Signal Space Coding over Rings

Chapter 1: Introduction 8
The N-dimensional hypercube constellation presented in Chapter 4 will be generalised
and studied in Chapter 5. On the other hand, a practical implementation of this N-
dimensional hypercube constellation also will be proposed, based on the use of
continuous-in-time orthogonal functions multiplied by discrete coefficients, as a way
of synthesising a given signal set S . This constellation is an N-dimensional
hypercube of energy signals, and is GU [1]. Three schemes based on this mapping are
presented. In a wavelet based ring-Trellis-Coded Mapping scheme (W-ring-TCM
scheme) each output of a ring-MCE is mapped into one of the N
2 signals of the set S .
Each signal is generated by adding N continuous-in-time functions taken from a
wavelet orthonormal basis. The normalised bit rate is 1 bit/sec. Performances of the
studied schemes are shown in terms of the parameter
2
A / 2 ) , where A is the
( σ
coefficient that multiplies the amplitude of each bit, and σ is the variance of the noise
in the channel.
An expression for the power spectral density of the transmission using this scheme is
derived. This expression shows that the spectral properties of the transmitted signal
can be determined by a proper selection of the wavelet basis, and its scale function.
The second scheme takes advantage of the fact that the resulting signal in the previous
scheme is a baseband signal, and applies the synthesised signal to an M-Quadrature
Amplitude Modulation (MQAM) system. This is an M-Quadrature Amplitude
Modulated W-ring-TCM scheme (MQAM W-ring-TCM scheme). The scheme shows
a performance that is close to that of the corresponding baseband scheme.
The third scheme is a concatenated wavelet based ring-TCM scheme. A ring-MCE
optimised over the wavelet orthonormal basis synthesised signal set S is concatenated
with a ring-MCE optimised over an MQAM constellation. The system involves the
concatenation of two ring-TCM schemes and their corresponding signal
constellations. The concatenation is an orthogonal in-time/frequency unequal error
correction combined coding and modulation scheme. However, and as will be seen in
Chapter 5, the hypercube of energy signals of dimension N behaves as an N-
dimensional constellation, but it does not provide a physical increase in the dimension
of the signal set.
Chapter 6 deals with signal space coding over rings based on block coding machines.
Ring-Block-Coded Modulation is a combined coding and modulation scheme based