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θ
s
(
g
1
)
,s
(
g
2
)
(
τ
)
N−
1
=
s
(
g
1
,
0
,t
+
τ
)
s
(
g
2
,
0
,t
)
t
=0
ı
u
1
(
t
+
τ
)
+2
ı
u
0
(
t
+
τ
)
ı
−v
1
(
t
)
+2
ı
−v
0
(
t
)
+
=2
t
even
ı
u
1
(
t
+
τ
)
2
ı
u
0
(
t
+
τ
)
ı
−v
1
(
t
)
2
ı
−v
0
(
t
)
2
t
odd
−
−
=2(
θ
u
1
,v
1
(
τ
)+2
θ
u
1
,v
0
(
τ
)+2
θ
u
0
,v
1
(
τ
)+4
θ
u
0
,v
0
(
τ
)) +
2(
θ
u
1
,v
1
(
τ
)
−
2
θ
u
1
,v
0
(
τ
)
−
2
θ
u
0
,v
1
(
τ
)+4
θ
u
0
,v
0
(
τ
))
=4(
θ
u
1
,v
1
(
τ
)+4
θ
u
0
,v
0
(
τ
))
Since
θ
u
0
,v
0
(
τ
) is orthogonal to
θ
u
1
,v
1
(
τ
) [1], we bound the magnitude of the
number
θ
s
(
g
1
)
,s
(
g
2
)
(
τ
) in the above expression as
θ
s
(
g
1
)
,s
(
g
2
)
(
τ
)
≤
4
|
Γ
(1)(1 + 4
ı
)
|
16
.
49
N/
2
11
.
66
√
N.
(7)
Now, if
τ
≡
1(mod2),weget
θ
s
(
g
1
)
,s
(
g
2
)
(
τ
)
N−
1
=
s
(
g
1
,
0
,t
+
τ
)
s
(
g
2
,
0
,t
)
t
=0
ı
u
1
(
t
+
τ
)
+2
ı
u
0
(
t
+
τ
)
ı
−v
1
(
t
)
2
ı
−v
0
(
t
)
+
2
ı
t
even
=
−
−
ı
u
1
(
t
+
τ
)
2
ı
u
0
(
t
+
τ
)
ı
−v
1
(
t
)
+2
ı
−v
0
(
t
)
2
ı
t
odd
−
−
2
ı
(
θ
u
1
,v
1
(
τ
)
−
2
θ
u
1
,v
0
(
τ
)+2
θ
u
0
,v
1
(
τ
)
−
4
θ
u
0
,v
0
(
τ
)) +
=
2
ı
(
θ
u
1
,v
1
(
τ
)+2
θ
u
1
,v
0
(
τ
)
−
2
θ
u
0
,v
1
(
τ
)
−
4
θ
u
0
,v
0
(
τ
))
=8
ı
(
θ
u
1
,v
0
(
τ
)
−
θ
u
0
,v
1
(
τ
))
.
Since we do not have any orthogonality relation between
θ
u
1
,v
0
(
τ
)and
θ
u
0
,v
1
(
τ
),
we assume the worst case and bound the magnitude of
θ
s
(
g
1
)
,s
(
g
2
)
(
τ
)intheabove
expression as
θ
s
(
g
1
)
,s
(
g
2
)
(
τ
)
≤
8
|
Γ
(1)(1 + 1)
|
16
N/
2
11
.
31
√
N.
(8)