Method and apparatus for higher rank CSI reporting in advanced wireless communication systems
A method of operating a user equipment (UE) for CSI feedback is provided. The method comprises receiving configuration information for the CSI feedback from a base station (BS) and identifying a number of antenna ports for the CSI feedback. The method comprises, if the number of antenna ports is <16, identifying a first codebook for the CSI feedback corresponding to a rank value of 3 or 4, and, if otherwise, identifying a second codebook for the CSI feedback corresponding to the rank value of 3 or 4. The method comprises generating the CSI feedback using the identified codebook and transmitting the generated CSI feedback to the BS. The first codebook has a structure that partitions the antenna ports into two equal partitions. The second codebook has a structure that partitions the antenna ports into four equal partitions by partitioning each partition into two equal sub-partitions.
1. A user equipment (UE) for channel state information (CSI) feedback, the UE comprising
a transceiver configured to receive, from a base station (BS), configuration information for the CSI feedback; and
a processor operably connected to the transceiver, the processor configured to:
identify a number of antenna ports for the CSI feedback;
if the number of antenna ports is <16, identify a first codebook for the CSI feedback corresponding to a rank value of 3 or 4;
if the number of antenna ports is ≥16, identify a second codebook for the CSI feedback corresponding to the rank value of 3 or 4; and
generate the CSI feedback using the identified codebook,
wherein the transceiver is further configured to transmit the generated CSI feedback to the BS,
wherein the first codebook has a structure that partitions the antenna ports into two equal partitions, P i where i=0,1, and the second codebook has a structure that partitions the antenna ports into four equal partitions by partitioning each partition P i further into two equal sub-partitions, P i,j where j=0,1, and
wherein the rank value corresponds to a plurality of layers indicated by a rank indicator (RI).
2. The UE of claim 1 , wherein:
the CSI feedback includes a pre-coding matrix indicator (PMI) that for the rank value of 3 or 4 includes PMI indices i 1,1 , i 1,2 , and i 1,3 ,
for the first codebook, the PMI indices i 1,1 and i 1,2 indicate a first beam v i 1,1 i 1,2 , and the PMI index i 1,3 indicates a distance (k 1 , k 2 ) of a second beam v i 1,1 k 1 ,i 1,2 +k 2 with respect to the first beam v i 1,1 i 1,2 , where v i 1,1 i 1,2 and v i 1,1 +k 1 ,i 1,2 +k 2 are used in common for the two partitions, P i , and
for the second codebook, the PMI indices i 1,1 , and i 1,2 indicate a beam {tilde over (v)} i 1,1 i 1,2 , and the PMI index i 1,3 indicates a co-phasing parameter θ p between the two sub-partitions P i,j of each partition P i such that {tilde over (v)} i 1,1 i 1,2 is used for a first of the sub-partitions P i,0 and θ p {tilde over (v)} i 1,1 i 1,2 is used for a second of the sub-partitions P i,1 .
3. The UE of claim 2 , wherein:
when the number of antenna ports is <16, mapping of the PMI index i 1,3 to k 1 and k 2 for the CSI feedback corresponding to the rank value of 3 or 4 is determined according to:
N 1 = 2, N 2 = 1
N 1 = 4, N 2 = 1
N 1 = 6, N 2 = 1
N 1 = 2, N 2 = 2
N 1 = 3, N 2 = 2
i 1,3
k 1
k 2
k 1
k 2
k 1
k 2
k 1
k 2
k 1
k 2
0
O 1
0
O 1
0
O 1
0
O 1
0
O 1
0
1
2O 1
0
2O 1
0
0
O 2
0
O 2
2
3O 1
0
3O 1
0
O 1
O 2
O 1
O 2
3
4O 1
0
2O 1
0,
N 1 and N 2 are higher layer signaled parameters that indicate a number of antenna ports with a same polarization in first and second dimensions, respectively, and
O 1 and O 2 are oversampling factors in first and second dimensions, respectively, that are used to obtain a oversampled two-dimensional discrete Fourier transform (DFT) codebook for the first and second beams, v i 1,1 i 1,2 and v i 1,1 +k 1 ,i 1,2 +k 2 , and for the beam {tilde over (v)} i 1,1 i 1,2 .
4. The UE of claim 2 , wherein the co-phasing parameter θ p =e jπp/4 , and i 1,3 =p=0,1,2,3.
5. The UE of claim 2 , wherein the PMI includes a PMI index i 2 that indicates a co-phasing parameter φ n between the two partitions P i , where φ n =e jπn/2 , and i 2 =n=0,1.
6. The UE of claim 2 , wherein, for the CSI feedback corresponding to the rank value of 3:
when the number of antenna ports is <16, the first codebook is determined based on a pre-coding matrix:
W
l
,
l
′
,
m
,
m
′
,
n
(
3
)
=
1
3
P
CSI
-
RS
[
v
l
,
m
v
l
′
,
m
′
v
l
,
m
φ
n
v
l
,
m
φ
n
v
l
′
,
m
′
-
φ
n
v
l
,
m
]
,
and
when the number of antenna ports is ≥16, the second codebook is determined based on a pre-coding matrix:
W
l
,
m
,
p
,
n
(
3
)
=
1
3
P
CSI
-
RS
[
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
θ
p
v
~
l
,
m
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
]
where rows of W l,l′,m,m′,n (3) correspond to the two partitions P i and rows of W l,m,p,n (3) correspond to the four sub-partitions P i,j , l=i 1,1 , m=i 1,2 , l′=i 1,1 +k 1 , m′=i 1,2 +k 2 , p=i 1,3 , n=i 2 , P CSI-RS is the number of antenna ports, and
φ
n
=
e
j
π
n
/
2
θ
p
=
e
j
π
p
/
4
u
m
=
{
[
1
e
j
2
π
m
O
2
N
2
…
e
j
2
π
m
(
N
2
-
1
)
O
2
N
2
]
N
2
>
1
1
N
2
=
1
.
v
l
,
m
=
[
u
m
e
j
2
π
l
O
1
N
1
u
m
…
e
j
2
π
l
(
N
1
-
1
)
O
1
N
1
u
m
]
T
v
~
l
,
m
=
[
u
m
e
j
4
π
l
O
1
N
1
u
m
…
e
j
4
π
l
(
N
1
/
2
-
1
)
O
1
N
1
u
m
]
T
7. The UE of claim 2 , wherein, for the CSI feedback corresponding to the rank value of 4:
when the number of antenna ports is <16, the first codebook is determined based on a pre-coding matrix:
W
l
,
l
′
,
m
,
m
′
,
n
(
4
)
=
1
4
P
CSI
-
RS
[
v
l
,
m
v
l
′
,
m
′
v
l
,
m
v
l
′
,
m
′
φ
n
v
l
,
m
φ
n
v
l
′
,
m
′
-
φ
n
v
l
,
m
-
φ
n
v
l
′
,
m
′
]
,
and
when the number of antenna ports is ≥16, the second codebook is determined based on a pre-coding matrix:
W
l
,
m
,
p
,
n
(
4
)
=
1
4
P
CSI
-
RS
[
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
θ
p
v
~
l
,
m
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
]
,
where rows of W l,l′,m,m′,n (4) correspond to the two partitions P i and rows of W l,m,p,n (4) correspond to the four sub-partitions P i,j , l=i 1,1 , m=i 1,2 , l′=i 1,1 +k 1 , m′=i 1,2 +k 2 , p=i 1,3 , n=i 2 , P CSI-RS is the number of antenna ports, and
φ
n
=
e
j
π
n
/
2
θ
p
=
e
j
π
p
/
4
u
m
=
{
[
1
e
j
2
π
m
O
2
N
2
…
e
j
2
π
m
(
N
2
-
1
)
O
2
N
2
]
N
2
>
1
1
N
2
=
1
.
v
l
,
m
=
[
u
m
e
j
2
π
l
O
1
N
1
u
m
…
e
j
2
π
l
(
N
1
-
1
)
O
1
N
1
u
m
]
T
v
~
l
,
m
=
[
u
m
e
j
4
π
l
O
1
N
1
u
m
…
e
j
4
π
l
(
N
1
/
2
-
1
)
O
1
N
1
u
m
]
T
8. A base station (BS), the BS comprising
a processor configured to generate configuration information for a channel state information (CSI) feedback; and
a transceiver operably connected to the processor, the transceiver configured to:
transmit, to a user equipment (UE), the configuration information for the CSI feedback; and
receive, from the UE, the CSI feedback generated by the UE,
wherein if a number of antenna ports for the CSI feedback is <16, the CSI feedback corresponding to a rank value of 3 or 4 is generated based on a first codebook,
wherein if the number of antenna ports is ≥16, the CSI feedback corresponding to the rank value of 3 or 4 is generated based on a second codebook,
wherein the first codebook has a structure that partitions the antenna ports into two equal partitions, P i where i=0,1, and the second codebook has a structure that partitions the antenna ports into four equal partitions by partitioning each partition P i further into two equal sub-partitions, P i,j where j=0,1, and
wherein the rank value corresponds to a plurality of layers indicated by a rank indicator (RI).
9. The BS of claim 8 , wherein:
the CSI feedback includes a pre-coding matrix indicator (PMI) that for the rank value of 3 or 4 includes PMI indices i 1,1 , i 1,2 , and i 1,3 ,
for the first codebook, the PMI indices i 1,1 and i 1,2 indicate a first v i 1,1 i 1,2 , and the PMI index i 1,3 indicates a distance (k 1 , k 2 ) of a second beam v i 1,1 +k 1 ,i 1,2 +k 2 with respect to the first beam v i 1,1 i 1,2 , where v i 1,1 i 1,2 and v i 1,1 +k 1 ,i 1,2 +k 2 are used in common for the two partitions, P i , and
for the second codebook, the PMI indices i 1,1 and i 1,2 indicate a beam {tilde over (v)} i 1,1 i 1,2 , and the PMI index i 1,3 indicates a co-phasing parameter θ p between the two sub-partitions P i,j of each partition P i such that {tilde over (v)} i 1,1 i 1,2 is used for a first of the sub-partitions P i,0 and θ p {tilde over (v)} i 1,1 i 1,2 is used for a second of the sub-partitions P i,1 .
10. The BS of claim 9 , wherein:
when the number of antenna ports is <16, mapping of the PMI index i 1,3 to k 1 and k 2 for the CSI feedback corresponding to the rank value of 3 or 4 is determined according to:
N 1 = 2, N 2 = 1
N 1 = 4, N 2 = 1
N 1 = 6, N 2 = 1
N 1 = 2, N 2 = 2
N 1 = 3, N 2 = 2
i 1,3
k 1
k 2
k 1
k 2
k 1
k 2
k 1
k 2
k 1
k 2
0
O 1
0
O 1
0
O 1
0
O 1
0
O 1
0
1
2O 1
0
2O 1
0
0
O 2
0
O 2
2
3O 1
0
3O 1
0
O 1
O 2
O 1
O 2
3
4O 1
0
2O 1
0,
N 1 and N 2 are higher layer signaled parameters that indicate a number of antenna ports with a same polarization in first and second dimensions, respectively, and
O 1 and O 2 are oversampling factors in first and second dimensions, respectively, that are used to obtain a oversampled two-dimensional discrete Fourier transform (DFT) codebook for the first and second beams, v i 1,1 i 1,2 and v i 1,1 +k 1 ,i 1,2 +k 2 , and for the beam {tilde over (v)} i 1,1 i 1,2 .
11. The BS of claim 9 , wherein the co-phasing parameter θ p =e jπp/4 , and i 1,3 =p=0,1,2,3.
12. The BS of claim 9 , wherein the PMI includes a PMI index i 2 that indicates a co-phasing parameter φ n between the two partitions P i , where φ n =e jπn/2 , and i 2 =n=0,1.
13. The BS of claim 9 , wherein, for the CSI feedback corresponding to the rank value of 3:
when the number of antenna ports is <16, the first codebook is determined based on a pre-coding matrix:
W
l
,
l
′
,
m
,
m
′
,
n
(
3
)
=
1
3
P
CSI
-
RS
[
v
l
,
m
v
l
′
,
m
′
v
l
,
m
φ
n
v
l
,
m
φ
n
v
l
′
,
m
′
-
φ
n
v
l
,
m
]
,
and
when the number of antenna ports is ≥16, the second codebook is determined based on a pre-coding matrix:
W
l
,
m
,
p
,
n
(
3
)
=
1
3
P
CSI
-
RS
[
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
θ
p
v
~
l
,
m
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
]
where rows of W l,l′,m,m′,n (3) correspond to the two partitions P i and rows of W l,m,p,n (3) correspond to the four sub-partitions P i,j , l=i 1,1 , m=i 1,2 , l′=i 1,1 +k 1 , m′=i 1,2 +k 2 , p=i 1,3 , n=i 2 , P CSI-RS is the number of antenna ports, and
φ
n
=
e
j
π
n
/
2
θ
p
=
e
j
π
p
/
4
u
m
=
{
[
1
e
j
2
π
m
O
2
N
2
…
e
j
2
π
m
(
N
2
-
1
)
O
2
N
2
]
N
2
>
1
1
N
2
=
1
.
v
l
,
m
=
[
u
m
e
j
2
π
l
O
1
N
1
u
m
…
e
j
2
π
l
(
N
1
-
1
)
O
1
N
1
u
m
]
T
v
~
l
,
m
=
[
u
m
e
j
4
π
l
O
1
N
1
u
m
…
e
j
4
π
l
(
N
1
/
2
-
1
)
O
1
N
1
u
m
]
T
14. The BS of claim 9 , wherein, for the CSI feedback corresponding to the rank value of 3:
when the number of antenna ports is <16, the first codebook is determined based on a pre-coding matrix:
W
l
,
l
′
,
m
,
m
′
,
n
(
4
)
=
1
4
P
CSI
-
RS
[
v
l
,
m
v
l
′
,
m
′
v
l
,
m
v
l
′
,
m
′
φ
n
v
l
,
m
φ
n
v
l
′
,
m
′
-
φ
n
v
l
,
m
-
φ
n
v
l
′
,
m
′
]
,
and
when the number of antenna ports is ≥16, the second codebook is determined based on a pre-coding matrix:
W
l
,
m
,
p
,
n
(
4
)
=
1
4
P
CSI
-
RS
[
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
θ
p
v
~
l
,
m
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
]
,
where rows of W l,l′,m,m′,n (4) correspond to the two partitions P i and rows of W l,m,p,n (4) correspond to the four sub-partitions P i,j , l=i 1,1 , m=i 1,2 , l′=i 1,1 +k 1 , m′=i 1,2 +k 2 , p=i 1,3 , n=i 2 , P CSI-RS is the number of antenna ports, and
φ
n
=
e
j
π
n
/
2
θ
p
=
e
j
π
p
/
4
u
m
=
{
[
1
e
j
2
π
m
O
2
N
2
…
e
j
2
π
m
(
N
2
-
1
)
O
2
N
2
]
N
2
>
1
1
N
2
=
1
.
v
l
,
m
=
[
u
m
e
j
2
π
l
O
1
N
1
u
m
…
e
j
2
π
l
(
N
1
-
1
)
O
1
N
1
u
m
]
T
v
~
l
,
m
=
[
u
m
e
j
4
π
l
O
1
N
1
u
m
…
e
j
4
π
l
(
N
1
/
2
-
1
)
O
1
N
1
u
m
]
T
15. A method of operating a user equipment (UE) for channel state information (CSI) feedback, the method comprising
receiving, from a base station (BS), configuration information for the CSI feedback;
identifying a number of antenna ports for the CSI feedback;
if the number of antenna ports is <16, identifying a first codebook for the CSI feedback corresponding to a rank value of 3 or 4;
if the number of antenna ports is ≥16, identifying a second codebook for the CSI feedback corresponding to the rank value of 3 or 4;
generating the CSI feedback using the identified codebook; and
transmitting the generated CSI feedback to the BS,
wherein the first codebook has a structure that partitions the antenna ports into two equal partitions, P i where i=0,1, and the second codebook has a structure that partitions the antenna ports into four equal partitions by partitioning each partition P i further into two equal sub-partitions, P i,j where j=0,1, and
wherein the rank value corresponds to a plurality of layers indicated by a rank indicator (RI).
16. The method of claim 15 , wherein:
the CSI feedback includes a pre-coding matrix indicator (PMI) that for the rank value of 3 or 4 includes PMI indices i 1,1 , i 1,2 , and i 1,3 ,
for the first codebook, the PMI indices i 1,1 and i 1,2 indicate a first beam v i 1,1 i 1,2 , and the PMI index i 1,3 indicates a distance (k 1 , k 2 ) of a second beam v i 1,1 +k 1 ,i 1,2 +k 2 with respect to the first beam v i 1,1 i 1,2 , where v i 1,1 i 1,2 and v i 1,1 +k 1 ,i 1,2 +k 2 are used in common for the two partitions, P i , and
for the second codebook, the PMI indices i 1,1 and i 1,2 indicate a beam {tilde over (v)} i 1,1 i 1,2 , and the PMI index i 1,3 indicates a co-phasing parameter θ p between the two sub-partitions P i,j of each partition P i such that {tilde over (v)} i 1,1 i 1,2 is used for a first of the sub-partitions P i,0 and θ p {tilde over (v)} i 1,1 i 1,2 is used for a second of the sub-partitions P i,1 .
17. The method of claim 16 , wherein:
when the number of antenna ports is <16, mapping of the PMI index i 1,3 to k 1 and k 2 for the CSI feedback corresponding to the rank value of 3 or 4 is determined according to:
N 1 = 2, N 2 = 1
N 1 = 4, N 2 = 1
N 1 = 6, N 2 = 1
N 1 = 2, N 2 = 2
N 1 = 3, N 2 = 2
i 1,3
k 1
k 2
k 1
k 2
k 1
k 2
k 1
k 2
k 1
k 2
0
O 1
0
O 1
0
O 1
0
O 1
0
O 1
0
1
2O 1
0
2O 1
0
0
O 2
0
O 2
2
3O 1
0
3O 1
0
O 1
O 2
O 1
O 2
3
4O 1
0
2O 1
0
N 1 and N 2 are higher layer signaled parameters that indicate a number of antenna ports with a same polarization in first and second dimensions, respectively, and
O 1 and O 2 are oversampling factors in first and second dimensions, respectively, that are used to obtain a oversampled two-dimensional discrete Fourier transform (DFT) codebook for the first and second beams, v i 1,1 i 1,2 and v i 1,1 +k 1 ,i 1,2 +k 2 , and for the beam {tilde over (v)} i 1,1 i 1,2 .
18. The method of claim 16 , wherein at least one of:
the co-phasing parameter θ p =e jπp/4 , and i 1,3 =p=0,1,2,3, or
the PMI includes a PMI index i 2 that indicates a co-phasing parameter φ n between the two partitions P i , where φ n =e jπn/2 , and i 2 =n=0,1.
19. The method of claim 16 , wherein, for the CSI feedback corresponding to the rank value of 3:
when the number of antenna ports is <16, the first codebook is determined based on a pre-coding matrix:
W
l
,
l
′
,
m
,
m
′
,
n
(
3
)
=
1
3
P
CSI
-
RS
[
v
l
,
m
v
l
′
,
m
′
v
l
,
m
φ
n
v
l
,
m
φ
n
v
l
′
,
m
′
-
φ
n
v
l
,
m
]
,
and
when the number of antenna ports is ≥16, the second codebook is determined based on a pre-coding matrix:
W
l
,
m
,
p
,
n
(
3
)
=
1
3
P
CSI
-
RS
[
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
θ
p
v
~
l
,
m
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
]
where rows of W l,l′,m,m′,n (4) correspond to the two partitions P i and rows of W l,m,p,n (3) correspond to the four sub-partitions P i,j , l=i 1,1 , m=i 1,2 , l′=i 1,1 +k 1 , m′=i 1,2 +k 2 , p=i 1,3 , n=i 2 , P CSI-RS is the number of antenna ports, and
φ
n
=
e
j
π
n
/
2
θ
p
=
e
j
π
p
/
4
u
m
=
{
[
1
e
j
2
π
m
O
2
N
2
…
e
j
2
π
m
(
N
2
-
1
)
O
2
N
2
]
N
2
>
1
1
N
2
=
1
.
v
l
,
m
=
[
u
m
e
j
2
π
l
O
1
N
1
u
m
…
e
j
2
π
l
(
N
1
-
1
)
O
1
N
1
u
m
]
T
v
~
l
,
m
=
[
u
m
e
j
4
π
l
O
1
N
1
u
m
…
e
j
4
π
l
(
N
1
/
2
-
1
)
O
1
N
1
u
m
]
T
20. The method of claim 16 , wherein, for the CSI feedback corresponding to the rank value of 4:
when the number of antenna ports is <16, the first codebook is determined based on a pre-coding matrix:
W
l
,
l
′
,
m
,
m
′
,
n
(
4
)
=
1
4
P
CSI
-
RS
[
v
l
,
m
v
l
′
,
m
′
v
l
,
m
v
l
′
,
m
′
φ
n
v
l
,
m
φ
n
v
l
′
,
m
′
-
φ
n
v
l
,
m
-
φ
n
v
l
′
,
m
′
]
,
and
when the number of antenna ports is ≥16, the second codebook is determined based on a pre-coding matrix:
W
l
,
m
,
p
,
n
(
4
)
=
1
4
P
CSI
-
RS
[
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
v
~
l
,
m
θ
p
v
~
l
,
m
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
-
θ
p
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
v
~
l
,
m
-
φ
n
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
-
φ
n
θ
p
v
~
l
,
m
φ
n
θ
p
v
~
l
,
m
]
,
where rows of W l,l′,m,m′,n (4) correspond to the two partitions P i and rows of W l,m,p,n (4) correspond to the four sub-partitions P i,j , l=i 1,1 , m=i 1,2 , l′=i 1,1 l+k 1 , m′=i 1,2 +k 2 , p=i 1,3 , n=i 2 , P CSI-RS is the number of antenna ports, and
φ
n
=
e
j
π
n
/
2
θ
p
=
e
j
π
p
/
4
u
m
=
{
[
1
e
j
2
π
m
O
2
N
2
…
e
j
2
π
m
(
N
2
-
1
)
O
2
N
2
]
N
2
>
1
1
N
2
=
1
.
v
l
,
m
=
[
u
m
e
j
2
π
l
O
1
N
1
u
m
…
e
j
2
π
l
(
N
1
-
1
)
O
1
N
1
u
m
]
T
v
~
l
,
m
=
[
u
m
e
j
4
π
l
O
1
N
1
u
m
…
e
j
4
π
l
(
N
1
/
2
-
1
)
O
1
N
1
u
m
]
T