Low Complexity Iterative Algorithms in Channel Coding and Compressed Sensing

Abstract

Iterative algorithms are now widely used in all areas of signal processing and digital communications. In modern communication systems, iterative algorithms are notably used for decoding low-density parity-check (LDPC) codes, a popular class of error-correction codes known to have exceptional error-rate performance under iterative decoding. In a more recent field known as compressed sensing, iterative algorithms are used as a method of reconstruction to recover a sparse signal from a linear set of measurements. This work primarily deals with the development of low-complexity iterative algorithms for the two aforementioned fields, namely, the design of low-complexity decoding algorithms for LDPC codes, and the development and analysis of a low complexity reconstruction algorithm for compressed sensing. In the first part of this dissertation, we focus on the decoding algorithms for LDPC codes. It is now well known that LDPC codes suffer from an error floor phenomenon in spite of their exceptional performance. This phenomenon originates from the failures of traditional iterative decoders, like belief propagation (BP), on certain low-noise configurations. Recently, a novel class of decoders, called finite alphabet iterative decoders (FAIDs), were proposed with the capability of surpassing BP in the error floor region at a much lower complexity. We show that numerous FAIDs can be designed, and among them only a few will have the ability of surpassing traditional decoders in the error floor region. In this work, we focus on the problem of the selection of good FAIDs for column-weight-three codes over the binary symmetric channel. Traditional methods for decoder selection use asymptotic techniques such as the density evolution method, but the designed decoders do not guarantee good performance for finite-length codes especially in the error floor region. Instead we propose a methodology to identify FAIDs with good error-rate performance in the error floor. This methodology relies on the knowledge of potentially harmful topologies that could be present in a code. The selection method uses the concept of noisy trapping set. Numerical results are provided to show that FAIDs selected based on our methodology outperform BP in the error floor on a wide range of codes. Moreover first results on column-weight-four codes demonstrate the potential of such decoders on codes which are more used in practice, for example in storage systems. In the second part of this dissertation, we address the area of iterative reconstruction algorithms for compressed sensing. This field has attracted a lot of attention since Donoho's seminal work due to the promise of sampling a sparse signal with less samples than the Nyquist theorem would suggest. Iterative algorithms have been proposed for compressed sensing in order to tackle the complexity of the optimal reconstruction methods which notably use linear programming. In this work, we modify and analyze a low complexity reconstruction algorithm that we refer to as the interval-passing algorithm (IPA) which uses sparse matrices as measurement matrices. Similar to what has been done for decoding algorithms in the area of coding theory, we analyze the failures of the IPA and link them to the stopping sets of the binary representation of the sparse measurement matrices used. The performance of the IPA makes it a good trade-off between the complex ℓ₁-minimization reconstruction and the very simple verification decoding. The measurement process has also a lower complexity as we use sparse measurement matrices. Comparison with another type of message-passing algorithm, called approximate message-passing, show the IPA can have superior performance with lower complexity. We also demonstrate that the IPA can have practical applications especially in spectroscopy.