Information Technology Reference
In-Depth Information
As explained in [1], the maximum non-trivial correlation magnitude of a se-
quence family is given the modified definition:
⎧
⎨
⎫
⎬
either
s
(
j, t
)
,s
(
k, t
)
have been assigned
to distinct users or
τ
θ
s
(
j
)
,s
(
k
)
(
τ
)
θ
max
:= max
⎩
⎭
=0
We use
θ
max
to denote the maximum correlation magnitude after energy nor-
malization and this is used as a basis for comparison across signal constellations.
Family
A
is an asymptotically optimal family of quaternary sequences (i.e.,
over
Z
4
) discovered independently by Sole[10]andBoztas, Hammons and Kumar
[4]. Let
2
r
i
=1
denote 2
r
{
γ
i
}
distinct elements in
T
, i.e., we have the alternate
.Thereare2
r
+ 1 cyclically distinct sequences
expression
T
=
{
γ
1
,γ
2
,...,γ
2
r
}
,eachofperiod2
r
in Family
A
−
1. For details on family
A
, the reader is referred
to [4,10,14].
There are some more related papers: Boztas [3] gives new lower bounds on
the periodic crosscorrelation; Udaya and Siddiqi [12] give families of biphase
sequences derived from families of interleaved maximal-length sequences over
Z
4
; Tang and Udaya [11] modify families
B
C
D
and
to obtain family
,whichis
Z
4
.
a larger family of optimal quadriphase sequences over
2 Family
I
2
SQ−B
2
Family
is constructed by interleaving 2 selected QAM sequences. It
is defined over the
M
2
-QAM constellation, with
M
=2
m
,m
I
SQ−B
≥
2. Compared to
2
family
, this family has a lower value of normalized
maximum correlation
θ
max
. The data rate for all these families is the same -
(
m
+ 1) bits of data per sequence period.
Let
m
SQ
and family
I
SQ−A
≥
2. Let
{
δ
0
=0
,δ
1
,δ
2
,...,δ
m−
1
}
be elements from
F
q
such that
tr
(
δ
k
)=1
,
∀
k
≥
1. Set
H
=
{
δ
0
,δ
1
···
,δ
m−
1
}
.Let
G
=
{
g
k
}
be the largest
subset of
F
q
having the property that
g
k
+
δ
p
=
g
l
+
δ
q
,g
k
,g
l
∈
G, δ
p
,δ
q
∈
H,
unless
g
k
=
g
l
and
δ
p
=
δ
q
.
Then the corresponding Gilbert-Varshamov and Hamming bounds [9] on the
size of
G
are given by
2
r
1+
m−
1
+
m−
2
≤|
2
r
1+
m−
1
G
|≤
(1)
2.1 Subspace-Based Construction for
G
and
H
[1]
Given constellation parameter
m
,let
l
denote the smallest power of 2 greater
than (
m
−
1), i.e.,
l
is defined by
2
l−
1
<
(
m
2
l
.
−
1)
≤
(2)
