System and method for transceiver design

1. A method for operating a controller of a multiple input, multiple output communications system, the method comprising:

formulating an objective function according to a resource allocation for a user equipment (UE) and a mean square error expression;

updating the objective function to generate an updated resource allocation for the UE, a transmit beamforming vector to precode a transmission to the UE, and a receive beamforming vector to adjust a receiver to receive the precoded transmission; and

transmitting allocation information about the updated resource allocation for the UE and the transmit beamforming vector to a communications controller serving the UE.

2. The method of claim 1 , wherein the updating the objective function comprises evaluating

min

U

,

V

,

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∑

k

=

1

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∑

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.

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k

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)

≤

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k

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∀

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∈

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∀

g

∈

𝒢

∑

g

=

1

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β

g

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1

β

g

≥

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∀

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,

where c i k (β, E i k ) is the objective function, E i k is the mean square error expression of a data vector intended for the UE as received by the UE, V i k g is a matrix of transmit beamforming vectors for UE i k of group g, β g is a resource allocation for UE group g, G is a number of UE groups, P k is a transmit power of communications controller k, (•) H is a complex conjugate transpose operator, and Tr(•) is a trace operator.

3. The method of claim 2 , wherein the evaluating comprises utilizing a block coordinate descent algorithm.

4. The method of claim 2 , wherein the evaluating comprises

determining the receive beamforming vector according to a previous value of the transmit beamforming vector;

updating a weighting matrix; and

determining the transmit beamforming vector according to the receive beamforming vector and the updated weighting matrix.

5. The method of claim 4 , wherein the determining the receive beamforming vector comprises evaluating

U i k g ←(Σ (j,l) H i k j V l j g V l j g H H i k j H +σ i k 2 I) −1 H i k k V i k g , ∀ i k ε , ∀ g ε ,

where V l j g is the previous value of the transmit beamforming vector, H xy is a channel response from transmitter y to receiver x, I is an identity matrix, σ ik 2 is a receiver noise power for user i at cell k, G is the number of UE groups, and (•) H is the complex conjugate transpose operator.

6. The method of claim 4 , wherein the updating the weighting matrix comprises evaluating

W i k g ←∇E i k g c i k (•), ∀ i k ε , ∀ g ε ,

where c i k (•) is the objective function, E i k g is a mean square error form of a received signal at UE i k in UE group g, G is the number of UE groups, and Δ(•) is a gradient map operator.

7. The method of claim 4 , wherein the determining the transmit beamforming vector comprises evaluating

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I

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←

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∑

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∀

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,

where U l j g is the receive beamforming vector, H xy is a channel response from transmitter y to receiver x, I is an identity matrix, μ k * is an optimal Lagrangian multiplier corresponding to a power budget constraint at cell k, G is the number of UE groups, and (•) H is the complex conjugate transpose operator.

8. The method of claim 4 , wherein the evaluating further comprises updating the resource allocation for the UE.

9. The method of claim 8 , wherein the updating the resource allocation for the UE comprises evaluating

min

β

⁢

∑

k

=

1

K

⁢

∑

i

=

1

I

k

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c

i

k

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k

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.

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=

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≥

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∀

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∈

𝒢

.

where c i k (β, E i k ) is the objective function, E i k is the mean square error expression of the data vector intended for the UE as received by the UE, β g is a resource allocation for UE group g, and G is the number of UE groups.

10. The method of claim 4 , further comprising repeating the determining the receive beamforming vector, the updating the weighting matrix, and the determining the transmit beamforming vector until a convergence criterion is met.

11. The method of claim 1 , further comprising transmitting the receive beamforming vector to the UE.

12. The method of claim 1 , wherein the updating the objective function comprises finding a globally optimal solution of the objective function.

13. The method of claim 1 , wherein the updating the objective function comprises finding a locally optimal solution of the objective function.

14. A controller comprising:

a processor configured to formulate an objective function according to a resource allocation for a user equipment (UE) and a mean square error expression, and to update the objective function to generate an updated resource allocation for the UE, a transmit beamforming vector to precode a transmission to the UE, and a receive beamforming vector to adjust a receiver to receive the precoded transmission; and

a transmitter operatively coupled to the processor, the transmitter configured to transmit allocation information about the updated resource allocation for the UE and the transmit beamforming vector to a communications controller serving the UE.

15. The controller of claim 14 , wherein the transmitter is configured to transmit the receive beamforming vector to the UE.

16. The controller of claim 14 , wherein the processor is configured to evaluate

min

U

,

V

,

β

⁢

∑

k

=

1

K

⁢

∑

i

=

1

I

k

⁢

c

i

k

⁡

(

β

,

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i

k

)

s

.

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.

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i

=

1

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k

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Tr

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i

k

g

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V

i

k

gH

)

≤

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k

,

∀

k

∈

K

,

∀

g

∈

𝒢

∑

g

=

1

G

⁢

β

g

=

1

β

g

≥

0

,

∀

g

∈

𝒢

,

where c i k (β,E i k ) is the objective function, E i k is the mean square error expression of a data vector intended for the UE as received by the UE, V i k g is a matrix of transmit beamforming vectors for UE i k of group g, β g is a resource allocation for UE group g, G is a number of UE groups, P k is a transmit power of communications controller k, (•) H is a complex conjugate transpose operator, and Tr(•) is a trace operator.

17. The controller of claim 16 , wherein the processor is configured to determine the receive beamforming vector according to a previous value of the transmit beamforming vector, to update a weighting matrix, and to determine the transmit beamforming vector according to the receive beamforming vector and the updated weighting matrix.

18. The controller of claim 17 , wherein the processor is configured to update the resource allocation for the UE.

19. The controller of claim 18 , wherein the processor configured to update the resource allocation for the UE comprises the processor configured to evaluate

min

β

⁢

∑

k

=

1

K

⁢

∑

i

=

1

I

k

⁢

c

i

k

⁡

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β

,

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i

k

)

s

.

t

.

⁢

∑

g

=

1

G

⁢

β

g

=

1

β

g

≥

0

,

∀

g

∈

G

,

where c i k (β,E i k ) is the objective function, E i k , is the mean square error expression of the data vector intended for the UE as received by the UE, β g is a resource allocation for UE group g, and G is the number of UE groups.

20. The controller of claim 17 , wherein the processor configured to determine the receive beamforming vector comprises the processor configured to evaluate

U i k g ←(Σ (j,l) H i k j V l j g V l j g H H i k j H+σ i k 2 I) −1 H i k k , V i k g , ∀ i k ε , ∀ g ε ,

where V l j g is the previous value of the transmit beamforming vector, H xy is a channel response from transmitter y to receiver x, I is an identity matrix, σ ik 2 is a receiver noise power for user i at cell k, G is the number of UE groups, and (•) H is the complex conjugate transpose operator.

21. The controller of claim 17 , wherein the processor configured to update the weighting matrix comprises the processor configured to evaluate

W i k g ←ΔE i k g c i k (•), ∀ i k ε , ∀ g ε ,

where c i k (•) is the objective function, E i k g is a mean square error form of a received signal at UE i k in UE group g, G is the number of UE groups, and Δ(•) is a gradient map operator.

22. The controller of claim 17 , wherein the processor configured to determine the transmit beamforming vector comprises the processor configured to evaluate

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i

k

g

I

,

∀

g

∈

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←

(

∑

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j

,

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)

⁢

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W

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+

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-

1

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i

k

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k

H

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i

k

g

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W

i

k

g

,

∀

i

k

∈

,

where U l j g is the receive beamforming vector, H xy is a channel response from transmitter y to receiver x, I is an identity matrix, μ k * is an optimal Lagrangian multiplier corresponding to a power budget constraint at cell k, G is the number of UE groups, and (•) H is the complex conjugate transpose operator.

23. The controller of claim 17 , wherein the processor is further configured to repeat the determine the receive beamforming vector, the update the weighting matrix, and the determine the transmit beamforming vector until a convergence criterion is met.

24. The controller of claim 16 , wherein the processor configured to evaluate comprises the processor configured to utilize a block coordinate descent algorithm.

25. The controller of claim 14 , wherein the processor configured to update the objective function comprises the processor configured to find a globally optimal solution of the objective function.

26. The controller of claim 14 , wherein the processor configured to update the objective function comprises the processor configured to find a locally optimal solution of the objective function.