Information Technology Reference
In-Depth Information
To transmit 1 the sequence
η
1
, ..., η
2
r
is split into a sequence of pairs
(
η
1
,η
2
),...,(
η
2
r−
1
,η
2
r
) and every pair is rearranged in ascending order. The num-
ber
ν
r
of increasing pairs for sequential count with possible overlaps in the output
sequence
ς
1
, ..., ς
2
r
is equal to
r−
1
ν
r
=
r
+
I
(
ς
2
i
≤
ς
2
i
+1
)
.
i
=1
Let us consider stochastic values
µ
(
t
)isequalto
I
(
ς
2
i
≤
ς
2
i
+1
)
,
if
t
=2
i
+1
,
and 0
,
if
t
=2
i
+1
.
and
D
t
=1
µ
(
t
)
It is obvious that for any
tEµ
2
(
t
)
<
∞
−→ ∞
if
T
−→ ∞
.
So [3] the distribution of
t
=1
µ
(
t
)
−
t
=1
Eµ
(
t
)
D
t
=1
µ
(
t
)
converges to Gaussian distribution with parameters 0 and 1 when
T
.
The situation with transmitting 0 is similar. A sequence
η
1
, ..., η
2
r
is split
into a sequence of pairs (
η
1
,η
2
)
, ...,
(
η
2
r−
1
,η
2
r
), and every pair is rearranged in
descending order. The number
ω
r
of decreasing pairs for sequential count with
possible overlaps in the output sequence
ς
1
, ..., ς
2
r
is equal to
−→ ∞
r−
1
I
(
ς
2
i
≥
ς
2
i
+1
)
.
ω
r
=
r
+
i
=1
r
after being centered
and normed converges to Gaussian distribution with parameters 0 and 1.
If we consider
ν
r
calculated on the base of the original sequence
η
1
, ..., η
2
r
,
after being centered and normed it will also converge to Gaussian distribution
with parameters 0 and 1.
Let us find expectations of
ν
r
and
ω
r
when we transmit 1, 0 and
x
.
Let the transmitted value be 1. To evaluate the estimation
ν
r
let us consider
stochastic values
η
2
i−
1
,η
2
i
,η
2
i
+1
,η
2
i
+2
.InourMarkovchain
Like in the previous case, the random number
ω
r
−
1
P
(
η
2
i−
1
=
s, η
2
i
=
k, η
2
i
+1
=
l, η
2
i
+2
=
n
)=
1)
3
.
m
(
m
−
So
Eµ
(2
i
+1)=
P
(max(
η
2
i−
1
,η
2
i
)
≤
min(
η
2
i
+1
,η
2
i
+2
)) =
=
m
4
m
3
9
m
2
−
−
−
18
m
−
6
1
6
(1 +
O
(
1
=
m
))
,
6
m
(
m
−
1)
3
when the value of
m
is large. Then
Eν
r
=
6
r
+
O
(
m
).
Similarly for 0 we have
Eω
r
=
6
r
+
O
(
m
), and for
xEν
r
=
r
+
O
(
m
).
Due to the fact that
ν
r
and
ω
r
are asymptotically Gaussian, deviations of
7
the above expectations are not greater than
√
r
ln
r
with probability converging
to 1. Hence we can recognize 1, 0 and
x
when
r
and
m
are large.
