Signal Space Coding over Rings

Chapter 1: Introduction 4
can be used as an N-dimensional orthogonal signal space. In this case the signal set is
composed of power signals, that is, signals of infinite energy. The set of wavelet
functions is also an N-dimensional signal space. It will be used in this work as a
signal set for ring-signal space codes. Advantages and disadvantages of this signal set
will be pointed out in Chapter 5. If a signal space is an N-dimensional hypercube
signal space composed of energy signals, such that it is constructed as a hypercube of
energy signals of dimension N , as is presented by Shannon in his paper [2], a
reduction of the gap to the Shannon limit is only obtained by increasing the
dimensionality of the signal space in comparison with the dimension of the message
space. This means that only some of all the points of the signal space are selected as
signals to be transmitted. The hypercube of dimension N can be constructed also by
using a wavelet set of functions. A novel technique is introduced in Chapter 5, for the
design of signal sets based on a set of wavelet functions. The wavelet based signal set
used in this Chapter is then combined with MQAM modulation, in a concatenated
scheme. As a result of conclusions obtained from the use of a wavelet based
orthonormal basis for synthesising signals of a signal set in ring-Trellis-Coded
Modulation (ring-TCM) schemes, a new signal set is proposed for ring-Block Coded
Modulation (ring-BCM) schemes in Chapter 6.
The aim of this research is to provide an improvement in performance of signal space
codes over rings, selecting as a parameter to be optimised the squared Euclidean free
distance, or equivalently, the Asymptotic Coding Gain. The thesis is divided in two
main parts. Chapters 2 and 3 deal with the background knowledge of combined
coding and modulation techniques, and signal space coding. A definition of a
multilevel signal space code is provided. Chapters 4, 5 and 6 are devoted to look for
an improvement in performance over signal space coding schemes over rings by
proposing modifications of the coding machine, and of the signal space used, though
Chapters 4 and 6 also contain some relevant background about coding over rings,
presented in some introductory sections.
This thesis is organised as follows:
Chapter 2 deals with combined coding and modulation techniques, and mainly with
Trellis-Coded Modulation and its parameters and performance analysis. An
introduction to bounds in communications is presented in this Chapter.