US 8,045,632 B2Granted

Systems and methods for dual-carrier modulation encoding and decoding

AssigneeTexas Instruments Incorporated

InventorsAnuj Batra, Deric W. Waters

Life of this patent

Estimating the term…

Abstract

Systems and methods for dual-carrier modulation (DCM) encoding and decoding for communication systems. Some embodiments comprise a DCM encoder for applying a pre-transmission function to at least one 16-QAM input symbol and mapping resulting transformed symbols onto at least one larger constellation prior to transmission. Some embodiments joint decode, by a DCM decoder, a predetermined number of received data elements and compute a set of log-likelihood ratio (LLR) values for at least eight bits from a resulting at least one transformed symbol.

Claims6

Dependent claims

A communications system, comprising:

a dual carrier modulation (DCM) decoder able to jointly decode a predetermined number of received data elements and compute a set of log-likelihood ratio (LLR) values for at least eight bits from at least one resulting transformed symbol, wherein the decoder, in computing the set of LLR values, employs a log likelihood function selected from the group of functions:
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
)
=
log
[
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
(
4
⁢
k
+
p
)
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
2
)
=
log
[
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
2
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
)
=
log
[
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
2
)
=
log
[
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
2
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
and
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
where b is a bit, k is a selected tone, p is a non-zero integer,
H
k
=
[
h
k
0
0
h
k
+
p
]
·
and
⁢
⁢
W
k
=
[
σ
k
2
2
0
0
σ
k
+
p
2
2
]
.

2,842 more elements

A method for communications, comprising:

joint decoding, by a dual carrier modulation (DCM) decoder, a predetermined number of received data elements;
and
computing a set of log-likelihood ratio (LLR) values for at least eight bits from a resulting at least one transformed symbol, wherein the computing further comprises employing a log likelihood function selected from the group of functions:
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
)
=
log
[
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
(
4
⁢
k
+
p
)
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
R
T
⁢
T
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
2
)
=
log
[
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
2
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
0
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
1
⁢
exp
⁡
[
2
σ
2
⁢
S
I
T
⁢
T
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
)
=
log
[
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
(
k
+
p
)
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
k
+
1
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
+
1
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
1
)
=
log
[
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
0
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
∑
b
4
⁢
k
⁢
b
4
⁢
k
+
1
⁢
b
4
⁢
(
k
+
p
)
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
1
=
1
⁢
exp
⁡
[
S
R
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
α
k
-

H
k

2
⁢
T
⁢
⁢
S
R
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
2
)
=
log
[
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
2
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
k
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
(
k
+
p
)
+
2
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
k
+
3
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
2
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
3
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
2
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
⁢
and
L
⁢
⁢
L
⁢
⁢
R
⁡
(
b
4
⁢
(
k
+
p
)
+
3
)
=
log
[
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
0
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
∑
b
4
⁢
k
+
2
⁢
b
4
⁢
k
+
3
⁢
b
4
⁢
(
k
+
p
)
+
2
∈
{
0
,
1
}
,
b
4
⁢
(
k
+
p
)
+
3
=
1
⁢
exp
⁡
[
S
I
T
⁢
T
⁢
⁢
W
k
-
1
⁡
(
2
⁢
β
k
-

H
k

2
⁢
T
⁢
⁢
S
I
)
]
]
,
where b is a bit, k is a selected tone, p is a non-zero integer,
H
k
=
[
h
k
0
0
h
k
+
p
]
·
and
⁢
⁢
W
k
=
[
σ
k
2
2
0
0
σ
k
+
p
2
2
]
.

2,844 more elements

Dependencies are inferred from the claim text.