Beamforming systems and methods for link layer multicasting
View Patent ↗Beamforming systems and methods are disclosed. In accordance with one method, an optimization problem that is based on channel matrices for a plurality of receivers is solved to obtain corresponding multipliers for each of the channel matrices. In addition, a determination of whether a maximum eigenvalue of a particular matrix that is based on the channel matrices and the multipliers has a multiplicity of one is made. Further, the method includes applying an eigenvector of the particular matrix for the maximum eigenvalue as a beamforming matrix to data symbols if the maximum eigenvalue has a multiplicity of one. Moreover, data signals are transmitted to the receivers in accordance with the beamforming matrix.
1. A beamforming method comprising:
solving an optimization problem that is based on channel matrices for a plurality of receivers to obtain corresponding multipliers for each of the channel matrices;
determining whether a maximum eigenvalue of a particular matrix that is based on the channel matrices and the multipliers has a multiplicity of one;
applying an eigenvector of the particular matrix for the maximum eigenvalue as a beamforming matrix to data symbols if the maximum eigenvalue has a multiplicity of one; and
transmitting data signals to the plurality of receivers in accordance with the beamforming matrix, wherein the eigenvector maximizes a lowest reception rate among reception rates of each receiver of the plurality of receivers if the maximum eigenvalue has a multiplicity of one.
2. The method of claim 1 , wherein the particular matrix is
∑
k
=
1
K
λ
k
H
k
*
H
k
,
wherein k denotes an index of one of the plurality of receivers, K denotes a total number of the plurality of receivers, H k is a channel matrix of a receiver denoted by k, H* k is the Hermitian matrix of H k and λ k is a multiplier for the receiver denoted by k.
3. The method of claim 2 , wherein the optimization problem is finding a set of K multipliers, λ 1 , λ 2 , . . . , λ K-1 , λ K , that minimizes an objective function (α) such that
∑
k
=
1
K
λ
k
≥
1
and
α
I
-
∑
k
=
1
K
λ
k
H
k
*
H
k
is a non-negative, definite matrix, wherein I is an identity matrix.
4. The method of claim 3 , wherein the solving comprises solving the optimization problem in accordance with Semi-Definite Programming.
5. The method of claim 1 , wherein the method further comprises:
applying a suboptimal approximation method to determine the beamforming matrix if the maximum eigenvalue has a multiplicity that is greater than one.
6. The method of claim 1 , further comprising:
obtaining indications of the channel matrices from the plurality of receivers.
7. A non-transitory computer readable storage medium comprising a computer readable program, wherein the computer readable program when executed on a computer causes the computer to:
solve an optimization problem that is based on channel matrices for a plurality of receivers to obtain corresponding multipliers for each of the channel matrices;
determine whether a maximum eigenvalue of a particular matrix that is based on the channel matrices and the multipliers has a multiplicity of one; and
apply an eigenvector of the particular matrix for the maximum eigenvalue as a beamforming matrix to data symbols if the maximum eigenvalue has a multiplicity of one to generate a transmit signal for the plurality of receivers, wherein the eigenvector maximizes a lowest reception rate among reception rates of each receiver of the plurality of receivers if the maximum eigenvalue has a multiplicity of one.
8. The computer readable storage medium of claim 7 , wherein the particular matrix is
∑
k
=
1
K
λ
k
H
k
*
H
k
,
wherein k denotes an index of one of the plurality of receivers, K denotes a total number of the plurality of receivers, H k is a channel matrix of a receiver denoted by k, H* k is the Hermitian matrix of H k and λ k is a multiplier for the receiver denoted by k.
9. The computer readable storage medium of claim 8 , wherein the optimization problem is finding a set of K multipliers, λ 1 , λ 2 , . . . , λ K-1 , λ K , that minimizes an objective function (α) such that
∑
k
=
1
K
λ
k
≥
1
and
α
I
-
(
∑
k
=
1
K
λ
k
H
k
*
H
k
)
is a non-negative, definite matrix, wherein I is an identity matrix.
10. The computer readable storage medium of claim 9 , wherein the computer readable program when executed on a computer further causes the computer to solve the optimization problem in accordance with Semi-Definite Programming.
11. The computer readable storage medium of claim 7 , wherein the computer readable program when executed on a computer further causes the computer to apply a suboptimal approximation method to determine the beamforming matrix if the maximum eigenvalue has a multiplicity that is greater than one.
12. A beamforming transmitter system comprising:
a storage medium configured to store channel matrices for a plurality of receivers;
a processor configured to solve an optimization problem that is based on the channel matrices to obtain corresponding multipliers for each of the channel matrices, to determine whether a maximum eigenvalue of a particular matrix that is based on the channel matrices and the multipliers has a multiplicity of one, and to apply an eigenvector of the particular matrix for the maximum eigenvalue as a beamforming matrix to data symbols if the maximum eigenvalue has a multiplicity of one; and
a transmitter configured to transmit data signals to the plurality of receivers in accordance with the beamforming matrix, wherein the eigenvector maximizes a lowest reception rate among reception rates of each receiver of the plurality of receivers if the maximum eigenvalue has a multiplicity of one.
13. The system of claim 12 , wherein the particular matrix is
∑
k
=
1
K
λ
k
H
k
*
H
k
,
wherein k denotes an index of one of the plurality of receivers, K denotes a total number of the plurality of receivers, H k is a channel matrix of a receiver denoted by k, H* k is the Hermitian matrix of H k and λ k is a multiplier for the receiver denoted by k.
14. The system of claim 13 , wherein the optimization problem is finding a set of K multipliers, λ 1 , λ 2 , . . . , λ K-1 , λ K , that minimizes an objective function (α) such that
∑
k
=
1
K
λ
k
≥
1
and
α
I
-
(
∑
k
=
1
K
λ
k
H
k
*
H
k
)
is a non-negative, definite matrix, wherein I is an identity matrix.
15. The system of claim 14 , wherein the processor is further configured to solve the optimization problem in accordance with Semi-Definite Programming.
16. The system of claim 12 , wherein the processor is further configured to apply a suboptimal approximation method to determine the beamforming matrix if the maximum eigenvalue has a multiplicity that is greater than one.
17. The system of claim 12 , further comprising:
a receiver configured to receive indications of the channel matrices from the plurality of receivers.