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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
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