Signal Space Coding over Rings

Chapter 2: Combined coding and modulation techniques 26
2.4 Representations and design for TCM
2.4.1 Introduction
There are basically two representations for a TCM encoder. The Ungerboeck
representation [45] is based on the use of a convolutional encoder that represents the
memory of the system, and also of a selector for the partition of the set of signals. The
Calderbank-Mazo representation [47] shows an analytical expression for the mapping
between source symbols and signals of the constellation.
2.4.2 The Ungerboeck representation
In a TCM encoder-modulator a source symbol a i corresponding to the time instant i
can be assigned to one of
k
2 values, represented as sequences of k binary digits
( 1)
( 2)
( k )
b i , bi
,..., bi
. The encoder output c i at the time instant i depends on the input,
represented by the binary digits, and on previous ν ≥ 0 bits , j = 1,
2,...,
k :
( 1)
( 1)
( 1)
( k ) ( k ) ( k )
c = f ( bi
, bi−1
,..., b − ,..., bi
, bi−1
,..., b − )
(2.21)
i i v1
i vk
The encoder presents two parts. One is the memory, implemented by using a binary
convolutional encoder with k binary inputs
( 1)
i
( 2)
i
( n)
i
j
( 1)
i
( 2)
i
( k )
i
b , b ,..., b and n binary outputs
c , c ,..., c . The other is the memoryless part, which is basically a modulator that
maps the binary n-tuple output into a signal s i of the constellation.