Method and device for codebook generation and downlink signal transmission in a wireless communication system supporting multiple antennas
The present invention relates to a method in which a base station transmits a downlink signal using 8 antennas may comprise the steps of: mapping a downlink signal to N (1≦N≦4) layers; precoding the signal mapped to the N layers, by using a specific precoding matrix selected from a pre-stored codebook; subjecting the precoded signal to processing for OFDM symbol generation; and transmitting OFDM symbols through 8 antennas. The pre-stored codebook may comprise precoding matrices for the respective ranks corresponding to N, the pre-stored codebook may consist of 16 codebook indexes, and a portion of the precoding matrices for high ranks in the 16 codebook indexes may be generated from the precoding matrices for low ranks.
1. A method of transmitting a downlink signal, which is transmitted by a base station using 8 antennas, comprising the steps of:
mapping the downlink signal to N layers (1≦N≦4);
precoding the signal mapped to the N layers using a specific precoding matrix selected from a previously stored codebook;
performing a processing for OFDM symbol configuration on the precoded signal; and
transmitting OFDM symbol via the 8 antennas,
wherein the previously stored codebook comprises precoding matrixes respectively corresponding to N ranks,
wherein the previously stored codebook comprises 16 codebook indexes,
wherein a portion of the precoding matrix for a high rank for each of the 16 codebook indexes comprises the precoding matrix for a low rank,
wherein the previously stored codebook comprises
Codebook
Index
Rank 1
Rank 2
Rank3
1
[W 1 (1)]
[W 1 (1) W 1 (2)]
[W 1 (1) W 1 (2) W 1 (3)]
2
[W 1 (2)]
[W 1 (2) W 1 (3)]
[W 1 (2) W 1 (3) W 1 (4)]
3
[W 1 (3)]
[W 1 (3) W 1 (4)]
[W 1 (3) W 1 (4) W 1 (5)]
4
[W 1 (4)]
[W 1 (4) W 1 (5)]
[W 1 (4) W 1 (5) W 1 (6)]
5
[W 1 (5)]
[W 1 (5) W 1 (6)]
[W 1 (5) W 1 (6) W 1 (7)]
6
[W 1 (6)]
[W 1 (6) W 1 (7)]
[W 1 (6) W 1 (7) W 1 (8)]
7
[W 1 (7)]
[W 1 (7) W 1 (8)]
[W 1 (7) W 1 (8) W 1 (1)]
8
[W 1 (8)]
[W 1 (8) W 1 (1)]
[W 1 (8) W 1 (1) W 1 (2)]
9
[W 4 (1)]
[W 2 (1) W 2 (2)]
[W 2 (1) W 2 (2) W 4 (2)]
10
[W 4 (2)]
[W 2 (3) W 2 (4)]
[W 2 (3) W 2 (4) W 4 (4)]
11
[W 4 (3)]
[W 2 (5) W 2 (6)]
[W 2 (5) W 2 (6) W 4 (6)]
12
[W 4 (4)]
[W 2 (7) W 2 (8)]
[W 2 (7) W 2 (8) W 4 (8)]
13
[W 4 (5)]
[W 3 (1) W 3 (2)]
[W 3 (1) W 3 (2) W 4 (1)]
14
[W 4 (6)]
[W 3 (3) W 3 (4)]
[W 3 (3) W 3 (4) W 4 (3)]
15
[W 4 (7)]
[W 3 (5) W 3 (6)]
[W 3 (5) W 3 (6) W 4 (5)]
16
[W 4 (8)]
[W 3 (7) W 3 (8)]
[W 3 (7) W 3 (8) W 4 (7)]
Codebook
Index
Rank 4
1
[W 1 (1) W 1 (2) W 1 (3) W 1 (4)]
2
[W 1 (2) W 1 (3) W 1 (4) W 1 (5)]
3
[W 1 (3) W 1 (4) W 1 (5) W 1 (6)]
4
[W 1 (4) W 1 (5) W 1 (6) W 1 (7)]
5
[W 1 (5) W 1 (6) W 1 (7) W 1 (8)]
6
[W 1 (6) W 1 (7) W 1 (8) W 1 (1)]
7
[W 1 (7) W 1 (8) W 1 (1) W 1 (2)]
8
[W 1 (8) W 1 (1) W 1 (2) W 1 (3)]
9
[W 2 (1) W 2 (2) W 2 (3) W 2 (4)]
10
[W 2 (3) W 2 (4) W 2 (5) W 2 (6)]
11
[W 2 (5) W 2 (6) W 2 (7) W 2 (8)]
12
[W 2 (7) W 2 (8) W 2 (1) W 2 (2)]
13
[W 3 (1) W 3 (2) W 3 (3) W 3 (4)]
14
[W 3 (3) W 3 (4) W 3 (5) W 3 (6)]
15
[W 3 (5) W 3 (6) W 3 (7) W 3 (8)]
16
[W 3 (7) W 3 (8) W 3 (1) W 3 (2)],
wherein W 1 , W 2 , W 3 and W 4 comprise
W
1
=
DFT
8
×
8
W
2
=
P
T
[
D
1
0
0
D
1
]
P
,
where
D
1
=
DFT
4
×
4
W
3
=
P
T
[
D
2
0
0
D
2
]
P
,
where
D
2
=
diag
[
1
,
ⅇ
π
j
/
4
,
ⅇ
π
j
/
2
,
ⅇ
3
π
j
/
4
]
DFT
4
×
4
,
and
W
4
=
P
T
W
1
,
wherein it is P=[e 1 e 5 e 2 e 6 e 3 e 7 e 4 e 8 ],
wherein e k (1≦k≦8) is a unitary vector having a non-zero value for a k-th element,
wherein DFT 4×4 and DFT 8×8 comprise
DFT
4
×
4
=
1
4
[
1
1
1
1
1
j
-
1
-
j
1
-
1
1
-
1
1
-
j
-
1
j
]
and
DFT
8
×
8
=
1
8
[
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
1
w
2
w
3
w
4
w
5
w
6
w
7
w
0
w
2
w
4
w
6
w
8
w
10
w
12
w
14
w
0
w
3
w
6
w
9
w
12
w
15
w
18
w
21
w
0
w
4
w
8
w
12
w
16
w
20
w
24
w
28
w
0
w
5
w
10
w
15
w
20
w
25
w
30
w
35
w
0
w
6
w
12
w
18
w
24
w
30
w
36
w
42
w
0
w
7
w
14
w
21
w
28
w
35
w
42
w
49
]
where
,
w
=
ⅇ
-
2
π
ⅈ
8
,
respectively.
2. The method of claim 1 , wherein the precoding matrixes of codebook indexes 1 to 8 for a rank 2 have linear configuration in the previously stored codebook and wherein the precoding matrixes of codebook indexes 9 to 16 for the rank 2 have block diagonal configuration in the previously stored codebook.
3. The method of claim 1 , wherein the precoding matrixes of the codebook indexes 1 to 8 for a rank 1 transmission comprise different column vectors of a prescribed 8×8 matrix in the previously stored codebook, respectively and wherein the precoding matrixes of the codebook indexes 9 to 16 for the rank 1 transmission comprise different column vectors of a matrix permutated from the prescribed 8×8 matrix in the previously stored codebook, respectively.
4. A method of receiving a downlink signal, which is received from a base station by a user equipment, comprising the steps of:
receiving the downlink signal transmitted via 8 antennas by being mapped to N layers (1≦N≦4) from the base station; and
processing the received signal using a specific precoding matrix selected from a previously stored codebook,
wherein the previously stored codebook comprises precoding matrixes respectively corresponding to N ranks,
wherein the previously stored codebook comprises 16 codebook indexes,
wherein a portion of the precoding matrix for a high rank for each of the 16 codebook indexes comprises the precoding matrix for a low rank,
wherein the previously stored codebook comprises
Codebook
Index
Rank 1
Rank 2
Rank3
Rank 4
1
[W 1 (1)]
[W 1 (1) W 1 (2)]
[W 1 (1) W 1 (2) W 1 (3)]
[W 1 (1) W 1 (2) W 1 (3) W 1 (4)]
2
[W 1 (2)]
[W 1 (2) W 1 (3)]
[W 1 (2) W 1 (3) W 1 (4)]
[W 1 (2) W 1 (3) W 1 (4) W 1 (5)]
3
[W 1 (3)]
[W 1 (3) W 1 (4)]
[W 1 (3) W 1 (4) W 1 (5)]
[W 1 (3) W 1 (4) W 1 (5) W 1 (6)]
4
[W 1 (4)]
[W 1 (4) W 1 (5)]
[W 1 (4) W 1 (5) W 1 (6)]
[W 1 (4) W 1 (5) W 1 (6) W 1 (7)]
5
[W 1 (5)]
[W 1 (5) W 1 (6)]
[W 1 (5) W 1 (6) W 1 (7)]
[W 1 (5) W 1 (6) W 1 (7) W 1 (8)]
6
[W 1 (6)]
[W 1 (6) W 1 (7)]
[W 1 (6) W 1 (7) W 1 (8)]
[W 1 (6) W 1 (7) W 1 (8) W 1 (1)]
7
[W 1 (7)]
[W 1 (7) W 1 (8)]
[W 1 (7) W 1 (8) W 1 (1)]
[W 1 (7) W 1 (8) W 1 (1) W 1 (2)]
8
[W 1 (8)]
[W 1 (8) W 1 (1)]
[W 1 (8) W 1 (1) W 1 (2)]
[W 1 (8) W 1 (1) W 1 (2) W 1 (3)]
9
[W 4 (1)]
[W 2 (1) W 2 (2)]
[W 2 (1) W 2 (2) W 4 (2)]
[W 2 (1) W 2 (2) W 2 (3) W 2 (4)]
10
[W 4 (2)]
[W 2 (3) W 2 (4)]
[W 2 (3) W 2 (4) W 4 (4)]
[W 2 (3) W 2 (4) W 2 (5) W 2 (6)]
11
[W 4 (3)]
[W 2 (5) W 2 (6)]
[W 2 (5) W 2 (6) W 4 (6)]
[W 2 (5) W 2 (6) W 2 (7) W 2 (8)]
12
[W 4 (4)]
[W 2 (7) W 2 (8)]
[W 2 (7) W 2 (8) W 4 (8)]
[W 2 (7) W 2 (8) W 2 (1) W 2 (2)]
13
[W 4 (5)]
[W 3 (1) W 3 (2)]
[W 3 (1) W 3 (2) W 4 (1)]
[W 3 (1) W 3 (2) W 3 (3) W 3 (4)]
14
[W 4 (6)]
[W 3 (3) W 3 (4)]
[W 3 (3) W 3 (4) W 4 (3)]
[W 3 (3) W 3 (4) W 3 (5) W 3 (6)]
15
[W 4 (7)]
[W 3 (5) W 3 (6)]
[W 3 (5) W 3 (6) W 4 (5)]
[W 3 (5) W 3 (6) W 3 (7) W 3 (8)]
16
[W 4 (8)]
[W 3 (7) W 3 (8)]
[W 3 (7) W 3 (8) W 4 (7)]
[W 3 (7) W 3 (8) W 3 (1) W 3 (2)],
wherein W 1 , W 2 , W 3 and W 4 comprise
W
1
=
DFT
8
×
8
W
2
=
P
T
[
D
1
0
0
D
1
]
P
,
where
D
1
=
DFT
4
×
4
W
3
=
P
T
[
D
2
0
0
D
2
]
P
,
where
D
2
=
diag
[
1
,
ⅇ
π
j
/
4
,
ⅇ
π
j
/
2
,
ⅇ
3
π
j
/
4
]
DFT
4
×
4
,
and
W
4
=
P
T
W
1
,
wherein it is P=[e 1 e 5 e 2 e 6 e 3 e 7 e 4 e 8 ],
wherein e k (1≦k≦8) is a unitary vector having a non-zero value for a k-th element,
wherein DFT 4×4 and DFT 8×8 comprise
DFT
4
×
4
=
1
4
[
1
1
1
1
1
j
-
1
-
j
1
-
1
1
-
1
1
-
j
-
1
j
]
and
DFT
8
×
8
=
1
8
[
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
1
w
2
w
3
w
4
w
5
w
6
w
7
w
0
w
2
w
4
w
6
w
8
w
10
w
12
w
14
w
0
w
3
w
6
w
9
w
12
w
15
w
18
w
21
w
0
w
4
w
8
w
12
w
16
w
20
w
24
w
28
w
0
w
5
w
10
w
15
w
20
w
25
w
30
w
35
w
0
w
6
w
12
w
18
w
24
w
30
w
36
w
42
w
0
w
7
w
14
w
21
w
28
w
35
w
42
w
49
]
where
,
w
=
ⅇ
-
2
π
ⅈ
8
,
respectively.
5. The method of claim 4 , wherein the precoding matrixes of codebook indexes 1 to 8 for a rank 2 have linear configuration in the previously stored codebook and wherein the precoding matrixes of codebook indexes 9 to 16 for the rank 2 have block diagonal configuration in the previously stored codebook.
6. The method of claim 4 , wherein the precoding matrixes of the codebook indexes 1 to 8 for a rank 1 transmission comprise different column vectors of a prescribed 8×8 matrix in the previously stored codebook, respectively and wherein the precoding matrixes of the codebook indexes 9 to 16 for the rank 1 transmission comprise different column vectors of a matrix permutated from the prescribed 8×8 matrix in the previously stored codebook, respectively.
7. A base station device, which transmitting a downlink signal, comprising:
8 antennas configured for signal transmission and reception;
a memory storing a codebook used to transmit a signal mapped to at least one layer via the 8 antennas; and
a processor configured to process a transmission of the downlink signal by being connected to the antennas and the memory, the processor comprising:
a layer mapper mapping the downlink signal to N layers (1≦N≦4);
a precoder precoding the signal mapped to the N layers using a specific precoding matrix selected from a previously stored codebook; and
a transmitting module performing a processing for OFDM symbol configuration on the precoded signal, the transmitting module transmitting the processed signal via the 8 antennas,
wherein the codebook comprises precoding matrixes respectively corresponding to N ranks,
wherein the codebook comprises 16 codebook indexes,
wherein a portion of the precoding matrix for a high rank for each of the 16 codebook indexes comprises the precoding matrix for a low rank,
wherein the codebook comprises
Codebook
Index
Rank 1
Rank 2
Rank3
Rank 4
1
[W 1 (1)]
[W 1 (1) W 1 (2)]
[W 1 (1) W 1 (2) W 1 (3)]
[W 1 (1) W 1 (2) W 1 (3) W 1 (4)]
2
[W 1 (2)]
[W 1 (2) W 1 (3)]
[W 1 (2) W 1 (3) W 1 (4)]
[W 1 (2) W 1 (3) W 1 (4) W 1 (5)]
3
[W 1 (3)]
[W 1 (3) W 1 (4)]
[W 1 (3) W 1 (4) W 1 (5)]
[W 1 (3) W 1 (4) W 1 (5) W 1 (6)]
4
[W 1 (4)]
[W 1 (4) W 1 (5)]
[W 1 (4) W 1 (5) W 1 (6)]
[W 1 (4) W 1 (5) W 1 (6) W 1 (7)]
5
[W 1 (5)]
[W 1 (5) W 1 (6)]
[W 1 (5) W 1 (6) W 1 (7)]
[W 1 (5) W 1 (6) W 1 (7) W 1 (8)]
6
[W 1 (6)]
[W 1 (6) W 1 (7)]
[W 1 (6) W 1 (7) W 1 (8)]
[W 1 (6) W 1 (7) W 1 (8) W 1 (1)]
7
[W 1 (7)]
[W 1 (7) W 1 (8)]
[W 1 (7) W 1 (8) W 1 (1)]
[W 1 (7) W 1 (8) W 1 (1) W 1 (2)]
8
[W 1 (8)]
[W 1 (8) W 1 (1)]
[W 1 (8) W 1 (1) W 1 (2)]
[W 1 (8) W 1 (1) W 1 (2) W 1 (3)]
9
[W 4 (1)]
[W 2 (1) W 2 (2)]
[W 2 (1) W 2 (2) W 4 (2)]
[W 2 (1) W 2 (2) W 2 (3) W 2 (4)]
10
[W 4 (2)]
[W 2 (3) W 2 (4)]
[W 2 (3) W 2 (4) W 4 (4)]
[W 2 (3) W 2 (4) W 2 (5) W 2 (6)]
11
[W 4 (3)]
[W 2 (5) W 2 (6)]
[W 2 (5) W 2 (6) W 4 (6)]
[W 2 (5) W 2 (6) W 2 (7) W 2 (8)]
12
[W 4 (4)]
[W 2 (7) W 2 (8)]
[W 2 (7) W 2 (8) W 4 (8)]
[W 2 (7) W 2 (8) W 2 (1) W 2 (2)]
13
[W 4 (5)]
[W 3 (1) W 3 (2)]
[W 3 (1) W 3 (2) W 4 (1)]
[W 3 (1) W 3 (2) W 3 (3) W 3 (4)]
14
[W 4 (6)]
[W 3 (3) W 3 (4)]
[W 3 (3) W 3 (4) W 4 (3)]
[W 3 (3) W 3 (4) W 3 (5) W 3 (6)]
15
[W 4 (7)]
[W 3 (5) W 3 (6)]
[W 3 (5) W 3 (6) W 4 (5)]
[W 3 (5) W 3 (6) W 3 (7) W 3 (8)]
16
[W 4 (8)]
[W 3 (7) W 3 (8)]
[W 3 (7) W 3 (8) W 4 (7)]
[W 3 (7) W 3 (8) W 3 (1) W 3 (2)],
wherein W 1 , W 2 , W 3 and W 4 comprise
W
1
=
DFT
8
×
8
W
2
=
P
T
[
D
1
0
0
D
1
]
P
,
where
D
1
=
DFT
4
×
4
W
3
=
P
T
[
D
2
0
0
D
2
]
P
,
where
D
2
=
diag
[
1
,
ⅇ
π
j
/
4
,
ⅇ
π
j
/
2
,
ⅇ
3
π
j
/
4
]
DFT
4
×
4
,
and
W
4
=
P
T
W
1
,
wherein it is P=[e 1 e 5 e 2 e 6 e 3 e 7 e 4 e 8 ],
wherein e k (1≦k≦8) is a unitary vector having a non-zero value for a k-th element,
wherein DFT 4×4 and DFT 8×8 comprise
DFT
4
×
4
=
1
4
[
1
1
1
1
1
j
-
1
-
j
1
-
1
1
-
1
1
-
j
-
1
j
]
and
DFT
8
×
8
=
1
8
[
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
1
w
2
w
3
w
4
w
5
w
6
w
7
w
0
w
2
w
4
w
6
w
8
w
10
w
12
w
14
w
0
w
3
w
6
w
9
w
12
w
15
w
18
w
21
w
0
w
4
w
8
w
12
w
16
w
20
w
24
w
28
w
0
w
5
w
10
w
15
w
20
w
25
w
30
w
35
w
0
w
6
w
12
w
18
w
24
w
30
w
36
w
42
w
0
w
7
w
14
w
21
w
28
w
35
w
42
w
49
]
where
,
w
=
ⅇ
-
2
πⅈ
8
,
respectively.
8. A user equipment device, which receives a downlink signal from a base station, comprising:
a signal transceiving antenna;
a memory storing a codebook used to receive the downlink signal transmitted via 8 antennas by being mapped to N layers (1≦N≦4) from the base station; and
a processor processing a reception of the downlink signal by being connected to the antenna and the memory, the processor configured to process the received signal using a specific precoding matrix selected from a codebook stored in the memory,
wherein the codebook comprises precoding matrixes respectively corresponding to N ranks,
wherein the previously stored codebook comprises 16 codebook indexes,
wherein a portion of the precoding matrix for a high rank for each of the 16 codebook indexes comprises the precoding matrix for a low rank,
wherein the previously stored codebook comprises
Codebook
Index
Rank 1
Rank 2
Rank3
Rank 4
1
[W 1 (1)]
[W 1 (1) W 1 (2)]
[W 1 (1) W 1 (2) W 1 (3)]
[W 1 (1) W 1 (2) W 1 (3) W 1 (4)]
2
[W 1 (2)]
[W 1 (2) W 1 (3)]
[W 1 (2) W 1 (3) W 1 (4)]
[W 1 (2) W 1 (3) W 1 (4) W 1 (5)]
3
[W 1 (3)]
[W 1 (3) W 1 (4)]
[W 1 (3) W 1 (4) W 1 (5)]
[W 1 (3) W 1 (4) W 1 (5) W 1 (6)]
4
[W 1 (4)]
[W 1 (4) W 1 (5)]
[W 1 (4) W 1 (5) W 1 (6)]
[W 1 (4) W 1 (5) W 1 (6) W 1 (7)]
5
[W 1 (5)]
[W 1 (5) W 1 (6)]
[W 1 (5) W 1 (6) W 1 (7)]
[W 1 (5) W 1 (6) W 1 (7) W 1 (8)]
6
[W 1 (6)]
[W 1 (6) W 1 (7)]
[W 1 (6) W 1 (7) W 1 (8)]
[W 1 (6) W 1 (7) W 1 (8) W 1 (1)]
7
[W 1 (7)]
[W 1 (7) W 1 (8)]
[W 1 (7) W 1 (8) W 1 (1)]
[W 1 (7) W 1 (8) W 1 (1) W 1 (2)]
8
[W 1 (8)]
[W 1 (8) W 1 (1)]
[W 1 (8) W 1 (1) W 1 (2)]
[W 1 (8) W 1 (1) W 1 (2) W 1 (3)]
9
[W 4 (1)]
[W 2 (1) W 2 (2)]
[W 2 (1) W 2 (2) W 4 (2)]
[W 2 (1) W 2 (2) W 2 (3) W 2 (4)]
10
[W 4 (2)]
[W 2 (3) W 2 (4)]
[W 2 (3) W 2 (4) W 4 (4)]
[W 2 (3) W 2 (4) W 2 (5) W 2 (6)]
11
[W 4 (3)]
[W 2 (5) W 2 (6)]
[W 2 (5) W 2 (6) W 4 (6)]
[W 2 (5) W 2 (6) W 2 (7) W 2 (8)]
12
[W 4 (4)]
[W 2 (7) W 2 (8)]
[W 2 (7) W 2 (8) W 4 (8)]
[W 2 (7) W 2 (8) W 2 (1) W 2 (2)]
13
[W 4 (5)]
[W 3 (1) W 3 (2)]
[W 3 (1) W 3 (2) W 4 (1)]
[W 3 (1) W 3 (2) W 3 (3) W 3 (4)]
14
[W 4 (6)]
[W 3 (3) W 3 (4)]
[W 3 (3) W 3 (4) W 4 (3)]
[W 3 (3) W 3 (4) W 3 (5) W 3 (6)]
15
[W 4 (7)]
[W 3 (5) W 3 (6)]
[W 3 (5) W 3 (6) W 4 (5)]
[W 3 (5) W 3 (6) W 3 (7) W 3 (8)]
16
[W 4 (8)]
[W 3 (7) W 3 (8)]
[W 3 (7) W 3 (8) W 4 (7)]
[W 3 (7) W 3 (8) W 3 (1) W 3 (2)],
wherein W 1 , W 2 , W 3 and W 4 comprise
W
1
=
DFT
8
×
8
W
2
=
P
T
[
D
1
0
0
D
1
]
P
,
where
D
1
=
DFT
4
×
4
W
3
=
P
T
[
D
2
0
0
D
2
]
P
,
where
D
2
=
diag
[
1
,
ⅇ
π
j
/
4
,
ⅇ
π
j
/
2
,
ⅇ
3
π
j
/
4
]
DFT
4
×
4
,
and
W
4
=
P
T
W
1
,
wherein it is P=[e 1 e 5 e 2 e 6 e 3 e 7 e 4 e 8 ],
wherein e k (1≦k≦8) is a unitary vector having a non-zero value for a k-th element,
wherein DFT 4×4 and DFT 8×8 comprise
DFT
4
×
4
=
1
4
[
1
1
1
1
1
j
-
1
-
j
1
-
1
1
-
1
1
-
j
-
1
j
]
and
DFT
8
×
8
=
1
8
[
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
0
w
1
w
2
w
3
w
4
w
5
w
6
w
7
w
0
w
2
w
4
w
6
w
8
w
10
w
12
w
14
w
0
w
3
w
6
w
9
w
12
w
15
w
18
w
21
w
0
w
4
w
8
w
12
w
16
w
20
w
24
w
28
w
0
w
5
w
10
w
15
w
20
w
25
w
30
w
35
w
0
w
6
w
12
w
18
w
24
w
30
w
36
w
42
w
0
w
7
w
14
w
21
w
28
w
35
w
42
w
49
]
where
,
w
=
ⅇ
-
2
πⅈ
8
,
respectively.