Multiple antenna transmission with per-antenna power constraints

1. A method for transmitting signals in a wireless communication system, the method comprising:

identifying a number of streams to be transmitted;

identifying a precoding algorithm based on whether a single stream or multiple streams are to be transmitted;

initializing the precoding algorithm to a complex matrix, the precoding algorithm for precoding signals transmitted by a plurality of antennas;

iteratively processing the precoding algorithm on a per-antenna basis by, at each iteration, sequentially updating a precoder for each of the plurality of antennas;

after each iteration, determining whether the precoding algorithm has converged based on a change in a rate of mutual information across iterations; and

in response to determining that the precoding algorithm converged, transmitting the signals using the precoding algorithm,

wherein sequentially updating the precoder comprises:

(i) in response to identifying that a single stream is to be transmitted, sequentially updating the precoder (v i ) of an i-th antenna according to:

v

i

=

ψ

(

∑

k

≠

i

⁢

f

ik

⁢

v

k

)

·

p

i

where ψ(w) represents a phase of a complex variable w, where ƒ ki is an element of F at a k-th row and an i-th column, F is a matrix obtained by multiplying the complex matrix by a conjugate transpose of the complex matrix on the left, v k is a precoder at the k-th antenna, and p i is a power constraint on the i-th antenna; or

(ii) in response to identifying that multiple streams are to be transmitted, sequentially updating the precoder (v ik ) of an i-th antenna and a k-th stream according to:

v

ik

=

ψ

(

∑

j

≠

i

⁢

g

ij

k

⁢

v

jk

)

·

p

ik

where ψ(w) represents a phase of a complex variable w, g ij k is the (i, j)-th entry of G k with G k being a matrix derived from the channel matrix and a current value of the precoder, v jk is a current value of the precoder at a j-th antenna and a k-th stream , and p ik is a power constraint on the k-th stream and the i-th antenna.

2. The method of claim 1 further comprising:

in response to determining that the precoding algorithm has not converged, continuing to iteratively process the precoding algorithm on the per-antenna basis at a next iteration.

3. The method of claim 1 , wherein sequentially updating the precoder of each of the plurality of antennas comprises:

sequentially updating the precoder (u i ) of an i-th antenna according to:

u i =Z i (ρ i opt Γ i 2 −I ) −1 Z i † w i

where Z i is a unitary matrix with columns being singular vectors of D i , where ρ i opt is a scalar, where Γ i 2 is a diagonal matrix with diagonal entries being singular values of D i , where I is an identity matrix, where Z i † is a conjugate transpose of Z i , where w i is a complex vector for the i-th antenna, and where D i is a positive semi-definite matrix derived based on the channel matrix and a current value of the precoder.

4. The method of claim 3 , wherein initializing the precoding algorithm comprises initializing the precoding algorithm to the complex matrix such that Σ k=1 N s ∥u ki 2 =p i , where N s is a number of streams to be transmitted, u ki is a precoder for a k-th stream and the i-th antenna, and p i is a power constraint on the i-th antenna.

5. The method of claim 1 , wherein, for an identification that a single stream is to be transmitted, initializing the precoding algorithm comprises initializing the precoding algorithm to the complex matrix such that ∥v i ∥ 2 =p i .

6. The method of claim 1 , wherein, for an identification that multiple streams are to be transmitted, initializing the precoding algorithm comprises initializing the precoding algorithm to the complex matrix such that ∥v ik ∥ 2 =p ik .

7. An apparatus configured to transmit signals in a wireless communication system, the apparatus comprising:

a controller configured to identify a number of streams to be transmitted; identify a precoding algorithm based on whether a single stream or multiple streams are to be transmitted; initialize the precoding algorithm to a complex matrix, the precoding algorithm for precoding signals transmitted by a plurality of antennas; iteratively process the precoding algorithm on a per-antenna basis by, at each iteration, sequentially updating a precoder for each of the plurality of antennas; and after each iteration, determine whether the precoding algorithm has converged based on a change in a rate of mutual information across iterations;

a precoding unit configured to, in response to a determination that the precoding algorithm converged, precode the signals using the precoding algorithm; and

the plurality of antennas configured to transmit the precoded signals,

wherein the controller is further configured to:

(i) in response to identification that a single stream is to be transmitted, sequentially update the precoder (v i ) of an i-th antenna according to:

v

i

=

ψ

(

∑

k

≠

i

⁢

f

ik

⁢

v

k

)

·

p

i

where ψ(w) represents a phase of a complex variable w, where ƒ ki is an element of F at a k-th row and an i-th column, F is a matrix obtained by multiplying the complex matrix by a conjugate transpose of the complex matrix on the left, v k is a precoder at the k-th antenna, and p i is a power constraint on the i-th antenna; or

(ii) in response to identification that multiple streams are to be transmitted, sequentially update the precoder (v ik ) of an i-th antenna and a k-th stream according to:

v

ik

=

ψ

(

∑

j

≠

i

⁢

g

ij

k

⁢

v

jk

)

·

p

ik

where ψ(w) represents a phase of a complex variable w, g ij k is the (i,j)-th entry of G k with G k being a matrix derived from the channel matrix and a current value of the precoder, v jk is a current value of the precoder at a j-th antenna and a k-th stream, and p ik is a power constraint on the k-th stream and the i-th antenna.

8. The apparatus of claim 7 , wherein the controller is configured to, in response to determining that the precoding algorithm has not converged, continue to iteratively process the precoding algorithm on the per-antenna basis at a next iteration.

9. The apparatus of claim 7 , wherein to sequentially update the precoder of each of the plurality of antennas, the controller is further configured to sequentially update the precoder (u i ) of an i-th antenna according to:

u i =Z i ( p i opt Γ i 2 −I ) −1 Z i † w i

where Z i is a unitary matrix with columns being singular vectors of D i , where ρ i opt is a scalar, where Γ i 2 is a diagonal matrix with diagonal entries being singular values of D i , where I is an identity matrix, where Z i † is a conjugate transpose of Z i , where w i is a complex vector for the i-th antenna, and where D i is a positive semi-definite matrix derived based on the channel matrix and a current value of the precoder.

10. The apparatus of claim 9 , wherein to initialize the precoding algorithm, the controller is further configured to initialize the precoding algorithm to the complex matrix such that Σ k=1 N s ∥u ki 2 =p i , where N s is a number of streams to be transmitted, u ki , is a precoder for a k-th stream and the i-th antenna, and p i is a power constraint on the i-th antenna.

11. The apparatus of claim 7 , wherein, for identification that a single stream is to be transmitted, to initialize the precoding algorithm, the controller is further configured to initialize the precoding algorithm to the complex matrix such that ∥v i ∥ 2 =p i .

12. The apparatus of claim 7 , wherein, for an identification that multiple streams are to be transmitted, to initialize the precoding algorithm, the controller is further configured to initialize the precoding algorithm to the complex matrix such that ∥v ik ∥ 2 =p ik .

13. A system comprising the apparatus of claim 7 and a receiver configured to receive the transmitted signals.

14. An apparatus configured to transmit signals in a wireless communication system, the apparatus comprising:

a controller configured to:

identify a number of streams to be transmitted;

identify a precoding algorithm based on whether a single stream or multiple streams are to be transmitted;

initialize the precoding algorithm to a complex matrix, the precoding algorithm for precoding signals transmitted by a plurality of antennas;

iteratively process the precoding algorithm on a per-antenna basis by, at each iteration, sequentially updating a precoder for each of the plurality of antennas, wherein to sequentially update the precoder of each of the plurality of antennas, the controller is further configured to sequentially update the precoder (u i ) of an i-th antenna according to:

u i =Z i (ρ i opt Γ i 2 −I ) −1 Z i † w i

where Z i is a unitary matrix with columns being singular vectors of D i , where ρ i opt is a scalar, where Γ i 2 is a diagonal matrix with diagonal entries being singular values of D i , where I is an identity matrix, where Z i † is a conjugate transpose of Z i , where w i is a complex vector for the i-th antenna, and where D i is a positive semi-definite matrix derived based on the channel matrix and a current value of the precoder;

after each iteration, determine whether the precoding algorithm has converged based on a change in a rate of mutual information across iterations; and

a precoding unit configured to, in response to a determination that the precoding algorithm converged, precode the signals using the precoding algorithm; and

the plurality of antennas configured to transmit the precoded signals.

15. The apparatus of claim 14 , wherein the controller is configured to, in response to determining that the precoding algorithm has not converged, continue to iteratively process the precoding algorithm on the per-antenna basis at a next iteration.

16. The apparatus of claim 14 , wherein to initialize the precoding algorithm, the controller is further configured to initialize the precoding algorithm to the complex matrix such that Σ k=1 N s ∥u ki ∥ 2 =p i , where N s is a number of streams to be transmitted, U ki is a precoder for a k-th stream and the i-th antenna, and p i is a power constraint on the i-th antenna.

17. A system comprising the apparatus of claim 14 and a receiver configured to receive the transmitted signals.

18. A method for transmitting signals in a wireless communication system, the method comprising:

identifying a number of streams to be transmitted;

identifying a precoding algorithm based on whether a single stream or multiple streams are to be transmitted;

initializing the precoding algorithm to a complex matrix, the precoding algorithm for precoding signals transmitted by a plurality of antennas;

iteratively processing the precoding algorithm on a per-antenna basis by, at each iteration, sequentially updating a precoder for each of the plurality of antennas, sequentially updating the precoder of each of the plurality of antennas comprises sequentially updating the precoder (u i ) of an i-th antenna according to:

u i =Z i (ρ i opt Γ i 2 −I ) −1 Z i † w i

where Z i is a unitary matrix with columns being singular vectors of D i , where ρ i opt is a scalar, where Γ i 2 a diagonal matrix with diagonal entries being singular values of D i , where I is an identity matrix, where Z † i is a conjugate transpose of Z i , where w i is a complex vector for the i-th antenna, and where D i is a positive semi-definite matrix derived based on the channel matrix and a current value of the precoder;

after each iteration, determining whether the precoding algorithm has converged based on a change in a rate of mutual information across iterations; and

in response to determining that the precoding algorithm converged, transmitting the signals using the precoding algorithm.

19. The method of claim 18 further comprising:

in response to determining that the precoding algorithm has not converged, continuing to iteratively process the precoding algorithm on the per-antenna basis at a next iteration.

20. The method of claim 18 , wherein initializing the precoding algorithm comprises initializing the precoding algorithm to the complex matrix such that Σ k=1 N s ∥u ki ∥ 2 =p i , where N S is a number of streams to be transmitted, u ki is a precoder for a k-th stream and the i-th antenna, and p i is a power constraint on the i-th antenna.