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In-Depth Information
Proof.
We have
2
n
−
2
1
P
i,j
(
τ, L
)=
P
i,j
(
τ, k, L
)
2
n
−
1
k
=0
2
n
−
2
L−
1
1
ω
s
i
(
k⊕t⊕τ
)
−s
j
(
k⊕t
)
=
2
n
−
1
k
=0
t
=0
2
n
L
−
1
−
2
1
ω
s
i
(
k⊕t⊕τ
)
−s
j
(
k⊕t
)
=
2
n
−
1
t
=0
k
=0
L−
1
1
L
=
C
i,j
(
τ
)=
1
C
i,j
(
τ
)
,
2
n
−
1
2
n
−
t
=0
where after interchanging the summations the resulting inner sum is clearly a
full period correlation sum which is independent of
t.
It is of interest in applications to consider only the nontrivial periodic auto-
correlation function, i.e.,
i
=
j, τ
=0(mod2
n
1), for estimating the false
self-synchronisation probability. Before addressing this, we need a definition.
−
Definition 4.
We define the Standard Normalized Correlation Distribution for
aquantity
θ
(
τ
)
as:
1. If
n
=2
t
+1
,
then
⎧
⎨
2
n
−
1
,
1
time,
1+2
t
+
ω
2
t
,
2
n−
2
+2
t−
1
−
times,
1+2
t
ω
2
t
,
2
n−
2
+2
t−
1
θ
(
τ
)=
−
−
times,
⎩
2
t
+
ω
2
t
,
2
n−
2
2
t−
1
−
1
−
−
times,
2
t
ω
2
t
,
2
n−
2
2
t−
1
−
1
−
−
−
times.
2. If
n
=2
t,
then
⎧
⎨
2
n
−
1
,
1
time,
1+2
t
,
2
n−
2
+2
t−
1
−
times,
2
t
,
2
n−
2
2
t−
1
θ
(
τ
)=
−
1
−
−
times,
⎩
1+
ω
2
t
,
2
n−
2
−
times,
ω
2
t
,
2
n−
2
−
1
−
times.
This definition is used in the proof below. The result follows from arguments
along the lines of [1] but the autocorrelation distribution was never computed
there; in that paper, the focus was on the global correlation distribution.
Lemma 5.
For Family A the full period autocorrelation function obeys:
1. If we consider the zero divisor sequence (binary
m
-sequence), then
C
i,i
(
τ
)=
2
n
−
1
,
1
time,
2
n
−
1
,
−
2
times.
