Mathematical approach to channel codes with a diagonal matrix structure

Digital communications have now become a fundamental part of modern society. In communications, channel coding is an effective way to reduce the information rate down to channel capacity so that the information can be transmitted reliably through the channel. This thesis is devoted to studying the mathematical theory and analysis of channel codes that possess a useful diagonal structure in the parity-check and generator matrices. The first aspect of these codes that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal structure using polynomials. Using this framework, an efficient new method is proposed to obtain a generator matrix G from certain types of parity-check matrices with a so-called defective cyclic block structure. By the nature of this method, G can also be completely described by a polynomial, which leads to efficient encoder design using shift registers. In addition, there is no need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination. Following this work, we proceed to explore some of the properties of diagonally structured lowdensity parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. The first crucial property studied is the minimum free distance of LDPC convolutional code ensembles, an important parameter contributing to the error-correcting capability of the code. Here, asymptotic methods are used to form lower bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of minimum distance to block length for corresponding LDPC block codes. Another interesting property of these codes is the way in which the structure affects the performance in the infamous error floor (which occurs at high signal to noise ratio) of the bit error rate curve. It has been suggested that “near-codewords” may be a significant factor affecting decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel. A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords can be associated with so-called ‘trapping sets’ that exist in the Tanner graph of a code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional codes are analysed. Here, asymptotic methods are used to calculate a lower bound for the trapping set growth rates for several ensembles of asymptotically good protograph-based LDPC convolutional codes. This value can be used to predict where the error floor will occur for these codes under iterative message-passing decoding

FPGE implementation of LDPC codes

Low density parity check LDPC Code is a type of Block Error Correction code discovered and performance very close to Shanon’s limit .Good error correcting performance enables reliable communication. Since its discovery by Gallagar there is more research going on for its efficient construction and implementation. Though there is no unique method for constructing LDPC codes. Implementation of LDPC Code is done by taking different factors in to consideration such as error rate, parallelism of decoder, ease in implementation etc. This thesis is about FPGA implementation of LDPC codes and their performance evaluation. Protograph codes were introduced and analyzed by NASA's Jet Propulsion Laboratory in the early years of this century. Part of this thesis continues that work, investigating the decoding of specific protograph codes and extending existing tools for analyzing codes to protograph codes In this thesis I have taken the performance of LDPC coded BPSK modulated signal which is transmitted through AWGN channel and the performance is tested using MATLAB Simulation

Design and Performance Analysis for LDPC Coded Modulation in Multiuser MIMO Systems

The channel capacity can be greatly increased by using multiple transmit and receive antennas, which is usually called multi-input multi-output (MIMO) systems. Iterative processing has achieved near-capacity on a single-antenna Gaussian or Rayleigh fading channel. How to use the iterative technique to exploit the capacity potential in single-user and/or multiuser MIMO systems is of great interest. We propose a low-density parity-check (LDPC) coded modulation scheme in multiuser MIMO systems. The receiver can be regarded as a serially concatenated iterative detection and decoding scheme, where the LDPC decoder performs the role of outer decoder and the multiuser demapper does that of the inner decoder. For the proposed scheme, appropriate selection of a bit-to-symbol mapping is crucial to achieve a good performance, so we investigate and find the best mapping under various cases.Analytical bound serves as a useful tool to assess system performance. The search for powerful codes has motivated the introduction of efficient bounding techniques tailored to some ensembles of codes. We then investigate combinatorial union bounding techniques for fast fading multiuser MIMO systems. The union upper bound on maximum likelihood (ML) decoding error probability provides a prediction for the system performance, with which the simulated system performance can be compared. Closed-form expression for the union bound is obtained, which can be evaluated efficiently by using a polynomial expansion. In addition, the constrained channel capacity and the threshold obtained from extrinsic information transfer (EXIT) chart can also serve as performance measures. Based on the analysis for fast fading case, we generalize the union upper bound to the block fading case

Near-capacity fixed-rate and rateless channel code constructions

Fixed-rate and rateless channel code constructions are designed for satisfying conflicting design tradeoffs, leading to codes that benefit from practical implementations, whilst offering a good bit error ratio (BER) and block error ratio (BLER) performance. More explicitly, two novel low-density parity-check code (LDPC) constructions are proposed; the first construction constitutes a family of quasi-cyclic protograph LDPC codes, which has a Vandermonde-like parity-check matrix (PCM). The second construction constitutes a specific class of protograph LDPC codes, which are termed as multilevel structured (MLS) LDPC codes. These codes possess a PCM construction that allows the coexistence of both pseudo-randomness as well as a structure requiring a reduced memory. More importantly, it is also demonstrated that these benefits accrue without any compromise in the attainable BER/BLER performance. We also present the novel concept of separating multiple users by means of user-specific channel codes, which is referred to as channel code division multiple access (CCDMA), and provide an example based on MLS LDPC codes. In particular, we circumvent the difficulty of having potentially high memory requirements, while ensuring that each user’s bits in the CCDMA system are equally protected. With regards to rateless channel coding, we propose a novel family of codes, which we refer to as reconfigurable rateless codes, that are capable of not only varying their code-rate but also to adaptively modify their encoding/decoding strategy according to the near-instantaneous channel conditions. We demonstrate that the proposed reconfigurable rateless codes are capable of shaping their own degree distribution according to the nearinstantaneous requirements imposed by the channel, but without any explicit channel knowledge at the transmitter. Additionally, a generalised transmit preprocessing aided closed-loop downlink multiple-input multiple-output (MIMO) system is presented, in which both the channel coding components as well as the linear transmit precoder exploit the knowledge of the channel state information (CSI). More explicitly, we embed a rateless code in a MIMO transmit preprocessing scheme, in order to attain near-capacity performance across a wide range of channel signal-to-ratios (SNRs), rather than only at a specific SNR. The performance of our scheme is further enhanced with the aid of a technique, referred to as pilot symbol assisted rateless (PSAR) coding, whereby a predetermined fraction of pilot bits is appropriately interspersed with the original information bits at the channel coding stage, instead of multiplexing pilots at the modulation stage, as in classic pilot symbol assisted modulation (PSAM). We subsequently demonstrate that the PSAR code-aided transmit preprocessing scheme succeeds in gleaning more information from the inserted pilots than the classic PSAM technique, because the pilot bits are not only useful for sounding the channel at the receiver but also beneficial for significantly reducing the computational complexity of the rateless channel decoder

Spatially Coupled Sparse Regression Codes for Single- and Multi-user Communications

Sparse regression codes (SPARCs) are a class of channel codes for efficient communication over the single-user additive white Gaussian noise (AWGN) channel at rates approaching the channel capacity. In a standard SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix, and the user message is encoded in the indices of those columns. Techniques such as power allocation and spatial coupling have been proposed to improve the performance of low-complexity iterative decoding algorithms such as approximate message passing (AMP). In this thesis we investigate spatially coupled SPARCs, where the design matrix has a block- wise band-diagonal structure, and modulated SPARCs, which generalise standard SPARCs by introducing modulation to the encoding of user messages. We introduce a base matrix framework which provides a unified way to construct power allocated and spatially coupled design matrices, and propose AMP decoders for modulated SPARCs constructed using base matrices. We prove that phase shift keying modulated and spatially coupled SPARCs with AMP decoding asymptotically achieve the capacity of the (complex) AWGN channel. We also show via numerical simulations that they can achieve lower error rates than standard coded modulation schemes at finite code lengths. A sliding window AMP decoder is proposed for spatially coupled SPARCs that significantly reduces the decoding latency and complexity. We then investigate coding schemes based on random linear models and AMP decoding for the multi-user Gaussian multiple access channel in the asymptotic regime where the number of users grows linearly with the code length. For a fixed target error rate and message size per user (in bits), we obtain the exact trade-off between energy-per-bit and the user density achievable in the large system limit. We show that a coding scheme based on spatially coupled Gaussian matrices and AMP decoding achieves near-optimal trade-off for a large range of user densities. To the best of our knowledge, this is the first efficient coding scheme to do so in this multiple access regime. Moreover, the spatially coupled coding scheme has a practical interpretation: it can be viewed as block-wise time-division with overlap.Funded by a Doctoral Training Partnership Award from the Engineering and Physical Sciences Research Council

Sparse graph-based coding schemes for continuous phase modulations

The use of the continuous phase modulation (CPM) is interesting when the channel represents a strong non-linearity and in the case of limited spectral support; particularly for the uplink, where the satellite holds an amplifier per carrier, and for downlinks where the terminal equipment works very close to the saturation region. Numerous studies have been conducted on this issue but the proposed solutions use iterative CPM demodulation/decoding concatenated with convolutional or block error correcting codes. The use of LDPC codes has not yet been introduced. Particularly, no works, to our knowledge, have been done on the optimization of sparse graph-based codes adapted for the context described here. In this study, we propose to perform the asymptotic analysis and the design of turbo-CPM systems based on the optimization of sparse graph-based codes. Moreover, an analysis on the corresponding receiver will be done