Raptor Codes for BIAWGN Channel: SNR Mismatch and the Optimality of the Inner and Outer Rates

Fountain codes are a class of rateless codes with two interesting properties, first, they can generate potentially limitless numbers of encoded symbols given a finite set of source symbols, and second, the source symbols can be recovered from any subset of encoded symbols with cardinality greater than the number of source symbols. Raptor codes are the first implementation of fountain codes with linear complexity and vanishing error floors on noisy channels. Raptor codes are designed by the serial concatenation of an inner Luby trans-form (LT) code, the first practical realization of fountain codes, and an outer low-density parity-check (LDPC) code. Raptor codes were designed to operate on the binary erasure channel (BEC), however, since their invention they received considerable attention in or-der to improve their performance on noisy channels, and especially additive white Gaussiannoise (AWGN) channels. This dissertation considers two issues that face Raptor codes on the binary input additive white Gaussian noise (BIAWGN) channel: inaccurate estimation of signal to noise ratio (SNR) and the optimality of inner and outer rates. First, for codes that use a belief propagation algorithm (BPA) in decoding, such as Raptor codes on the BIAWGN channel, accurate estimation of the channel SNR is crucial to achieving optimal performance by the decoder. A difference between the estimated SNR and the actual channel SNR is known as signal to noise ratio mismatch (SNRM). Using asymptomatic analysis and simulation, we show the degrading effects of SNRM on Raptor codes and observe that if the mismatch is large enough, it can cause the decoding to fail. Using the discretized density evolution (DDE) algorithm with the modifications required to simulate the asymptotic performance in the case of SNRM, we determine the decoding threshold of Raptor codes for different values of SNRM ratio. Determining the threshold under SNRM enables us to quantify its effects which in turn can be used to reach important conclusions about the effects of SNRM on Raptor codes. Also, it can be used to compare Raptor codes with different designs in terms of their tolerance to SNRM. Based on the threshold response to SNRM, we observe that SNR underestimation is slightly less detrimental to Raptor codes than SNR overestimation for lower levels of mismatch ratio, however, as the mismatch increases, underestimation becomes more detrimental. Further, it can help estimate the tolerance of a Raptor code, with certain code parameters when transmitted at some SNR value, to SNRM. Or equivalently, help estimate the SNR needed for a given code to achieve a certain level of tolerance to SNRM. Using our observations about the performance of Raptor codes under SNRM, we propose an optimization method to design output degree distributions of the LT part that can be used to construct Raptor codes with more tolerance to high levels of SNRM. Second, we study the effects of choosing different values of inner and outer code rate pairs on the decoding threshold and performance of Raptor codes on the BIAWGN channel. For concatenated codes such as Raptor codes, given any instance of the overall code rate R, different inner (Ri) and outer (Ro) code rate combinations can be used to share the available redundancy as long asR=RiRo. Determining the optimal inner and outer rate pair can improve the threshold and performance of Raptor codes. Using asymptotic analysis, we show the effect of the rate pair choice on the threshold of Raptor codes on the BIAWGN channel and how the optimal rate pair is decided. We also show that Raptor codes with different output degree distributions can have different optimal rate pairs, therefore, by identifying the optimal rate pair we can further improve the performance and avoid suboptimal use of the code. We make the observation that as the outer rate of Raptor codes increases the potential of achieving better threshold increases, and provide the reason why the optimal outer rate of Raptor codes cannot occur at lower values. Finally, we present an optimization method that considers the optimality of the inner and outer rates in designing the output degree distribution of the inner LT part of Raptor codes. The designed distributions show improvement in both the decoding threshold and performance compared to other code designs that do not consider the optimality of the inner and outer rates

Exit charts based analysis and design of rateless codes for the erasure and Gaussian channels

Luby Transform Codes were the first class of universal erasure codes introduced to fully realize the concept of scalable and fault‐tolerant distribution of data over computer networks, also called Digital Fountain. Later Raptor codes, a generalization of the LT codes were introduced to trade off complexity with performance. In this work, we show that an even broader class of codes exists that are near optimal for the erasure channel and that the Raptor codes form a special case. More precisely, Raptorlike codes can be designed based on an iterative (joint) decoding schedule wherein information is transferred between the LT decoder and an outer decoder in an iterative manner. The design of these codes can be formulated as a LP problem using EXIT Charts and density evolution. In our work, we show the existence of codes, other than the Raptor codes, that perform as good as the existing ones. We extend this framework of joint decoding of the component codes to the additive white Gaussian noise channels and introduce the design of Rateless codes for these channels. Under this setting, for asymptotic lengths, it is possible to design codes that work for a class of channels defined by the signal‐to‐noise ratio. In our work, we show that good profiles can be designed using density evolution and Gaussian approximation. EXIT charts prove to be an intuitive tool and aid in formulating the code design problem as a LP problem. EXIT charts are not exact because of the inherent approximations. Therefore, we use density evolution to analyze the performance of these codes. In the Gaussian case, we show that for asymptotic lengths, a range of designs of Rateless codes exists to choose from based on the required complexity and the overhead. Moreover, under this framework, we can design incrementally redundant schemes for already existing outer codes to make the communication system more robust to channel noise variations

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