Signal Space Coding over Rings

Chapter 2: Combined coding and modulation techniques 28
The parameter b~ m identifies a variable m ~ that defines the number of input bits related
with the memory part of the scheme. The output function has as argument the binary
n-tuple ( a a ,..., a )
1 , 2
n
m
:
n
c(
a , a ,..., a ) = c + ∑c
a + ∑ c a a + ... + ∑ c
1
2
0
=
i
i 1
i
> =
j i
j i
ij
, 1
i
j
> >
=
ijl
i,
j,
l 1
l j i
a a a + ... + c
i
j
l
12...
n 1
a a ... a
2
n
(2.22)
The Analytic Description is more conveniently used when coefficients are of the form
of ± 1. Therefore a conversion as shown in Fig. 2.13 is done by using the following
expression:
bi = 1−
2ai
i = 1,
2,...,
n
(2.23)
The expression of the output as a function of the input vector is now of the form:
n
n
c(
b , b ,..., b ) = d + ∑ d b + ∑ d b b + ... + ∑ d
1
2
0
i=
1
i
i
i,
j=
1
j>
i
ij
i
j
i,
j,
l=
1
l>
j>
i
This equation can be expressed in matrix form as:
l
l
c
ijl
b b b + ... + d
i
j
l
12...
n 1
b b ... b
2
n
(2.24)
C = B D
(2.25)
where
⎡ c(
1,
1,...,
1)
⎤
⎢ ⎥
⎢
c(
-1,
1,...,
1)
⎥
⎢ . ⎥
= ⎢ ⎥
⎢ . ⎥
⎢ . ⎥
⎢ ⎥
⎢⎣
c(
-1,
-1,...,
-1)
⎥⎦
C l (2.26)