Constellation-rotating orthogonal space-time block coding technique
View Patent ↗A coding technique for achieving a maximum diversity gain in a wireless communication system having a transmitter for transmitting data on a fading channel through a plurality of transmit antennas and a receiver for receiving the data through a plurality of receive antennas. In the coding technique, P m th -order sub-rotated vectors {right arrow over (r)} i are generated by constellation-rotating an L th -order input vector {right arrow over (x)} including a plurality of input symbols modulated in at least one modulation scheme. m p th -order sub-vectors {right arrow over (u)} i are generated using the sub-rotated vectors {right arrow over (r)} i . Codeword matrices A i ({right arrow over (u)} i ) are generated by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i and then transmitted through the transmit antennas.
1. A transmitter for transmitting data through a plurality of transmit antennas in a wireless communication system including the transmitter and a receiver for receiving the data through a plurality of receive antennas, the transmitter comprising:
a constellation-rotating precoder for generating P m th -order sub-rotated vectors {right arrow over (r)} i by constellation-rotating an L th -order input vector {right arrow over (x)} including a plurality of input symbols modulated in at least one modulation scheme;
an accumulator for generating m p th -order sub-vectors {right arrow over (u)} i using the sub-rotated vectors {right arrow over (r)} i ; and
a space-time block coder for generating codeword matrices A i ({right arrow over (u)} i ) by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i .
2. The transmitter of claim 1 , wherein the constellation-rotating precoder comprises:
a first accumulation module for generating P m th -order sub-input vectors {right arrow over (x)} i by grouping the symbols of the input vector {right arrow over (x)} into P groups, each having m symbols, and outputting the P m th -order sub-input vectors {right arrow over (x)} i in parallel; and
a plurality of constellation rotators for generating the P m th -order sub-rotated vectors {right arrow over (r)} i by multiplying the parallel P m th -order sub-input vectors {right arrow over (x)} i by a constellation-rotating matrix Θ.
3. The transmitter of claim 2 , wherein the accumulator comprises:
a plurality of extraction modules for extracting signals r i,j included in each of the sub-rotated vectors {right arrow over (r)} i received from the constellation-rotating precoder; and
a second accumulation module for generating the m p th -order sub-vectors {right arrow over (u)} i by grouping the extracted signals r i,j .
4. The transmitter of claim 1 , wherein the accumulator comprises:
a plurality of extraction modules for extracting signals r i,j included in each of the sub-rotated vectors {right arrow over (r)} i received from the constellation-rotating precoder; and
a second accumulation module for generating the m p th -order sub-vectors {right arrow over (u)} i by grouping the extracted signals r i,j .
5. The transmitter of claim 4 , wherein the space-time block coder comprises:
a plurality of orthogonal space-time coder modules for generating the codeword matrices A i ({right arrow over (u)} i ) by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i received from the second accumulation module; and
a mapper for mapping the codeword matrices A i ({right arrow over (u)} i ) to the plurality of the transmit antennas.
6. The transmitter of claim 1 , wherein the space-time block coder comprises:
a plurality of orthogonal space-time coder modules for generating the codeword matrices A i ({right arrow over (u)} i ) by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i received from the accumulator; and
a mapper for mapping the codeword matrices A i ({right arrow over (u)} i ) to the plurality of the transmit antennas.
7. The transmitter of claim 6 , wherein the space-time block coder transmits columns of each of the codeword matrices A i ({right arrow over (u)} i ) through different transmit antennas for a same symbol period, and transmits columns of different codeword matrices A i ({right arrow over (u)} i ) through different transmit antennas for different symbol periods.
8. The transmitter of claim 6 , wherein the mapper maps a total of n columns of each of the codeword matrices A i ({right arrow over (u)} i ) to different transmit antennas for the same symbol period and columns in different codeword matrices A i ({right arrow over (u)} i ) to different transmit antennas for different symbol periods.
9. The transmitter of claim 1 , wherein a T×N codeword matrix C({right arrow over (x)})={c ti } having the codeword matrices A i ({right arrow over (u)} i ) as entries is formed as:
C
(
x
→
)
=
Δ
[
c
→
1
,
…
,
c
→
N
]
=
[
A
1
(
u
→
1
)
0
n
×
T
0
⋯
0
n
×
T
0
0
n
×
T
0
A
2
(
u
→
2
)
⋯
0
n
×
T
0
⋮
⋮
⋰
⋮
0
n
×
T
0
0
n
×
T
0
⋯
A
m
(
u
→
m
)
]
where 0 n×T 0 denotes an n×T 0 zero matrix and {right arrow over (c)} i denotes an ith column of C({right arrow over (x)}).
10. The transmitter of claim 9 , wherein the space-time block coder generates a modified T×N codeword matrix
C
(
x
→
)
′
=
Δ
U
C
(
x
→
)
V
by multiplying the codeword matrix C({right arrow over (x)}) by a T×T unitary matrix U and an N×N unitary matrix V at a left and right of C({right arrow over (x)}), and transmits the modified codeword matrix through N transmit antennas for T symbol periods.
11. The transmitter of claim 10 , wherein N=2 n , the space-time block coder uses Alamouti's code, and the constellation-rotating precoder uses a matrix expressed as:
Θ
=
1
N
[
1
α
0
…
α
0
N
-
1
1
α
1
…
α
1
N
-
1
⋮
⋮
⋰
⋮
1
α
N
-
1
…
α
N
-
1
N
-
1
]
,
α i =exp(j2π(i+1/4)/N) and i=0,1, . . . , N−1.
12. The transmitter of claim 11 , wherein the unitary matrix U is expressed as:
U
=
[
I
2
I
2
I
2
…
I
2
I
2
θ
θ
2
…
θ
N
-
1
I
2
θ
2
θ
4
…
θ
(
N
-
1
)
2
⋮
⋮
⋮
⋰
⋮
I
2
θ
N
-
1
θ
2
(
N
-
1
)
…
θ
(
N
-
1
)
2
]
,
and the matrix V is expressed as:
V
=
[
I
2
I
2
I
2
⋯
I
2
I
2
θ
θ
2
⋯
θ
N
-
1
I
2
θ
2
θ
4
⋯
θ
(
N
-
1
)
2
⋮
⋮
⋮
⋰
⋮
I
2
θ
N
-
1
θ
2
(
N
-
1
)
⋯
θ
(
N
-
1
)
2
]
where I 2 is a 2×2 unitary matrix and
θ
=
Δ
[
ⅇ
ⅈ
4
π
/
N
0
0
ⅇ
-
ⅈ
4
π
/
N
]
.
13. The transmitter of claim 12 , wherein the codeword matrix C({right arrow over (x)}) 40 =UC({right arrow over (x)})V is formed as:
C
(
x
_
)
′
=
U
[
C
Ala
(
r
1
,
r
N
/
2
+
1
)
0
2
…
0
2
0
2
C
Ala
(
r
1
,
r
N
/
2
+
2
)
…
0
2
⋮
⋮
⋰
⋮
0
2
0
2
…
C
Ala
(
r
N
/
2
,
r
N
)
]
V
=
[
C
Ala
(
s
1
,
s
N
/
2
+
1
)
C
Ala
(
s
N
/
2
,
s
N
)
…
C
Ala
(
s
2
,
s
N
/
2
+
2
)
C
Ala
(
s
2
,
s
N
/
2
+
2
)
C
Ala
(
s
1
,
s
N
/
2
+
1
)
…
C
Ala
(
s
3
,
s
N
/
2
+
3
)
⋮
⋮
⋰
⋮
C
Ala
(
s
N
/
2
,
s
N
)
C
Ala
(
s
N
/
2
-
1
,
s
N
-
1
)
⋯
C
Ala
(
s
1
,
s
N
/
2
+
1
)
]
where
C
Ala
(
a
,
b
)
=
Δ
[
a
b
-
b
*
a
*
]
,
s
l
=
ⅇ
ⅈ
(
l
-
1
)
π
/
N
x
l
,
and
s
N
/
2
+
l
=
ⅇ
ⅈ
(
l
-
1
)
π
/
N
x
N
/
2
+
l
(
l
=
1
,
…
,
N
/
2
)
.
14. The transmitter of claim 9 , wherein the space-time block coder simultaneously transmits other T 0 ×n codeword matrices B i (•), C i (•), and D i (•) generated independently of a total of m T 0 ×n codeword matrices A i (•) in a manner expressed as
[
A
1
(
·
)
B
1
(
·
)
…
C
1
(
·
)
D
1
(
·
)
…
0
m
0
m
0
m
A
2
(
·
)
B
2
(
·
)
…
C
2
(
·
)
D
2
(
·
)
…
0
m
⋮
⋮
⋰
⋰
⋰
⋰
⋰
⋮
0
m
0
m
…
A
m
(
·
)
B
m
(
·
)
…
C
m
(
·
)
D
m
(
·
)
]
T
,
[
A
1
(
·
)
B
1
(
·
)
…
C
1
(
·
)
D
1
(
·
)
D
2
(
·
)
A
2
(
·
)
B
2
(
·
)
…
C
2
(
·
)
⋰
⋰
⋰
⋰
⋰
…
C
m
(
·
)
D
m
(
·
)
A
m
(
·
)
B
m
(
·
)
]
T
,
[
A
1
(
·
)
B
1
(
·
)
…
C
1
(
·
)
D
1
(
·
)
…
A
2
(
·
)
B
2
(
·
)
…
C
2
(
·
)
D
2
(
·
)
…
⋮
⋮
…
⋮
⋮
…
A
m
(
·
)
B
m
(
·
)
…
C
m
(
·
)
D
m
(
·
)
…
]
T
[
A
1
(
·
)
A
2
(
·
)
…
A
m
(
·
)
B
1
(
·
)
B
2
(
·
)
…
B
m
(
·
)
⋮
⋮
⋮
⋮
C
1
(
·
)
C
2
(
·
)
…
C
m
(
·
)
D
1
(
·
)
D
2
(
·
)
…
D
m
(
·
)
]
T
.
15. The transmitter of claim 1 , wherein if a number of the transmit antennas is (N−1) being an odd number, the space-time block coder generates a new (N−1)×T codeword matrix by deleting a last of N rows of the N×T codeword matrix and transmits the (N−1)×T codeword matrix through (N−1) transmit antennas for the T symbol periods.
16. The transmitter of claim 1 , wherein the space-time block coder groups a total of m codeword matrices A i ({right arrow over (u)} i ), where m=a×b, a, and b≧1 into b codeword matrix groups, each having a codeword matrix, transmits the b codeword matrix groups after one of interleaving and using different sub-carriers, and transmits a codeword matrix in each codeword matrix group through n×a (≦N) transmit antennas for T 0 ×a (≦T) symbol periods.
17. A transmitter for transmitting data through a plurality of transmit antennas in a wireless communication system including the transmitter and a receiver for receiving the data through a plurality of receive antennas, comprising:
a constellation-rotating precoder for generating P m th -order sub-rotated vectors {right arrow over (r)} i (i=1, . . . , m) by constellation-rotating an input vector {right arrow over (x)} including L (=p×m≦N, p≧2) quadrature amplitude modulation (QAM) signals having Q constellation sizes;
an accumulator for grouping the L QAM signals r i (i=1, . . . , L) into m signal groups {right arrow over (u)} i (i=1, . . . , m) each having p signals; and
a space-time block coder for generating m n×T 0 codeword matrices A i ({right arrow over (u)} i ) (i=1,2, . . . , m) by encoding the m signal groups {right arrow over (u)} i , forming a T×N codeword matrix C({right arrow over (x)}) using the n×T 0 codeword matrices A i ({right arrow over (u)} i ), and transmitting the T×N codeword matrix C({right arrow over (x)}) through N transmit antennas for T (=T 0 ×m) symbol periods.
18. The transmitter of claim 17 , wherein the space-time block coder transmits n columns of each codeword matrix A i ({right arrow over (u)} i ) through n different antennas for each same symbol period of the T 0 symbol periods and columns in different codeword matrices A i ({right arrow over (u)} i ) through n different antennas for the T 0 different symbol periods.
19. The transmitter of claim 17 , wherein the space-time block coder generates a modified T×N codeword matrix
C
(
x
→
)
′
=
Δ
U
C
(
x
→
)
V
by multiplying the codeword matrix C({right arrow over (x)}) by a T×T unitary matrix U and an N×N unitary matrix V at a left and right of C({right arrow over (x)}), and transmits the modified codeword matrix through the N transmit antennas for the T symbol periods.
20. The transmitter of claim 19 , wherein N=2 n , the space-time block coder uses Alamouti's code, and the constellation-rotating precoder uses a matrix expressed as:
Θ
=
1
N
[
1
α
0
…
α
0
N
-
1
1
α
1
…
α
1
N
-
1
⋮
⋮
⋰
⋮
1
α
N
-
1
…
α
N
-
1
N
-
1
]
,
α i =exp(j2,π(i+1/4)/N) and i=0,1, . . . , N−1.
21. The transmitter of claim 20 , wherein the unitary matrix U is expressed as:
U
=
[
I
2
I
2
I
2
…
I
2
I
2
θ
θ
2
…
θ
N
-
1
I
2
θ
2
θ
4
…
θ
(
N
-
1
)
2
⋮
⋮
⋮
⋰
⋮
I
2
θ
N
-
1
θ
2
(
N
-
1
)
…
θ
(
N
-
1
)
2
]
and the unitary matrix V is expressed as:
V
=
[
I
2
I
2
I
2
⋯
I
2
I
2
θ
θ
2
⋯
θ
N
-
1
I
2
θ
2
θ
4
⋯
θ
(
N
-
1
)
2
⋮
⋮
⋮
⋰
⋮
I
2
θ
N
-
1
θ
2
(
N
-
1
)
⋯
θ
(
N
-
1
)
2
]
where I 2 is a 2×2 unitary matrix and
θ
=
Δ
[
ⅇ
ⅈ
4
π
/
N
0
0
ⅇ
-
ⅈ
4
π
/
N
]
.
22. The transmitter of claim 21 , wherein the codeword matrix
C
(
x
→
)
′
=
Δ
U
C
(
x
→
)
V
is formed as:
C
(
x
→
)
′
=
U
[
C
Ala
(
r
1
,
r
N
/
2
+
1
)
0
2
⋯
0
2
0
2
C
Ala
(
r
1
,
r
N
/
2
+
2
)
⋯
0
2
⋮
⋮
⋰
⋮
0
2
0
2
⋯
C
Ala
(
r
N
/
2
,
r
N
)
]
V
=
[
C
Ala
(
s
1
,
s
N
/
2
+
1
)
C
Ala
(
s
N
/
2
,
s
N
)
⋯
C
Ala
(
s
2
,
s
N
/
2
+
2
)
C
Ala
(
s
2
,
s
N
/
2
+
2
)
C
Ala
(
s
1
,
s
N
/
2
+
1
)
⋯
C
Ala
(
s
3
,
s
N
/
2
+
3
)
⋮
⋮
⋰
⋮
C
Ala
(
s
N
/
2
,
s
N
)
C
Ala
(
s
N
/
2
-
1
,
s
N
-
1
)
⋯
C
Ala
(
s
1
,
s
N
/
2
+
1
)
]
where
C
Ala
(
a
,
b
)
=
Δ
[
a
b
-
b
*
a
*
]
,
s
l
=
ⅇ
ⅈ
(
l
-
1
)
π
/
N
x
l
,
and
s
N
/
2
+
l
=
ⅇ
ⅈ
(
l
-
1
)
π
/
N
x
N
/
2
+
l
(
l
=
1
,
⋯
,
N
/
2
)
.
23. The transmitter of claim 17 , wherein if a number of the transmit antennas is (N−1) being an odd number, the space-time block coder generates a new (N−1)×T codeword matrix by deleting a last of N rows of the N×T codeword matrix and transmits the (N−1)×T codeword matrix through (N−1) transmit antennas for the T symbol periods.
24. The transmitter of claim 17 , wherein the space-time block coder groups the m codeword matrices A i ({right arrow over (u)} i ), where m=a×b, a and b≧1, into b codeword matrix groups, each having a codeword matrix, transmits the b codeword matrix groups after one of interleaving and using different sub-carriers, and transmits a codeword matrix in each codeword matrix group through n×a (≦N) transmit antennas for T 0 ×a (≦T) symbol periods.
25. A coding method in a wireless communication system including a transmitter for transmitting data through a plurality of transmit antennas and a receiver for receiving the data through a plurality of receive antennas, comprising:
(a) generating P m th -order sub-rotated vectors {right arrow over (r)} i by constellation-rotating an L th -order input vector {right arrow over (x)} including a plurality of input symbols modulated in at least one modulation scheme;
(b) generating m p th -order sub-vectors {right arrow over (u)} i using the sub-rotated vectors {right arrow over (r)} i ; and
(c) generating codeword matrices A i ({right arrow over (u)} i ) by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i .
26. The coding method of claim 25 , wherein the step (a) comprises the steps of:
generating P m th -order sub-input vectors {right arrow over (x)} i by grouping the symbols of the input vector {right arrow over (x)} into P groups, each having m symbols, and outputting the P m th -order sub-input vectors {right arrow over (x)} i in parallel; and
generating the P m th -order sub-rotated vectors {right arrow over (r)} i by multiplying the parallel P m th -order sub-input vectors {right arrow over (x)} i by a constellation-rotating matrix Θ.
27. The coding method of claim 26 , wherein the step (b) comprises the step of:
extracting signals r i,j included in each of the sub-rotated vectors {right arrow over (r)} i ; and
generating the m p th -order sub-vectors {right arrow over (u)} i by grouping the extracted signals r i,j .
28. The coding method of claim 27 , wherein the step of (c) comprises the steps of:
generating the codeword matrices A i ({right arrow over (u)} i ) by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i ; and
mapping the codeword matrices A i ({right arrow over (u)} i ) to the plurality of the transmit antennas.
29. The coding method of claim 25 , wherein the step (b) comprises the steps of:
extracting signals r i,j included in each of the sub-rotated vectors {right arrow over (r)}; and
generating the m p th -order sub-vectors {right arrow over (u)} i by grouping the extracted signals r i,j .
30. The coding method of claim 25 , wherein the step (c) comprises the steps of:
generating the codeword matrices A i ({right arrow over (u)} i ) by performing orthogonal space-time block coding on the sub-vectors {right arrow over (u)} i ; and
mapping the codeword matrices A i ({right arrow over (u)} i ) to the plurality of the transmit antennas.
31. The coding method of claim 30 , wherein the step (c) further comprises the steps of:
transmitting columns of each of the codeword matrices A i ({right arrow over (u)} i ) through different transmit antennas for a same symbol period; and
transmitting columns of different codeword matrices A i ({right arrow over (u)} i ) through different transmit antennas for different symbol periods.
32. The coding method of claim 25 , wherein the step (c) comprises the step of generating a T×N codeword matrix C({right arrow over (x)}) using the codeword matrices A i ({right arrow over (u)} i ), to be transmitted for N transmit antennas for T symbol periods.
33. The coding method of claim 32 , wherein the T×N codeword matrix C({right arrow over (x)}) is formed as:
C
(
x
→
)
=
Δ
[
c
→
1
,
⋯
,
c
→
N
]
=
[
A
1
(
u
→
1
)
0
n
×
T
0
⋯
0
n
×
T
0
0
n
×
T
0
A
2
(
u
→
2
)
⋯
0
n
×
T
0
⋮
⋮
⋰
⋮
0
n
×
T
0
0
n
×
T
0
⋯
A
m
(
u
→
m
)
]
where 0 n×T 0 denotes an n×T 0 zero matrix and {right arrow over (c)} i denotes an ith column of C({right arrow over (x)}).
34. The coding method of claim 33 , wherein the step (c) comprises the steps of:
if the number of the transmit antennas is (N−1) being an odd number, generating a new (N−1)×T codeword matrix by deleting the last of N rows of the N×T codeword matrix; and
transmitting the (N−1)×T codeword matrix through (N−1) transmit antennas for the T symbol periods.
35. The coding method of claim 33 , wherein the step (c) comprises the steps of:
grouping a total of m codeword matrices A i ({right arrow over (u)} i ), where m=a×b, a, and b≧1, into b codeword matrix groups each having a codeword matrix;
transmitting the b codeword matrix groups after interleaving or using different sub-carriers; and
transmitting a codeword matrices in each codeword matrix group through n×a (≦N) transmit antennas for T 0 ×a (≦T) symbol periods.
36. The coding method of claim 33 , wherein the step (c) comprises the step of simultaneously transmitting other T 0 ×n codeword matrices B i (•), C i (•), and D i (•) generated independently of a total of m T 0 ×n codeword matrices A i (•) in a manner expressed as:
[
A
1
(
·
)
B
1
(
·
)
…
C
1
(
·
)
D
1
(
·
)
…
0
m
0
m
0
m
A
2
(
·
)
B
2
(
·
)
…
C
2
(
·
)
D
2
(
·
)
…
0
m
⋮
⋮
⋰
⋰
⋰
⋰
⋰
⋮
0
m
0
m
…
A
m
(
·
)
B
m
(
·
)
…
C
m
(
·
)
D
m
(
·
)
]
T
,
[
A
1
(
·
)
B
1
(
·
)
…
C
1
(
·
)
D
1
(
·
)
D
2
(
·
)
A
2
(
·
)
B
2
(
·
)
…
C
2
(
·
)
⋰
⋰
⋰
⋰
⋰
…
C
m
(
·
)
D
m
(
·
)
A
m
(
·
)
B
m
(
·
)
]
T
,
[
A
1
(
·
)
B
1
(
·
)
…
C
1
(
·
)
D
1
(
·
)
…
A
2
(
·
)
B
2
(
·
)
…
C
2
(
·
)
D
2
(
·
)
…
⋮
⋮
…
⋮
⋮
…
A
m
(
·
)
B
m
(
·
)
…
C
m
(
·
)
D
m
(
·
)
…
]
T
[
A
1
(
·
)
A
2
(
·
)
…
A
m
(
·
)
B
1
(
·
)
B
2
(
·
)
…
B
m
(
·
)
⋮
⋮
⋮
⋮
C
1
(
·
)
C
2
(
·
)
…
C
m
(
·
)
D
1
(
·
)
D
2
(
·
)
…
D
m
(
·
)
]
T
.
37. The coding method of claim 32 , wherein the step (c) further comprises the step of generating a modified T×N codeword matrix
C
(
x
→
)
′
=
Δ
U
C
(
x
→
)
V
by multiplying the codeword matrix C({right arrow over (x)}) by a T×T unitary matrix U and an N×N unitary matrix V at a left and right of C({right arrow over (x)}), and transmitting the modified codeword matrix through the N transmit antennas for the T symbol periods.
38. The coding method of claim 37 , wherein N=2 n , Alamouti's code is used for the orthogonal space-time block coding, and
Θ
=
1
N
[
1
α
0
…
α
0
N
-
1
1
α
1
…
α
1
N
-
1
⋮
⋮
⋰
⋮
1
α
N
-
1
…
α
N
-
1
N
-
1
]
,
is used for the constellation rotation, the following matrix is used, α i =exp(j2π(i+1/4)/N) and i=0,1, . . . , N−1.
39. The coding method of claim 38 , wherein the unitary matrix U is expressed as:
U
=
[
I
2
I
2
I
2
…
I
2
I
2
θ
θ
2
…
θ
N
-
1
I
2
θ
2
θ
4
…
θ
(
N
-
1
)
2
⋮
⋮
⋮
⋰
⋮
I
2
θ
N
-
1
θ
2
(
N
-
1
)
…
θ
(
N
-
1
)
2
]
,
and the unitary matrix V is expressed as:
V
=
[
I
2
I
2
I
2
⋯
I
2
I
2
θ
θ
2
⋯
θ
N
-
1
I
2
θ
2
θ
4
⋯
θ
(
N
-
1
)
2
⋮
⋮
⋮
⋰
⋮
I
2
θ
N
-
1
θ
2
(
N
-
1
)
⋯
θ
(
N
-
1
)
2
]
where I 2 is a 2×2 unitary matrix and
θ
=
Δ
[
ⅇ
ⅈ
4
π
/
N
0
0
ⅇ
-
ⅈ
4
π
/
N
]
.
40. The coding method of claim 39 , wherein the codeword matrix
C
(
x
→
)
′
=
Δ
U
C
(
x
→
)
V
is formed as:
C
(
x
→
)
′
=
U
[
C
Ala
(
r
1
,
r
N
/
2
+
1
)
0
2
⋯
0
2
0
2
C
Ala
(
r
1
,
r
N
/
2
+
2
)
⋯
0
2
⋮
⋮
⋰
⋮
0
2
0
2
⋯
C
Ala
(
r
N
/
2
,
r
N
)
]
V
=
[
C
Ala
(
s
1
,
s
N
/
2
+
1
)
C
Ala
(
s
N
/
2
,
s
N
)
⋯
C
Ala
(
s
2
,
s
N
/
2
+
2
)
C
Ala
(
s
2
,
s
N
/
2
+
2
)
C
Ala
(
s
1
,
s
N
/
2
+
1
)
⋯
C
Ala
(
s
3
,
s
N
/
2
+
3
)
⋮
⋮
⋰
⋮
C
Ala
(
s
N
/
2
,
s
N
)
C
Ala
(
s
N
/
2
-
1
,
s
N
-
1
)
⋯
C
Ala
(
s
1
,
s
N
/
2
+
1
)
]
where
C
Ala
(
a
,
b
)
=
Δ
[
a
b
-
b
*
a
*
]
,
s
l
=
ⅇ
ⅈ
(
l
-
1
)
π
/
N
x
l
,
and
s
N
/
2
+
l
=
ⅇ
ⅈ
(
l
-
1
)
π
/
N
x
N
/
2
+
l
(
l
=
1
,
⋯
,
N
/
2
)
.