Signal Space Coding over Rings

Chapter 2: Combined coding and modulation techniques 30
2.4.4 Ungerboeck and Turgeon rules
The design procedure based on expression (2.29) applies rules stated by Ungerboeck
and Turgeon, for the signal assignment to trellis transitions.
A description of minimal complexity is one with the minimum number of
d coefficients so that each bit of the input sliding block appears only once in the
encoder formula.
Ungerboeck rules for assigning signals of the constellation to a given transition in the
corresponding trellis are [44, 45]:
• All signals of the constellation appear the same number of times in the assignment,
making the code be composed of equiprobable signals.
• When the trellis that defines the TCM encoder has parallel transitions, signals
corresponding to the same subset of highest intrasubset distance should be assigned
to those parallel transitions.
• The m
2 transitions that emerge from or return to a given state should be assigned
signals from one subset at the first level of the partitioning.
The Maximum Signal Value (MSV) is the maximum numerical value of a signal.
When the signal has a vectorial expression, the MSV is the maximum sum of the
vector co-ordinates.
A signal difference represents the difference between any two signals when only one
bit of the input sliding block is changed. The absolute signal difference is the absolute
value of the signal difference. Thus, the signal difference is given by the following
expression:
δ = x b , b ,..., b ,..., b , b + ,..., b ) − x(
b , b ,..., −b
,..., b , b + ,..., b ) (2.30)
i
( 1 2 i k k 1 n 1 2 i k k 1 n
Turgeon rules are useful for minimising the encoder formula: