Signal Space Coding over Rings

Chapter 2: Combined coding and modulation techniques 18
Euclidean distance between any two sequences s b0
, sb1
,..., sbK
−1
, and s a0
, sa1,...,
saK
−1
is
given by the following expression:
∑ − K 1
i=
0
2
2
d = || s − s ||
(2.10)
bi
ai
If a given code C is constituted from a set of sequences, the minimum squared
Euclidean distance between any two sequences will be considered as the minimum
2
squared Euclidean distance d min
2.3.4 Coded and uncoded sequences
of the code C .
When there is no coding procedure over the labels that correspond to the signals of the
constellation, the resulting transmitted sequence is composed of independent signals.
In this case a signal can be followed by any other of the constellation, so that the
minimum squared Euclidean distance of the sequence can be calculated by minimising
2
the terms || − s || ; i = 1,
2,...,
K −1
independently. Therefore:
d
s
2
min
b
≠ s
= min||s
a
∀
sbi ai
bi
s
− s
a
,s
b
ai
||
2
The symbol error probability is upper bounded by:
M −1
⎛
≤ ⎜ d
P(
e)
erfc
2 ⎜
⎝ 2.
N
0
⎞
⎟
⎟
⎠
(2.11)
min (2.12)
Two parameters are defined for comparison proposes; the bandwidth efficiency:
log 2 M
R = (2.13)
N