Signal Space Coding over Rings

Abstract
The aim of this thesis is to report research into the optimisation of Signal Space
coding schemes defined over the ring of integers modulo-Q. There are three main
entities that are subject to an optimisation in a given signal space coding scheme: The
encoding machine, the signal space, and the relationship (mapping) between these two
entities.
An optimisation of ring-Trellis-Coded Modulation (ring-TCM) schemes and ring-
Block-Coded Modulation (ring-BCM) schemes, considered as signal space coding
schemes over rings, is developed in this thesis.
Regarding ring-TCM schemes a new topology for the corresponding ring-Multilevel
Convolutional Encoder (ring-MCE) is presented in Chapter 4, together with upper
bound estimations for the squared Euclidean free distance
xii
2
d free . It is concluded that
ring-MCE topologies described by input-output transfer functions with numerator and
denominator of equal degree are optimum in terms of the parameter
2
d free , and the new
topology is in agreement with this condition. As the new ring-MCE is found to be
optimum in terms of the parameter d , a further improvement of a ring-TCM
2
free
scheme is provided by using a different signal space. Results are provided to show
that N-dimensional hypercube signal set ring-TCM schemes perform better than
MPSK ring-TCM schemes.
The design of a new signal set for signal space coding schemes is presented in
Chapter 5. An N-dimensional hypercube energy-signal set is constructed using a set
of Haar wavelet functions. A new concatenated ring-TCM scheme is also designed in
Chapter 5, which can be thought of as an in-time/frequency unequal error correction
signal space coding scheme.
Another novel signal set is proposed in Chapter 6 to improve the characteristics of the
one proposed in Chapter 5. It is the Modulated (Q/2)-dimensional signal set, which is
used as the signal space of a ring-BCM scheme. These ring-BCM schemes show an
improvement over MPSK ring-BCM schemes.
Thus, this thesis has contributed to improving signal space coding schemes over rings
for AWGN channels, by optimising its three main entities, and mainly by improving
the geometrical characteristics of the corresponding signal space.