Applications such as satellite and digital mobile radio systems (DMR) have gained widespread acceptance in recent years, and efficient digital processing techniques are gradually replacing the older analogue systems. An important subsystem of these applications is voiceband communication, especially digital speech encoding. Digital encoding of speech has been a focus of speech processing research for many years, and recently this activity together with the rapid advances in digital hardware, has begun to produce realistic working algorithms. This is typified by the Pan-European DMR system which operates at 13Kbit/s. For applications operating below this coding capacity, sophisticated algorithms have been developed. A particular class of these, termed Analysis-by-Synthesis Linear Predictive Coding (ABS-LPC), has been a subject of active world-wide research. In this thesis, ABS-LPC algorithms are investigated with particular emphasis on the Code-Excited Linear Predictive coding (CELP) variant. The aim of the research is to produce high communication quality speech at 8Kbit/s and below by considering aspects of quantisation, computational complexity and robustness. The ABS-LPC algorithms operate by exploiting short-term and long-term correlations of speech signals. Line Spectral Frequency (LSF) representation of the short-term correlation is examined and various LSF derivations and quantisation procedures are presented. The variants of ABS-LPC are compared for their advantages and disadvantages to determine an algorithm suitable for in-depth analysis. The particular chosen variant, CELP, was pursued. A study on the importance of the long-term prediction, and the simplification of CELP without sacrificing speech quality is presented. The derived alternative approaches for the computation of the long-term predictor and the filter excitation have enabled the previously unpractical CELP algorithm to produce high communication quality speech at rates below 8Kbit/s, and yet remain implement able in real-time on a single chip. Refinements of the CELP algorithm followed in order to improve the coder towards higher speech quality at 4.8Kbit/s and below. This involved the examination of the weaknesses of the basic CELP algorithm, and alternative strategies to overcome these limitations are presented.