By understanding the local behaviour over one single-step
channel transformation, conclusions can be reached about the
19

overall channel transformation from
W
N
to
W
(1)
N
, . . . , W
(
N
)
N
.
•
We begin with the rate parameter.
Theorem 6.1
Suppose
(
W, W
)
→
(
W , W
)
for some B-
DMC
W
. Then
I
(
W
) +
I
(
W
) = 2
I
(
W
)
(10)
I
(
W
)
≤
I
(
W
)
(11)
with equality iff
I
(
W
)
is 0 or 1.
•
(10) indicates that the single-step channel transformation pre-
serves the symmetric capacity.
(11) together with (10) implies that
–
I
(
W
) =
I
(
W
) =
I
(
W
)
iff
W
is either a noiseless chan-
nel (i.e.,
I
(
W
) = 1
) or a useless channel (i.e.,
I
(
W
) = 0
).
–
If
W
is neither noiseless nor useless (so that
I
(
W
)
<
I
(
W
)
), the single-step channel transformation shifts the
symmetric capacity of
W
and
W
away from the center
in the sense that
I
(
W
)
< I
(
W
)
< I
(
W
)
.
20

•
Next, we turn to the reliability parameter.
Theorem 6.2
Suppose
(
W, W
)
→
(
W , W
)
for some B-
DMC
W
. Then
Z
(
W
) =
Z
(
W
)
2
(12)
Z
(
W
)
≤
2
Z
(
W
)
−
Z
(
W
)
2
(13)
Z
(
W
)
≥
Z
(
W
)
≥
Z
(
W
)
(14)
Equality holds in (13) iff W is a BEC. We have
Z
(
W
) =
Z
(
W
)
iff
Z
(
W
)
is 0 or 1, or equivalently, if
I
(
W
)
is 1 or
0.
•
(13) together with (12) imply that reliability improves under
a single-step channel transformation in the sense that
Z
(
W
) +
Z
(
W
)
≤
2
Z
(
W
)
with equality iff
W
is a BEC.
•
In the special case where
W
,
W
or
W
is a BEC, we have
Theorem 6.3
If
W
is a BEC with erasure probability
, then
W
and
W
are BECs with erasure probabilities
2
−
2
and
2
, respectively. Conversely, if
W
or
W
is a BEC, then
W
is a BEC.
21

•
We now turn to the single-step channel transformation of the
form (7).
Theorem 6.4 follows from Theorems 6.1 and 6.2 as a special
case.
Theorem 6.4
For any B-DMC
W
,
N
= 2
n
,
n
≥
0
,
1
≤
i
≤
N
, the transformation
W
(
i
)
N
, W
(
i
)
N
→
W
(2
i
−
1)
2
N
, W
(2
i
)
2
N
is
rate-preserving and reliability-improving in the sense that
I
(
W
(2
i
−
1)
2
N
) +
I
(
W
(2
i
)
2
N
) = 2
I
(
W
(
i
)
N
)
(15)
Z
(
W
(2
i
−
1)
2
N
) +
Z
(
W
(2
i
)
2
N
)
≤
2
Z
(
W
(
i
)
N
)
(16)
with equality in (16) iff
W
is a BEC. Channel splitting moves
the rate and reliability away from the center in the sense that
I
(
W
(2
i
−
1)
2
N
)
≤
I
(
W
(
i
)
N
)
≤
I
(
W
(2
i
)
2
N
)
(17)
Z
(
W
(2
i
−
1)
2
N
)
≥
Z
(
W
(
i
)
N
)
≥
Z
(
W
(2
i
)
2
N
)
(18)
with equality in (17) and (18) iff
I
(
W
)
equals 0 or 1. The
reliability parameter further satisfies
Z
(
W
(2
i
−
1)
2
N
)
≤
2
Z
(
W
(
i
)
N
)
−
Z
(
W
(
i
)
N
)
2
(19)
Z
(
W
(2
i
)
2
N
) =
Z
(
W
(
i
)
N
)
2
(20)
22

with equality in (19) iff
W
is a BEC. The cumulative rate
and reliability satisfy
N
i
=1
I
(
W
(
i
)
N
) =
NI
(
W
)
(21)
N
i
=1
Z
(
W
(
i
)
N
)
≤
NZ
(
W
)
(22)
with equality in (22) iff
W
is a BEC.
•
The cumulative relations (21) and (22) follow by repeated
application of (15) and (16).
For example, for
N
= 8
, we have for the rate parameter,
I
(
W
(7)
8
) +
I
(
W
(8)
8
) = 2
I
(
W
(4)
4
)
I
(
W
(5)
8
) +
I
(
W
(6)
8
) = 2
I
(
W
(3)
4
)
I
(
W
(3)
8
) +
I
(
W
(4)
8
) = 2
I
(
W
(2)
4
)
I
(
W
(1)
8
) +
I
(
W
(2)
8
) = 2
I
(
W
(1)
4
)
I
(
W
(3)
4
) +
I
(
W
(4)
4
) = 2
I
(
W
(2)
2
)
I
(
W
(1)
4
) +
I
(
W
(2)
4
) = 2
I
(
W
(1)
2
)
I
(
W
(1)
2
) +
I
(
W
(2)
2
) = 2
I
(
W
(1)
1
) = 2
I
(
W
)
23

Combining these equations, we have
8
i
=1
I
(
W
(
i
)
8
) = 2
4
i
=1
I
(
W
(
i
)
4
) = 4
2
i
=1
I
(
W
(
i
)
2
) = 8
I
(
W
)
as expected.
•
Further, the conditions for equality in Theorem 6.4 are stated
in terms of
W
rather than
W
(
i
)
N
, which is possible since
–
by Theorem 6.3,
W
is a BEC iff
W
(
i
)
N
is a BEC,
–
by Theorem 6.1,
I
(
W
)
∈ {
0
,
1
}
iff
I
(
W
(
i
)
N
)
∈ {
0
,
1
}
.

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- Information Theory, Polarization, Wn, Linear code