Bit mapping scheme for an LDPC coded 32APSK system
A digital communication system, having a transmitter to transmit a digital signal; and a receiver to receive the digital signal; wherein the digital signal utilizes a 32APSK system with FEC coding, and the signal is bit-mapped using gray mapping, and bits of the digital signal are ordered based on the values of a log likelihood ratio from a communications channel.
1. A method of bit mapping in a 32APSK system, the method comprising:
transmitting a digital signal from a transmitter; and
receiving the digital signal at a receiver;
wherein the digital signal utilizes a 32APSK system, and the signal is bit-mapped prior to the transmitting according to the following formula:
(
I
(
i
)
,
Q
(
i
)
)
=
(
(
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
R
3
,
0
)
,
(
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
1
)
{
(
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
0
,
-
R
3
)
,
(
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
-
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
-
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
-
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
-
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
1
)
(
(
-
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
-
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
0
,
R
3
)
,
(
-
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
-
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
-
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
-
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
-
R
3
,
0
)
,
(
-
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
-
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
-
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
-
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
1
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
0
)
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
1
)
wherein R1 is a radius of an inner ring, R2 is a radius of an intermediate ring and R3 is a radius of an outer ring.
2. The method of claim 1 , wherein an FEC code is used.
3. The method of claim 2 , wherein the FEC code is regular LDPC code.
4. The method of claim 2 , wherein the FEC code is irregular LDPC code.
5. The method of claim 2 , wherein the FEC code is regular repeat-accumulate code.
6. The method of claim 2 , wherein the FEC code is irregular repeat-accumulate code.
7. A digital communications system, comprising: a transmitter to modulate at least one mapping group having five bits (b5i, b5i+1, b5i+2, b5i+3, b5i+4), for i=0, 1, 2, . . . , to 32APSK symbols based on formula:
(
I
(
i
)
,
Q
(
i
)
)
=
(
(
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
0
)
(
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
1
)
(
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
0
)
(
R
3
,
0
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
1
)
(
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
0
)
(
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
1
)
(
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
0
)
(
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
1
)
(
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
0
)
(
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
1
)
{
(
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
0
)
(
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
1
)
(
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
0
)
(
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
1
)
(
0
,
-
R
3
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
0
)
(
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
1
)
(
-
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
0
)
(
-
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
1
)
(
-
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
0
)
(
-
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
1
)
(
(
-
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
0
)
(
-
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
1
)
(
0
,
R
3
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
0
)
(
-
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
1
)
(
-
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
0
)
(
-
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
1
)
(
-
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
0
)
(
-
R
3
,
0
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
1
)
(
-
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
0
)
(
-
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
1
)
(
-
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
0
)
(
-
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
1
)
wherein R 1 is a radius of an inner ring, R 2 is a radius of an intermediate ring, and R 3 is a radius of an outer ring.
8. A digital communications system, comprising: a receiver to demodulate at least one mapped 32APSK symbol to an estimating message group having five bits (b5i, b5i+1, b5i+2, b5i+3, b5i+4), for i=0, 1, 2, . . . , based on a 32APSK constellation specification:
(
I
(
i
)
,
Q
(
i
)
)
=
(
(
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
0
)
(
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
1
)
(
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
0
)
(
R
3
,
0
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
1
)
(
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
0
)
(
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
1
)
(
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
0
)
(
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
1
)
(
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
0
)
(
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
1
)
{
(
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
0
)
(
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
1
)
(
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
0
)
(
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
1
)
(
0
,
-
R
3
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
0
)
(
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
1
)
(
-
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
0
)
(
-
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
1
)
(
-
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
0
)
(
-
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
1
)
(
(
-
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
0
)
(
-
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
1
)
(
0
,
R
3
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
0
)
(
-
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
1
)
(
-
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
0
)
(
-
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
1
)
(
-
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
0
)
(
-
R
3
,
0
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
1
)
(
-
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
0
)
(
-
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
1
)
(
-
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
0
)
(
-
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
1
)
wherein R 1 is a radius of an inner ring, R 2 is a radius of an intermediate ring, and R 3 is a radius of an outer ring.
9. A non-transitory computer readable medium to store a computer program, executable by a processor, to map at least one group of five bits (b5i, b5i+1, b5i+2, b5i+3, b5i+4), for i=0, 1, 2, . . . , to a 32APSK symbol based on a formula:
(
I
(
i
)
,
Q
(
i
)
)
=
(
(
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
0
)
(
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
0
,
1
)
(
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
0
)
(
R
3
,
0
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
0
,
1
,
1
)
(
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
0
)
(
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
0
,
1
)
(
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
0
)
(
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
0
,
1
,
1
,
1
)
(
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
0
)
(
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
0
,
1
)
{
(
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
0
)
(
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
0
,
1
,
1
)
(
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
0
)
(
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
0
,
1
)
(
0
,
-
R
3
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
0
)
(
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
0
,
1
,
1
,
1
,
1
)
(
-
R
2
cos
(
π
/
4
)
,
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
0
)
(
-
R
2
cos
(
π
/
12
)
,
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
0
,
1
)
(
-
R
3
cos
(
π
/
4
)
,
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
0
)
(
-
R
3
cos
(
π
/
8
)
,
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
0
,
1
,
1
)
(
(
-
R
2
sin
(
π
/
12
)
,
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
0
)
(
-
R
1
cos
(
π
/
4
)
,
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
0
,
1
)
(
0
,
R
3
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
0
)
(
-
R
3
sin
(
π
/
8
)
,
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
0
,
1
,
1
,
1
)
(
-
R
2
cos
(
π
/
4
)
,
-
R
2
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
0
)
(
-
R
2
cos
(
π
/
12
)
,
-
R
2
sin
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
0
,
1
)
(
-
R
3
cos
(
π
/
8
)
,
-
R
3
sin
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
0
)
(
-
R
3
,
0
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
0
,
1
,
1
)
(
-
R
2
sin
(
π
/
12
)
,
-
R
2
cos
(
π
/
12
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
0
)
(
-
R
1
cos
(
π
/
4
)
,
-
R
1
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
0
,
1
)
(
-
R
3
sin
(
π
/
8
)
,
-
R
3
cos
(
π
/
8
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
0
)
(
-
R
3
cos
(
π
/
4
)
,
-
R
3
sin
(
π
/
4
)
)
,
(
b
5
i
,
b
5
i
+
1
,
b
5
i
+
2
,
b
5
i
+
3
,
b
5
i
+
4
)
=
(
1
,
1
,
1
,
1
,
1
)
wherein R 1 is a radius of an inner ring, R 2 is a radius of an intermediate ring, and R 3 is a radius of an outer ring.