Biomedical Engineering Reference
In-Depth Information
Appendix A: Proofs of
Lemmas in
Chapter 4
ProofofLemma4.1
From. Definition. 4.4. in.
Chapter. 4
,. the. following. identities. can. be. easily.
obtained:
d x y
(
,
)
=
d x x
(
,
⊕
y
⊕
x
)
.
(A.1a)
i
i
j
i
i
j
.
(
)
(
)
=
d x
, *
d x
, *
.
(A.1b)
i
j
.
(
)
(
)
=
d
*,
y
d
*,
x
⊕
y
⊕
x
.
(A.1c)
i
i
i
j
.
where.
i i j
∈ ,.and.⊕.is.the.logical.XOR.operator.
Based.on.
Equation.(A.1)
,
.it.can.be.proved.that
x y x
,
,
{0, 1}
∆ ξ
(
,
V
;
ξ
,
G
)
= ∆ ξ
(
,
V
;
ξ
,
G
)
.
(A.2)
.
i
k
m
c k
,
i
k
m
c k
,
and
∆ ξ
(
,
V
′ ξ
;
,
V
′ = ∆ ξ
)
(
,
V
′ ξ
;
,
V
′
)
.
(A.3)
.
i
k
n
k
j
k
n
k
where
ξ
( )
l
′ ⊕ ξ
( )
l
⊕ ξ
( )
l
′
l G and
∈
ξ
( )
l
≠ ∗
and
ξ
( )
l
′ ≠ ∗
ξ
i
m
j
c k
,
m
i
( )
l
=
.
(A.4a)
m
ξ
( )
l
otherwise
m
.
and
ξ
( )
l
⊕ ξ
( )
l
⊕ ξ
( )
l
l V and
∈
′
ξ
≠ ∗
ξ
i
n
j
k
n
( )
l
=
.
(A.4b)
n
ξ
( )
l
otherwise
n
.
with.
shown.in
.
Figure.A.1
.
l
k
(
l
c
)
203
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