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as the overall number of Hamming autocorrelation (auto-hits) and Hamming
crosscorrelation (cross-hits) of S respectively, and call
S a ( S )
M ( L
A a ( S )=
1) ,
2 S c ( S )
LM ( M
A c ( S )=
1)
as the average Hamming autocorrelation (average of auto-hits) and the average
Hamming crosscorrelation (average of cross-hits) of S respectively.
Obviously, the average Hamming correlation (average of hits) indicates the aver-
age error (or interference) performance of the hopping systems. However, gener-
ally speaking, it is very dicult to derive the average Hamming correlation for a
given hopping sequence. For simplicity, we will denote S a = S a ( S ), S c = S c ( S ),
A a = A a ( S )and A c = A c ( S ).
We now give the theoretical limits which set a bounded relation among the
parameters q , L , M , A a and A c .
Theorem 1. Let S be a set of M hopping sequences of length L over a given fre-
quency slot set F with size q, A a and A c be the average Hamming autocorrelation
and the average Hamming crosscorrelation of S respectively, then
A a
L ( M
A c
( L
LM
q
1) +
1)
1) .
(2)
q ( L
1)( M
Proof. We first have
H ( x,y ; τ )= x∈S
H ( x,x ;0) +
H ( x,y ; τ )+
H ( x,y ; τ )
x,y∈S,τ
x∈S,τ =0
x = y∈S,τ
= LM + S a +2 S c .
In [5], a lower bound L 2 M 2 / q on x,y∈S, 0 ≤τ≤L− 1 H ( x,y ; τ ) was given.
Therefore,
L 2 M 2
q
LM + S a +2 S c
,
and
1
S a
2 S c
LM
1)
1) ,
1) +
1) +
( L
1)( M
LM ( L
1)( M
LM ( L
1)( M
q ( L
1)( M
1
A a
L ( M
A c
( L
LM
1) +
1) +
1)
1) ,
( L
1)( M
q ( L
1)( M
and (2) follows immediately. This completes the proof.
If the parameters q , L , M , A a ,and A c of the FH sequence set S satisfy inequality
(2) with equality, then it is said that sequence set S is an optimal average
Hamming correlation family.
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