection. In this paper, we show how to improve the performance of given LDPC codes by mapping the variable nodes of certain degree in a special way to the different bit levels. Since the equivalent binary-input component channels for each individual bit level are not symmetric, we use the tool of i.i.d. channel adapters to force symmetry. After that, we optimize the mapping of the LDPC code to the different bit levels by a downhill algorithm that uses a generalized discretized density evolution to calculate the thresholds. We show by thresholds and by simulations that a special mapping of LDPC codes lowers the bit error rates compared to a random mapping of LDPC codes.

Orthogonal frequency division multiplexing (OFDM) system suffers extra performance degradation in fast fading channels due to intercarrier interference (ICI). Combining frequency domain equalization and bit-interleaved coded modulation (BICM), the iterative receiver is able to harvest both temporal and frequency diversity. Realizing that ICI channels are intrinsically ISI channels, this paper proposes a soft-in soft-out (SISO) maximum a posteriori (MAP) equalizer by extending Ungerboeck's maximum likelihood sequence estimator (MLSE) formulation to ICI channels. The SISO MAP equalizer employs BCJR algorithm and computes the bit log-likelihood ratios (LLR) for the entire received sequence by efficiently constructing a trellis that takes into account of the ICI channel structure. A reduced state (RS) formulation of the SISO MAP equalizer which provides good performance/complexity tradeoff is also described. Utilizing the fact that ICI energy is clustered in adjacent subcarriers, frequency domain equalization is made localized. This paper further proposes two computational efficient linear minimum mean square error (LMMSE) based equalization methods: recursive q-tap SIC-LMMSE equalizer and recursive Sliding-Window (SW) SIC-LMMSE equalizer respectively. Simulations results demonstrate that the iterative SISO RS-MAP equalizer achieves the performance of no ICI with normalized Doppler frequency f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> up to 20.46% in realistic mobile WiMAX environment.