Optimal user pairing for multiuser MIMO
A method includes operating a receiver connected with n r receive antennas to receive uplink signals from K users having n t transmit antennas each; and scheduling which users are to transmit in a same time-frequency-code slot. A particular user pairing configuration is denoted by π within a set of all configurations Π, and scheduling includes expressing mutual information as an additive objective function and maximizing the additive objective function over a particular choice of πεΠ.
1. A method, comprising:
operating a receiver connected with a plurality of receive antennas to receive uplink signals from a plurality of users each having a plurality of transmit antennas; and
scheduling which ones of the plurality of users are to transmit in a same time-frequency-code slot, where a particular user pairing configuration is denoted by π within a set of all configurations Π, and where scheduling comprises expressing mutual information as an additive objective function and maximizing the additive objective function over a particular choice of πεΠ,
wherein the additive objective function that is maximized comprises:
max
π
∈
Π
∑
(
k
1
,
k
2
)
∈
π
(
N
)
f
k
1
,
k
2
(
π
)
,
where f is measure of performance for a pairing of users k 1 and k 2 and N is an integer total number of distinct pairs of users.
2. The method of claim 1 , wherein maximizing comprises using an n×n matrix W=[w k, ]=[w(x k y )] where w k, is the weight of the k th , th element of the matrix W and w(x k , y ) is the weight for a pairing of user x k and user y in the matrix W, and finding a permutation σεS n , where S n denotes the group of all permutations of n number of elements, for which
∑
k
=
1
n
w
(
x
k
y
σ
(
k
)
)
is a maximum, where W is a total bandwidth occupied and xy is an edge.
3. The method of claim 1 , wherein there are paired users and unpaired users, wherein a pairing configuration π={π pair ,π sing } is mapped to a permutation σ of K total number of elements of the form:
π
⇒
σ
:
(
1
2
…
K
σ
(
1
)
σ
(
2
)
…
σ
(
K
)
)
,
where σ(k) is a user paired with user k by the permutation σ.
4. The method of claim 2 , wherein for a case of an even number of users, maximizing the matrix W with zero entries on the diagonal and symmetric entries for w k1,k2 =w k2,k1 =f (k1,k2) , and obtaining a result by taking only the pairs (k,σ(k)), for k=1, . . . , K/2, where K is a total number of pairs of users for a given permutation and σ(k) is a user paired with user k by the permutation σ.
5. The method of claim 3 , where the additive objective function that is maximized is expressed as:
max
σ
∈
S
K
∑
(
k
,
σ
(
k
)
)
(
K
)
f
(
k
,
σ
(
k
)
)
where S K denotes a group of all permutations and k is an individual user from the K total number of elements.
6. The method of claim 5 , where maximizing comprises, given an n×n matrix W=[w k, ]=[w(x k y )], where W k, is a weight of the k th , th element of the matrix W and w(x k , y ) is a weight for a pairing of user x k and user y in the matrix W, finding a permutation σεS n , where S n denotes the group of all permutations of n number of elements, for which
∑
k
=
1
n
w
(
x
k
y
σ
(
k
)
)
is a maximum, and by setting w(x k y σ(k) )=f (k,σ(k)) , and where a result is found by taking only the pairs (k,σ(k)), for k=1, . . . , K/2, where W is a total bandwidth occupied and xy is an edge.
7. An apparatus, comprising:
at least one processor;
at least one memory including computer program code;
the at least one memory and the computer program code configured to, with the at least one processor, cause the apparatus at least to:
operate a receiver connected with a plurality of receive antennas to receive uplink signals from a plurality of users each having a plurality of transmit antennas; and
schedule which ones of the plurality of users are to transmit in a same time-frequency-code slot, where a particular user pairing configuration is denoted by π within a set of all configurations Π, and where scheduling comprises expressing mutual information as an additive objective function and maximizing the additive objective function over a particular choice of πεΠ,
wherein the additive objective function that is maximized comprises:
max
π
∈
Π
∑
(
k
1
,
k
2
)
∈
π
(
N
)
f
k
1
,
k
2
(
π
)
,
where f is measure of performance for a pairing of users k 1 and k 2 and N is an integer total number of distinct pairs of users.
8. The apparatus of claim 7 , wherein maximizing comprises using an n×n matrix W=[w k, ]=[w(x k y )], where w k, is the weight of the k th , th element of the matrix W and w (x k , y ) is the weight for a pairing of user x k and user y in the matrix W, and finding a permutation σεS n , where S n denotes the group of all permutations of n number of elements, for which
∑
k
=
1
n
w
(
x
k
y
σ
(
k
)
)
is a maximum, where W is a total bandwidth occupied and xy is an edge.
9. The apparatus of claim 7 , wherein there are paired users and unpaired users, wherein a pairing configuration π={π pair ,π sing } is mapped to a permutation σ of K total number of elements of the form:
π
⇒
σ
:
(
1
2
…
K
σ
(
1
)
σ
(
2
)
…
σ
(
K
)
)
,
where σ(k) is a user paired with user k by the permutation σ.
10. The apparatus of claim 7 , wherein the apparatus is embodied as at least one integrated circuit.
11. The apparatus of claim 8 , wherein for a case of an even number of users, maximizing the matrix W with zero entries on the diagonal and symmetric entries for w k 1 ,k 2 =w k 2 ,k 1 =f (k 1 ,k 1) , and obtaining a result by taking only the pairs (k,σ(k)), for k=1, . . . , K/2, where K is a total number of pairs of users for a given permutation and σ(k) is a user paired with user k by the permutation σ.
12. The apparatus of claim 9 , where the additive objective function that is maximized is expressed as:
max
σ
∈
S
K
∑
(
k
,
σ
(
k
)
)
(
K
)
f
(
k
,
σ
(
k
)
)
where S K denotes a group of all permutations and k is an individual user from the total number of users K, and where maximizing comprises, given an n×n matrix W=[w k, ]=[w(x k y )], where w k, is the weight of the k th , th element of the matrix W and w(x k , y ) is the weight for a pairing of user x k and user y in the matrix W, finding a permutation σεS n , where S n denotes the group of all permutations of n number of elements, for which
∑
k
=
1
n
w
(
x
k
y
σ
(
k
)
)
is a maximum, and by setting w(x k y σ(k) )=f k,σ(k)) , and where a result is found by taking only the pairs (k,σ(k)), for k=1, . . . , K/2, where W is a total bandwidth occupied and xy is an edge.
13. A non-transitory computer-readable storage medium tangibly embodying computer program code configured to, with at least one processor, perform operations, the operations comprising:
operating a receiver connected with a plurality of receive antennas to receive uplink signals from a plurality of users each having a plurality of transmit antennas; and
scheduling which ones of the plurality of users are to transmit in a same time-frequency-code slot, where a particular user pairing configuration is denoted by π within a set of all configurations Π, and where scheduling comprises expressing mutual information as an additive objective function and maximizing the additive objective function over a particular choice of πεΠ,
wherein the additive objective function that is maximized comprises:
max
π
∈
Π
∑
(
k
1
,
k
2
)
∈
π
(
N
)
f
k
1
,
k
2
(
π
)
,
where f is measure of performance for a pairing of users k 1 and k 2 and N is an integer total number of distinct pairs of users.
14. The non-transitory computer-readable storage medium of claim 13 , wherein maximizing comprises using an n×n matrix W=[w k, ]=[w(x k y )], where w k, is the weight of the k th , th element of the matrix W and w(x k , y ) is the weight for a pairing of user x k and user y in the matrix W, and finding a permutation σεS n , where S n denotes the group of all permutations of n number of elements, for which
∑
k
=
1
n
w
(
x
k
y
σ
(
k
)
)
is a maximum, where W is a total bandwidth occupied and xy is an edge.
15. The non-transitory computer-readable storage medium of claim 13 , wherein there are paired users and unpaired users, wherein a pairing configuration π={π pair ,π sing } is mapped to a permutation σ of K total number of elements of the form:
π
⇒
σ
:
(
1
2
…
K
σ
(
1
)
σ
(
2
)
…
σ
(
K
)
)
,
where σ(k) is a user paired with user k by the permutation σ.
16. The non-transitory computer-readable storage medium of claim 14 , wherein for a case of an even number of users, maximizing the matrix W with zero entries on the diagonal and symmetric entries for w k 1 ,k 2 =w k 2 ,k 1 =f (k 1 ,k 1) , and obtaining a result by taking only the pairs (k,σ(k)), for k=1, . . . , K/2, where K is a total number of pairs of users for a given permutation and σ(k) is a user paired with user k by the permutation σ.
17. The non-transitory computer-readable storage medium of claim 15 , where the additive objective function that is maximized is expressed as:
max
σ
∈
S
K
∑
(
k
,
σ
(
k
)
)
(
K
)
f
(
k
,
σ
(
k
)
)
where S K denotes a group of all permutations and k is an individual user from the total number of users K, and where maximizing comprises, given an n×n matrix W=[w k, ]=[w(x k y )], where w k, is the weight of the k th , th element of the matrix W and w(x k , y ) is the weight for a pairing of user x k and user y in the matrix W, finding a permutation σεS n , where S n denotes the group of all permutations of n number of elements, for which
∑
k
=
1
n
w
(
x
k
y
σ
(
k
)
)
is a maximum, and by setting w(x k y σ(k) )=f (k,σ(k)) , and where a result is found by taking only the pairs (k,σ(k)), for k=1, . . . , K/2, where W is a total bandwidth occupied and xy is an edge.