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relevant properties; Section 4 gives several illustrative examples; then the final
section concludes with a brief summary.
2
The Average Hamming Correlation for Frequency
Hopping Sequences
We first introduce the necessary notations. Let
p
be a prime number, GF(
p
)bea
finite field with elements
.Let
f
(
x
) be a polynomial over the finite
field GF(
p
), we will use
N
(
f
)=
N
(
f
(
x
)=0) to denote the number of solutions
of the equation
f
(
x
)=0 in GF(
p
).
Let
F
=
{
0,1, . . . ,
p
-1
}
=q
,and
S
be a set
of
M
frequency hopping sequences of length
L
. For any two frequency slots
f
i
,
f
j
∈
{
f
1
,
f
2
,. . . ,
f
q
}
|
F
|
be a frequency slot set with size
F
,let
h
(
f
i
,
f
j
)=
1, if
f
i
=
f
j
,
0, otherwise.
For any two FH sequences
x
=(
x
0
,
x
1
,...,
x
L−
1
),
y
=(
y
0
,
y
1
,...,
y
L−
1
)
∈
S
,
and any integer
τ
0, the periodic Hamming correlation function
H
(
x, y
;
τ
)of
x
and
y
at time delay
τ
is defined as follows:
≥
L−
1
H
(
x
,
y
;
τ
)=
h
(
x
i
,
y
i
+
τ
), (
τ
=0,1,
...
,
L
−
1)
i
=
0
where the subscript addition
i+τ
is performed modulo
L
.Moreover,the
H
(
x, x
;
τ
)
is called the periodic Hamming autocorrelation function when
x
=
y
and the
periodic Hamming crosscorrelation function when
x
=
y
. For any given FH se-
quence set
S
, the maximum periodic Hamming autocorrelation sidelobe
H
a
(
S
)
and the maximum periodic Hamming crosscorrelation
H
c
(
S
) are defined by
H
a
(
S
)=max
{H
(
x,x
;
τ
)
|x ∈ S,τ
=1,2,
...
,
L −
1)
H
c
(
S
)=max
{
H
(
x,y
;
τ
)
|
x,y
∈
S,x
=
y,τ
=0,1,
...
,
L
−
1)
An important performance indicator of the hopping sequences is the average
Hamming correlation defined as follows.
Definition 1 ([13]).
Let S be a set of M hopping sequences of length L over a
given frequency slot set F with size q,wecall
S
a
(
S
)=
H(x,x
;
τ),
x
∈
S,
1
≤
τ
≤
L
−
1
S
c
(
S
)=
1
2
H
(
x,y
;
τ
)
x,y
∈
S,x
=
y,
0
≤
τ
≤
L
−
1