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Title:
METHOD AND APPARATUS FOR BEAMFORMING IN MIMO SYSTEMS
Document Type and Number:
WIPO Patent Application WO/2010/040190
Kind Code:
A1
Abstract:
Methods, apparatus and systems for estimating transmit and receive beamforming vectors in MIMO- OFDM and related systems are described. These beamforming vectors are for use over the set of OFDM subcarriers and an OFDM symbol period and are referred to as symbolwise beamforming vectors. Symbolwise beamforming has the advantage of low complexity as only 1 IDFT/DFT is required in each transmitter and receiver rather than 1 per antenna. In one aspect the proposed method and systems perform joint estimatation of a pair of transmit and receive beamforming vectors by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpre). Simulations using OFDM MIMO systems operating at 60GHz show the proposed algorithm is both computationally feasible and convergent. Further the approach performs well in the presence of multiple access or co channel interference.

Inventors:
POLLOCK, Andre (Building GP, Level 1 Room 1.15,University of South Autralia,Mawson Lakes Campu, Mawson Lakes South Australia 5095, AU)
COWLEY, William,G. (Building GP, Level 1 Room 1.15,University of South Autralia,Mawson Lakes Campu, Mawson Lakes South Australia 5095, AU)
LETZEPIS, Nicholas, Alexander (Building GP, Level 1 Room 1.15,University of South Autralia,Mawson Lakes Campu, Mawson Lakes South Australia 5095, AU)
Application Number:
AU2009/001345
Publication Date:
April 15, 2010
Filing Date:
October 09, 2009
Export Citation:
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Assignee:
UNIVERSITY OF SOUTH AUSTRALIA (North Terrace, Adelaide, South Australia 5000, AU)
POLLOCK, Andre (Building GP, Level 1 Room 1.15,University of South Autralia,Mawson Lakes Campu, Mawson Lakes South Australia 5095, AU)
COWLEY, William,G. (Building GP, Level 1 Room 1.15,University of South Autralia,Mawson Lakes Campu, Mawson Lakes South Australia 5095, AU)
LETZEPIS, Nicholas, Alexander (Building GP, Level 1 Room 1.15,University of South Autralia,Mawson Lakes Campu, Mawson Lakes South Australia 5095, AU)
International Classes:
H04B7/02; H04J4/00
Domestic Patent References:
WO2008115650A1
Foreign References:
US20080165875A1
US20080158057A1
Other References:
PALOMAR ET AL.: 'Joint Transmit - Receive Space - Time Equalization in Spatially Correlated MIMO Channels: A Beamforming Approach' IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS vol. 21, no. 5, June 2003, pages 730 - 743
Attorney, Agent or Firm:
MADDERNS (First Floor, 64 Hindmarsh SquareAdelaide, South Australia 5000, AU)
Download PDF:
Claims:
THE CLAIMS

1. A method for joint estimation of a pair of transmit and receive beamforming vectors for use in a subcarrier-based multiple input multiple output (MEVIO) communication system comprising: jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MDVIO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MEVIO channel at the input of the receive side demodulator (SINRpre).

2. A method as claimed in claim 1, wherein maximisation of the expression for SINRpre is expressed as a generalised eigenproblem in which the receive beamforming vector is an eigenvector and a standard eigenproblem in which the transmit beamforming vector is an eigenvector, and the step of jointly estimating comprises: initialising an estimate of at least one of the pair of transmit and receive beamforming vectors; updating an estimate of the receive beamforming vector by solving the generalised eigenproblem using the current estimate of the transmit beamforming vector; updating an estimate of the transmit beamforming vector by solving the standard eigenproblem using the current estimate of the receive beamforming vector; and repeating the two updating steps until a predefined stopping criterion is met, wherein the updating steps may be performed in any order and the initialisation step is performed to obtain an estimate for use in the first updating step performed.

3. A method as claimed in claim 2, wherein the receive and transmit beamforming vectors are each estimated as the dominant eigenvectors of their respective generalised and standard eigenproblems and a power method is used to obtain the dominant eigenvectors in each update step.

4. A method as claimed in claim 3, wherein the step of initialising further comprises initialising an estimate of both of the transmit beamforming vector and the receive beamforming vector, and the step of updating an estimating of the receive beamforming vector further comprises iteratively estimating the dominant receive beamforming eigenvector for the current estimate of the transmit beamforming vector for NPM u iterations, and the step of updating an estimate of the transmit vector further comprises iteratively estimating the dominant transmit beamforming eigenvector for the current estimate of the receive beamforming vector for NPM v iterations.

5. A method as claimed in claim 4 wherein NPM u and NPM,v are identical.

6. A method as claimed in any one of claims 2 to 5 in which the stopping criterion is that the difference in SINRpn. between successive iterations is less than a predefined value.

7. A method as claimed in any one of claims 1 to 6, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are the set of OFDM subcarriers and the block of time-domain symbols corresponds to one OFDM symbol.

8. A method as claimed in any one of claims 1 to 6 wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division Multiple Access (OFDMA) MIMO system in which the set of OFDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

9. A method as claimed in any one of claims 1 to 6 wherein the subcarrier-based MIMO communication system is Single Carrier Frequency Division Multiple Access (SC-FDMA) MEvIO system in which the set of SC-FDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

10. A method as claimed in any one of claims 2 to 6, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are OFDM subcarriers and the jointly estimated pair of transmit and receive beamforming vectors are for use over a block of time-domain symbols (one OFDM symbol), and the standard eigenproblem has the form A(u)v = ζ'v where ζ' = (uHRwu) ζ is the eigenvalue and v is the eigenvector, and the generalised eigenproblem has the form B(v)u = γRwu where γ is the eigenvalue and u is the eigenvector and where u is the pre OFDM demodulation receive beamforming vector; v is the post OFDM modulation transmit beamforming vector;

L-I I ,,H Tj „ 12

Hi is the / Λ time lagged estimate of the MMO channel where I = 0..L- 1 ; Rw is an estimate of the noise plus interference co- variance matrix in the time domain; Ps is the available transmit power; ζ is the Lagrangian multiplier associated with the constraint vHv ≤ 1 and the Lagrangian C = SINRpre - ζ(vHv -l) used to maximise SINRpre;

A(u) = P5YHfUIfH1,

1=0

B(v) = Ps∑H,vv«H?; ι=o and the estimates of « and v are the dominant eigenvectors of the generalised and standard eigenproblems respectively.

11. A computer readable medium comprising instructions stored therein comprising: a first instruction set for jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpn.); a second instruction set for providing the transmit beamforming vector to a transmitter comprising a plurality of transmit antennas and the receive beamforming vector to a receiver comprising a plurality of receive for use over the block of time-domain symbols.

12. A computer readable medium as claimed in claim 11, wherein maximisation of the expression for SINRpre is expressed as a generalised eigenproblem in which the receive beamforming vector is an eigenvector and a standard eigenproblem in which the transmit beamforming vector is an eigenvector, and the instructions for jointly estimating comprises: initialising an estimate of at least one of the pair of transmit and receive beamforming vectors; updating an estimate of the receive beamforming vector by solving the generalised eigenproblem using the current estimate of the transmit beamforming vector; updating an estimate of the transmit beamforming vector by solving the standard eigenproblem using the current estimate of the receive beamforming vector; and repeating the two updating instructions until a predefined stopping criterion is met, wherein the updating may be performed in any order and the initialisation is performed to obtain an estimate for use in the first updating step performed.

13. A computer readable medium as claimed in claim 12, wherein the receive and transmit beamforming vectors are each estimated as the dominant eigenvectors of their respective generalised and standard eigenproblems and a power method is used to obtain the dominant eigenvectors in each update instruction.

14. A computer readable medium as claimed in claim 13, wherein the instructions for initialising an estimate further comprises initialising an estimate of both of the transmit beamforming vector and the receive beamforming vector, and updating the estimate of the receive beamforming vector further comprises iteratively estimating the dominant receive beamforming eigenvector for the current estimate of the transmit beamforming vector for NPM,U iterations, and the instructions for updating an estimate of the transmit vector further comprises iteratively estimating the dominant transmit beamforming eigenvector for the current estimate of the receive beamforming vector for NPM,V iterations.

15. A computer readable medium as claimed in claim 14 wherein NPMjU and NPM,V are identical.

16. A computer readable medium as claimed in any one of claims 12 to 15 in which the stopping criterion is that the difference in SINRpre between successive iterations is less than a predefined value.

17. A computer readable medium as claimed in any one of claims 11 to 16, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are the set of OFDM subcarriers and the block of time-domain symbols corresponds to one OFDM symbol.

18. A computer readable medium as claimed in any one of claims 11 to 16 wherein the subcarrier- based MIMO communication system is an Orthogonal Frequency Division Multiple Access (OFDMA) MIMO system in which the set of OFDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

19. A computer readable medium as claimed in any one of claims 11 to 16 wherein the subcarrier- based MIMO communication system is Single Carrier Frequency Division Multiple Access (SC- FDMA) MDvIO system in which the set of SC-FDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

20. A computer readable medium as claimed in any one of claims 12 to 16, wherein the subcarrier-based MMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are OFDM subcarriers and the jointly estimated pair of transmit and receive beamforming vectors are for use over a block of time-domain symbols (one OFDM symbol), and the standard eigenproblem has the form A(u) v = ζ'v where ζ' - (u*Rwu} ζ is the eigenvalue and v is the eigenvector, and the generalised eigenproblem has the form B{v)u = γRwu where γ is the eigenvalue and u is the eigenvector and where M is the pre OFDM demodulation receive beamforming vector; v is the post OFDM modulation transmit beamforming vector;

Z=O U KwU

Hi is the / * time lagged estimate of the MIMO channel where I = 0..L- 1; Rw is an estimate of the noise plus interference co-variance matrix in the time domain; P5 is the available transmit power; ζ is the Lagrangian multiplier associated with the constraint vHv < 1 and the Lagrangian

L = SINRpre - ^(vHv -l) used to maximise SINRpre;

and the estimates of u and v are the dominant eigenvectors of the generalised and standard eigenproblems respectively.

21. An apparatus for use in a subcarrier-based multiple input multiple output (MIMO) communication system comprising: a processor operative to jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpn.) ; a communication means for providing the transmit beamforming vector to a transmitter comprising a plurality of transmit antennas and the receive beamforming vector to a receiver comprising a plurality of receive antennas for use over the block of time-domain symbols.

22. An apparatus as claimed in claim 21, wherein maximisation of the expression for SINRpre is expressed as a generalised eigenproblem in which the receive beamforming vector is an eigenvector and a standard eigenproblem in which the transmit beamforming vector is an eigenvector, and the processor jointly estimates the transmit and receive beamforming by: initialising an estimate of at least one of the pair of transmit and receive beamforming vectors; updating an estimate of the receive beamforming vector by solving the generalised eigenproblem using the current estimate of the transmit beamforming vector; updating an estimate of the transmit beamforming vector by solving the standard eigenproblem using the current estimate of the receive beamforming vector; and repeating the two updating steps until a predefined stopping criterion is met, wherein the updating steps may be performed in any order and the initialisation step is performed to obtain an estimate for use in the first updating step performed.

23. An apparatus as claimed in claim 22, wherein the receive and transmit beamforming vectors are each estimated as the dominant eigenvectors of their respective generalised and standard eigenproblems and a power method is used to obtain the dominant eigenvectors in each update step.

24. An apparatus as claimed in claim 23, wherein the step of initialising further comprises initialising an estimate of both of the transmit beamforming vector and the receive beamforming vector, and the step of updating an estimating of the receive beamforming vector further comprises iteratively estimating the dominant receive beamforming eigenvector for the current estimate of the transmit beamforming vector for NpM,u iterations, and the step of updating an estimate of the transmit vector further comprises iteratively estimating the dominant transmit beamforming eigenvector for the current estimate of the receive beamforming vector for NPM,V iterations.

25. An apparatus as claimed in claim 24 wherein NpM|U and NPM>V are identical.

26. An apparatus as claimed in any one of claims 22 to 25 in which the stopping criterion is that the difference in SINRpre between successive iterations is less than a predefined value.

27. An apparatus as claimed in any one of claims 21 to 26, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are the set of OFDM subcarriers and the block of time-domain symbols corresponds to one OFDM symbol.

28. An apparatus as claimed in any one of claims 21 to 26 wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division Multiple Access (OFDMA) MIMO system in which the set of OFDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

29. An apparatus as claimed in any one of claims 21 to 26 wherein the subcarrier-based MIMO communication system is Single Carrier Frequency Division Multiple Access (SC-FDMA) MIMO system in which the set of SC-FDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

30. An apparatus as claimed in any one of claims 22 to 26, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MDvIO system, and the set of subcarriers are OFDM subcarriers and the jointly estimated pair of transmit and receive beamforming vectors are for use over a block of time-domain symbols (one OFDM symbol), and the standard eigenproblem has the form A(α)v = ζ'v where ζ' = (uHRwuj ζ is the eigenvalue and v is the eigenvector, and the generalised eigenproblem has the form B(v) u = γRwu where γ is the eigenvalue and u is the eigenvector and where u is the pre OFDM demodulation receive beamforming vector; v is the post OFDM modulation transmit beamforming vector;

L-I I ,,H ττ 12 y=SiNR = Psl u HHι l ; r pre s£t uHRwu

Hi is the / * time lagged estimate of the MIMO channel where / = 0..L-1 ;

R w is an estimate of the noise plus interference co-variance matrix in the time domain;

Ps is the available transmit power; ζ is the Lagrangian multiplier associated with the constraint vHv < 1 and the Lagrangian

£ = SINRpre - ζ ( vHv - 1) used to maximise SINRpn.;

Mu) = PsJflf UU^H1; ι=o

B(v) = Ps∑H[Vv»H?;

1=0 and the estimates of u and v are the dominant eigenvectors of the generalised and standard eigenproblems respectively.

31. A subcarrier-based multiple input multiple output (MIMO) communication system comprising: a processor operative to jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpre); a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the transmit beamforming vector is provided to the transmit beamforming module for use over the block of time-domain symbols by the plurality of transmit antennas and the receive beamforming vector is provided to the receive beamforming module for use over the block of time- domain symbols by the plurality of receive antennas.

32. A system as claimed in claim 31 , wherein maximisation of the expression for SINRpre is expressed as a generalised eigenproblem in which the receive beamforming vector is an eigenvector and a standard eigenproblem in which the transmit beamforming vector is an eigenvector, and the step of jointly estimating comprises: initialising an estimate of at least one of the pair of transmit and receive beamforming vectors; updating an estimate of the receive beamforming vector by solving the generalised eigenproblem using the current estimate of the transmit beamforming vector; updating an estimate of the transmit beamforming vector by solving the standard eigenproblem using the current estimate of the receive beamforming vector; and repeating the two updating steps until a predefined stopping criterion is met, wherein the updating steps may be performed in any order and the initialisation step is performed to obtain an estimate for use in the first updating step performed.

33. A system as claimed in claim 32, wherein the receive and transmit beamforming vectors are each estimated as the dominant eigenvectors of their respective generalised and standard eigenproblems and a power method is used to obtain the dominant eigenvectors in each update step.

34. A system as claimed in claim 33, wherein the step of initialising further comprises initialising an estimate of both of the transmit beamforming vector and the receive beamforming vector, and the step of updating an estimating of the receive beamforming vector further comprises iteratively estimating the dominant receive beamforming eigenvector for the current estimate of the transmit beamforming vector for NPM u iterations, and the step of updating an estimate of the transmit vector further comprises iteratively estimating the dominant transmit beamforming eigenvector for the current estimate of the receive beamforming vector for NPM v iterations.

35. A system as claimed in claim 34 wherein NPM,U and NPM,V are identical.

36. A system as claimed in any one of claims 32 to 35 in which the stopping criterion is that the difference in SINRpre between successive iterations is less than a predefined value.

37. A system as claimed in any one of claims 31 to 36, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are the set of OFDM subcarriers and the block of time-domain symbols corresponds to one OFDM symbol.

38. A system as claimed in any one of claims 31 to 36 wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division Multiple Access (OFDMA) MIMO system in which the set of OFDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

39. A system as claimed in any one of claims 31 to 36 wherein the subcarrier-based MIMO communication system is Single Carrier Frequency Division Multiple Access (SC-FDMA) MIMO system in which the set of SC-FDMA subcarriers is divided into a plurality of disjoint groups of subcarriers, and the jointly estimated pair of transmit and receive beamforming vectors are for use by one of the disjoint groups of subcarriers.

40. A system as claimed in any one of claims 32 to 36, wherein the subcarrier-based MIMO communication system is an Orthogonal Frequency Division multiplexing (OFDM) MIMO system, and the set of subcarriers are OFDM subcarriers and the jointly estimated pair of transmit and receive beamforming vectors are for use over a block of time-domain symbols (one OFDM symbol), and the standard eigenproblem has the form A(κ) v = ζ'v where ζ' = («"/?„,«) ζ is the eigenvalue and v is the eigenvector, and the generalised eigenproblem has the form B(v) u = yRwu where γ is the eigenvalue and u is the eigenvector and where u is the pre OFDM demodulation receive beamforming vector; v is the post OFDM modulation transmit beamforming vector;

L-X I H Ti 12

' pre SU uHRwu

H1 is the I * time lagged estimate of the MIMO channel where Z = 0..L-1; Rw is an estimate of the noise plus interference co-variance matrix in the time domain; Ps is the available transmit power; ζ is the Lagrangian multiplier associated with the constraint vHv < 1 and the Lagrangian

C = SINRpre - ζ{yHv -l) used to maximise SESfRpre;

A(u) = Ps∑H?uu»Hr, ι=o and the estimates of u and v are the dominant eigenvectors of the generalised and standard eigenproblems respectively.

41. A transmitter in a subcarrier-based multiple input multiple output (MMO) communication system comprising: a plurality of antennas; a transmit beamforming module for applying a transmit beamforming vector estimated by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpre) for use over a block of time domain symbols by the plurality of antennas.

42. The transmitter of claim 14 further comprising: a processor operative to jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpre); a communication means for providing the transmit beamforming vector to the transmit beamforming module and the receive beamforming vector to a receiver comprising a plurality of receive antennas.

43. A receiver in a subcarrier-based multiple input multiple output (MIMO) communication system comprising: a plurality of antennas; a receive beamforming module for applying a receive beamforming vector estimated by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SENRpn.) for use over a block of time domain symbols by the plurality of antennas.

44. The receiver of claim 16 further comprising: a processor operative to jointly estimating a pair of transmit and receive beamfoiming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MDvIO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINRpn.); a communication means for providing the receive beamforming vector to the receive beamforming module and the transmit beamforming vector to a transmitter comprising a plurality of receive antennas.

45. A method for joint estimation of a pair of transmit and receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: jointly estimating a pair of transmit and receive beamforming vectors for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol, in a OFDM MIMO system by maximising an expression for the mutual information (MI) in the MIMO channel, wherein the expression for MI is a function of at least the transmit and receive beamforming vectors, a MIMO channel response and a interference plus noise covariance matrix, and the maximisation is performed assuming a constant interference plus noise covariance matrix for all subcarriers and uniform power allocation, and the joint estimation is performed using a modified power method (MPM).

46. A method for joint estimation of a transmit beamforming vector and a plurality of receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: jointly estimating a transmit beamforming vector and a plurality of receive beamforming vectors by maximizing an expression for the mutual information, wherein the transmit beamforming vector is estimated using a modified power method (MPM) and the plurality of receive beamforming vectors is estimated using a closed-form expression that is a function of the transmit beamforming vector and the transmit beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of receive beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

47. A method for joint estimation of a plurality of transmit beamforming vectors and a receive beamforming vector for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MLVIO) communication system comprising: jointly estimating a receive beamforming vector and a plurality of transmit beamforming vectors by maximising an expression for the mutual information, wherein the receive beamforming vector is estimated using a modified power method (MPM) and the plurality of transmit beamforming vectors is estimated using a closed-form expression that is a function of the receive beamforming vector and the receive beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

48. An orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: a processor for jointly estimating a pair of transmit and receive beamforming vectors for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol, in a OFDM MIMO system by maximising an expression for the mutual information (MI) in the MIMO channel, wherein the expression for MI is a function of at least the transmit and receive beamforming vectors, a MIMO channel response and a interference plus noise covariance matrix, and the maximisation is performed assuming a constant interference plus noise covariance matrix for all subcarriers and uniform power allocation, and the joint estimation is performed using a modified power method (MPM); a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the transmit beamforming vector is provided to the transmit beamforming module for use over the block of time-domain symbols by the plurality of transmit antennas and the receive beamforming vector is provided to the receive beamforming module for use over the block of time- domain symbols by the plurality of receive antennas..

49. An orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: a processor for jointly estimating a transmit beamforming vector and a plurality of receive beamforming vectors by maximizing an expression for the mutual information, wherein the transmit beamforming vector is estimated using a modified power method (MPM) and the plurality of receive beamforming vectors is estimated using a closed-form expression that is a function of the transmit beamforming vector; a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the transmit beamforming vector is provided to the transmit beamforming for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and the plurality of receive beamforming vectors is provided to the receive beamforming module and each of the plurality of receive beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

50. An orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MDvIO) communication system comprising: a processor for jointly estimating a receive beamforming vector and a plurality of transmit beamforming vectors by maximising an expression for the mutual information, wherein the receive beamforming vector is estimated using a modified power method (MPM) and the plurality of transmit beamforming vectors is estimated using a closed-form expression that is a function of the receive beamforming vector and the receive beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers; a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the receive beamforming vector is provided to the receive beamforming for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and the plurality of transmit beamforming vectors is provided to the transmit beamforming module and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

Description:
METHOD AND APPARATUS FOR BEAMFORMING IN MIMO SYSTEMS

PRIORITY DOCUMENTS

The present application claims priority from Australian Provisional Patent Application No. 2008905282 entitled "Method and apparatus for beamforming in MIMO-OFDM systems" and filed on 10 October 2008. The content of this application is hereby incorporated by reference in its entirety.

INCORPORATION BY REFERENCE

The following publications are referred to in the present application and their contents are herby incorporated by reference in their entirety:

Andre Pollok, "Multi-Antenna Techniques for Millimetre-Wave Radios", PhD Thesis, University of South Australia, 2009;

A. Pollok, W. G. Cowley, and N. Letzepis, "Symbol-Wise beamforming for MIMO-OFDM transceivers in the presence of Co-Channel interference and spatial correlation," IEEE Trans. Wireless Commun., accepted for publication;

A. Pollok, N. Letzepis, and W. G. Cowley, "Symbol-Wise beamforming for Co-Channel interference reduction in MLMO-OFDM systems", in Proc. IEEE Global Commun. Conf. (Globecom), Honolulu, Hawaii, USA, Nov. 2009, accepted for publication.

FIELD OF THE INVENTION

The present invention relates to subcarrier-based multiple input multiple output (MIMO) wireless communication systems. In one particular form the invention relates to joint estimation of a pair of transmit and receive beamforming vectors for use in a subcarrier-based multiple input multiple output (MIMO) communication system.

BACKGROUND OF THE INVENTION

An ongoing goal in the field of wireless communications is to develop systems and method capable of providing high throughput, robust, and preferably low cost wireless communications to facilitate applications such as wireless Local Area Networks (WLANs).

Wideband communication systems are susceptible to frequency selective fading, which is characterized by different amounts of attenuation across the system bandwidth. Frequency selective fading is typically caused by multipath effects, in which multiple copies of a transmitted signal arrive at a receiver at different times. This leads to inter-symbol interference (ISI) whereby each symbol in a received signal acts as distortion to subsequent symbols in the received signal. This distortion degrades performance by impacting the ability to correctly detect the received symbols. One technique that has proved successful in combating the problem of ISI is Orthogonal Frequency Division Nultiplexing (OFDM). An OFDM system effectively partitions the overall system bandwidth into a number of (N F ) frequency subchannels (also referred to as subbands or frequency bins), each of which is associated with a respective subcarrier on which data may be modulated. OFDM can be viewed as an approach which uses many slowly-modulated subcarriers, rather than one rapidly- modulated wideband carrier that may be susceptible to frequency selective fading. The advantage of low data rate channels is that they make the use of a guard interval between symbols affordable, thereby combating any ISI effects. Division of the bandwidth into N F low data rate channels (frequency diversity) allows an overall high data rate to be supported.

Typically the guard interval comprises a cyclic prefix, in which a portion of each OFDM symbol is appended to each OFDM symbol so as to combat any ISI that may arise. Ideally the subchannels are sufficiently narrow so as to be frequency flat. However this is not always the case, and thus OFDM systems typically also use coding and interleaving to further improve data throughput.

Multiple antenna techniques can also be used to provide increased throughput through approaches such as array gain in which adapative beamforming is used to control the shape of the antenna array patterns, and spatial diversity gain which provides increased reliability relative to a single antenna system. One system of particular interest are MIMO communication system which employ multiple (N TX ) transmit antennas and multiple (N RX ) receive antennas for transmission of multiple independent data streams. It has been shown that MIMO systems can potentially deliver large gains in channel capacity (known as the multiplexing gain) over Single Input Single Output (SISO), Single Input Multiple Output (SIMO), or Multiple Input Single Output (MISO) systems in which a single antenna is used by at least one of the transmitter or receiver.

MIMO systems can achieve both array gain and spatial multiplexing gain.Array gain can be achieved by adaptive beamforming. Traditionally, beamforming is a technique which takes advantage of superposition to change the directionality of the array. When transmitting, a transmit beamformer uses a series of transmit weights to controls the phase and relative amplitude of the signal at each transmitter, in order to create a pattern of constructive and destructive interference in the wavefront. When receiving, a receive beamformer applies receive weights to signals from different receivers such that an expected pattern of radiation is preferentially observed. Phased array beamforming is the simplest form of beamforming in which only the differential phase shift between antenna elements in adjusted whilst keeping the magnitude of weights constant. Higher flexibility is achieved by an adaptive antenna which adjusts both the phase and magnitude of individual antenna weights (beamforming vectors) independently. This can be used to obtain a higher antenna gain than each of the individual elements or reduce the influence of an interfering source by forming an array pattern null in its the direction.

Spatial multiplexing refers to a technique in which a high rate signal is split into multiple lower rate streams and each stream is transmitted from a different transmit antenna in the same frequency channel. Provided antenna elements are spaced sufficiently far apart, random fades at each element are uncorrelated, and thus the combined probability of all elements experiencing a fade is much smaller. If these signals arrive at the receiver antenna array with sufficiently different spatial signatures, the receiver can separate these streams, or spatial subchannels, to obtain a spatial multiplexing gain. The RF channel between the multiple-antenna array at the transmitting station and the multiple-antenna array at a receiving station is referred to as a MIMO channel. In spatial multiplexing, the MIMO channel (the RF channel between the multiple-antenna array at the transmitting station and the multiple-antenna array at a receiving station) is comprised of Ns independent channels, where N 5 < min {N τ , N R } . Each of the N s independent channels is referred to as a spatial subchannel of the MDvIO channel.

MMO systems may also utilise space diversity techniques, also known as diversity combining or diversity coding. Space diversity techaniques, refers to techniques in which the same signal is transmitted by multiple spaced antennas to provide robustness against fading. A single stream is coded using space-time coding and transmitted from each of the transmit antennas using full or near orthogonal coding. Space diversity techniques exploits the independent fading in the multiple antenna links to enhance signal diversity and provide signal robustness. Beamforming is related to space diversity or diversity combining, and is sometimes incorrectly used interchangeably with these terms. The distinction between the terms is not clear cut although there exist a fundamental trade-off between beamforming (or array) gain and diversity gain in that the amount of available array and diversity gain depends on the degree of spatial correlation in the MIMO subchannels.

MDvIO-OFDM systems thus have the potential to provide high throughput systems which are resistant to multipath fading effects. However MEvIO-OFDM systems present additionally complexity over SISO, SDvIO or MISO systems. Firstly the MEvIO channel is more complex and additional data processing is required to both estimate the channel and decode transmitted symbols. Further the complexity is such that extending previous approaches used in SISO, SDvIO or MISO systems to MDvIO systems is not always a straight forward or easy task. Studies of MIMO-OFDM have shown that the available multiplexing gain strongly depends on the channel conditions, and correlation between different transmit and receive branches has been shown to be the limiting factor (H. Bδlcskei, D. Gesbert, and A. J. Paulraj, 'On the capacity of OFDM-based spatial multiplexing systems," IEEE Trans. Commun., vol. 50, no. 2, pp. 225-234, Feb. 2002; D.-S. Shiu, G. Foschini, M. Gans, and J. Kahn, "Fading correlation and its effect on the capacity of multielement antenna systems," IEEE Trans. Commun., vol. 48, no. 3, pp. 502-513, Mar. 2000.). Depending on the correlation as well as on the acceptable complexity of a practical implementation, the joint use of transmit and receive beamforming may be preferred over spatial multiplexing. Interest in MIMO and OFDM MIMO systems has increased due to the recent release of approximately 7GHz of unlicensed spectrum in the 60GHz frequency band. The 60GHz band is appealing as it allows transmission rates in the order of Gigabits per second. However implementation of MIMO systems in the 60GHZ band is more challenging than at lower (eg GHz) frequencies, as high propagation losses limit transmissions to a few metres (reducing multipath effects) and renders non line of sight operation difficult. Further, operating a wireless multi-Gbps link places strict requirements on processing delay and transceiver complexity which is a further hurdle in achieving the potential gains MIMO systems represent.

The observation of non rich multipath propagation at the 60GHz band suggests a potentially high degree of spatial correlation and thus spatial multiplexing appears less desirable than beamforming as a means for obtaining the potential multiplexing gain. It has been shown that the beamforming approach which maximises the mutual information, and is thus the optimal beamforming approach is to calculate independent beamforming vectors for all OFDM subcarriers (K.-K. Wong et al., "Adaptive antennas at the mobile and base stations in an OFDM/TDMA system," BEEE Trans. Commun., vol. 49, no. 1, pp. 195-206, Jan. 2001.). In this approach antenna weight vectors are calculated for each independent OFDM subcarrier by maximising the signal-to-interference-plus-noise ratio on each subcarrier (SINR n ). Whilst this approach does yields optimal beamforming weights, the method has relatively high computational requirements, as a dedicated inverse discrete Fourier transform (IDFT) processor is required for each transmit antenna element, and a separate DFT processor is required per receive antenna. The overall computational complexity amount to 0((M 1x + M n )N Iog 2 N). Further, for each subcarrier an eigenvalue decomposition is required plus additional matrix vector arithmetic. In practice the computational complexity of this subcarrier wise method is likely to be prohibitive for a practical implementation. Further, there is considerable interest in developing low-cost low-power transceivers that facilitate high data rates, and the additional cost and power consumption associated with extra IDFT/DFT processors makes this approach infeasible. To overcome the computational and cost hurdles associated with subcarrier wise beamforming, a time domain symbol based beamforming approach has been proposed (D. Huang and K. B. Letaief, " v Symbol-based space diversity for coded OFDM systems," IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 117—127, Jan. 2004). In time domain beamforming, the processing for transmit beamforming is carried out in the time domain after the IDFT which realises the OFDM modulation. Similarly beamforming at the receive end is performed prior to OFDM demodulation. The transmit and receive beamforming vectors are then kept fixed for the entire OFDM symbol period.

In most cases this approach is sub-optimal compared to subcarrier wise beamforming, as it requires finding the best weight across all subcarrriers, rather than choosing an individual weight in each subcarrier (the optimal case). However the trade off is that no additional DFT processors are required in addition to the DFT processor which is already present and used to perform OFDM modulation, thereby providing a (M a + M n )/2 fold saving in DFT processors and making the approach feasible to implement in a MIMO-OFDM system.

In the system proposed by Huang and Letaief, the transmit and receive antenna weights or beamforming vectors are chosen that optimise the average pair-wise codeword distance for the duration of an OFDM symbol. However a problem with this approach is that it was developed only in the context of Additive Gaussian White Noise (AWGN) in which case there is a simple relationship between the pair-wise code word distance and the pairwise error probability. The pair-wise error probability is the probability of deciding erroneously in favour of a coded sequence, instead of the transmitted sequence conditioned upon the channel.

However in many environments, such as office WLAN environments, there may be multiple independent communication systems (including other MIMO-OFDM systems) operating and sharing the same bandwidth. For a given MIMO-OFDM system, the presence of such systems is in the form of noise known as multiple-access interference. Multiple access interference differs significantly from a simple AWGN noise environment as the interference is typically a function of the channel conditions. The proposed solution of Huang and Letaief fundamentally relies on the pairwise error probability following a Gaussian distribution, which is only valid in the context of AWGN noise, and there was no consideration of the effect of other noise sources such as coloured noise or more importantly, multiple access interference. Unfortunately, in the presence of such noise environments the relationship between the average pair-wise code word distance and the pairwise error probability breaks down, and thus the estimated beam weights will, in most cases be extremely suboptimal thus making the method impractical for most realistic transmission environments. Accordingly, there exists a need to develop alternative methods and systems for estimating and performing symbol wise beamforming in MIMO-OFDM systems.

SUMMARY OF THE INVENTION

According to a first aspect of the present invention, there is provided a method for joint estimation of a pair of transmit and receive beamforming vectors for use in a subcarrier-based multiple input multiple output (MIMO) communication system comprising: jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINR pn .).

According to a second aspect of the present invention, there is provided a computer readable medium comprising instructions stored therein comprising: a first instruction set for jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINR pn .); a second instruction set for providing the transmit beamforming vector to a transmitter comprising a plurality of transmit antennas and the receive beamforming vector to a receiver comprising a plurality of receive for use over the block of time-domain symbols.

According to a third aspect of the present invention, there is provided an apparatus for use in a subcarrier-based multiple input multiple output (MIMO) communication system comprising: a processor operative to jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MEvIO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MEvIO channel at the input of the receive side demodulator (SESTR pre ); a communication means for providing the transmit beamforming vector to a transmitter comprising a plurality of transmit antennas and the receive beamforming vector to a receiver comprising a plurality of receive antennas for use over the block of time-domain symbols.

According to a fourth aspect of the present invention, there is provided a subcarrier-based multiple input multiple output (MEvIO) communication system comprising: a processor operative to jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time-domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINR pn .); a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the transmit beamforming vector is provided to the transmit beamforming module for use over the block of time-domain symbols by the plurality of transmit antennas and the receive beamforming vector is provided to the receive beamforming module for use over the block of time- domain symbols by the plurality of receive antennas.

According to a fifth aspect of the present invention, there is provided a transmitter in a subcarrier- based multiple input multiple output (MEMO) communication system comprising: a plurality of antennas; a transmit beamforming module for applying a transmit beamforming vector estimated by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINR pn .) for use over a block of time domain symbols by the plurality of antennas.

According to a sixth aspect of the present invention, there is provided a receiver in a subcarrier-based multiple input multiple output (MIMO) communication system comprising: a plurality of antennas; a receive beamforming module for applying a receive beamforming vector estimated by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINR pn .) for use over a block of time domain symbols by the plurality of antennas.

According to a seventh aspect of the present invention, there is provided a method for joint estimation of a pair of transmit and receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MMO) communication system comprising: jointly estimating a pair of transmit and receive beamforming vectors for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol, in a OFDM MMO system by maximising an expression for the mutual information (MI) in the MMO channel, wherein the expression for MI is a function of at least the transmit and receive beamforming vectors, a MMO channel response and a interference plus noise covariance matrix, and the maximisation is performed assuming a constant interference plus noise covariance matrix for all subcarriers and uniform power allocation, and the joint estimation is performed using a modified power method (MPM).

According to a eighth aspect of the present invention, there is provided a method for joint estimation of a transmit beamforming vector and a plurality of receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: jointly estimating a transmit beamforming vector and a plurality of receive beamforming vectors by maximizing an expression for the mutual information, wherein the transmit beamforming vector is estimated using a modified power method (MPM) and the plurality of receive beamforming vectors is estimated using a closed-form expression that is a function of the transmit beamforming vector and the transmit beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of receive beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

According to a ninth aspect of the present invention, there is provided a method for joint estimation of a plurality of transmit beamforming vectors and a receive beamforming vector for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: jointly estimating a receive beamforming vector and a plurality of transmit beamforming vectors by maximising an expression for the mutual information, wherein the receive beamforming vector is estimated using a modified power method (MPM) and the plurality of transmit beamforming vectors is estimated using a closed-form expression that is a function of the receive beamforming vector and the receive beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

According to a tenth aspect of the present invention, there is provided an orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: a processor for jointly estimating a pair of transmit and receive beamforming vectors for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol, in a OFDM MIMO system by maximising an expression for the mutual information (MI) in the MIMO channel, wherein the expression for MI is a function of at least the transmit and receive beamforming vectors, a MIMO channel response and a interference plus noise covariance matrix, and the maximisation is performed assuming a constant interference plus noise covariance matrix for all subcarriers and uniform power allocation, and the joint estimation is performed using a modified power method (MPM); a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the transmit beamforming vector is provided to the transmit beamforming module for use over the block of time-domain symbols by the plurality of transmit antennas and the receive beamforming vector is provided to the receive beamforming module for use over the block of time- domain symbols by the plurality of receive antennas..

According to a eleventh aspect of the present invention, there is provided an orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: a processor for jointly estimating a transmit beamforming vector and a plurality of receive beamforming vectors by maximizing an expression for the mutual information, wherein the transmit beamforming vector is estimated using a modified power method (MPM) and the plurality of receive beamforming vectors is estimated using a closed-form expression that is a function of the transmit beamforming vector; a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the transmit beamforming vector is provided to the transmit beamforming for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and the plurality of receive beamforming vectors is provided to the receive beamforming module and each of the plurality of receive beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

According to a twelfth aspect of the present invention, there is provided an orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system comprising: a processor for jointly estimating a receive beamforming vector and a plurality of transmit beamforming vectors by maximising an expression for the mutual information, wherein the receive beamforming vector is estimated using a modified power method (MPM) and the plurality of transmit beamforming vectors is estimated using a closed-form expression that is a function of the receive beamforming vector and the receive beamforming vector is for use over a set of

OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of

OFDM subcarriers; a transmitter comprising a plurality of transmit antennas and a transmit beamforming module; and a receiver comprising a plurality of receive antennas and a receive beamforming module; wherein the receive beamforming vector is provided to the receive beamforming for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and the plurality of transmit beamforming vectors is provided to the transmit beamforming module and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

BRIEF DESCRIPTION OF THE DRAWINGS

An illustrative embodiment of the present invention will be discussed with reference to the accompanying drawings wherein:

FIGURE 1 - shows an exemplary structure of a MIMO-OFDM system using sub-carrier wise beamforming;

FIGURE 2 - shows a block diagram of a transmitter and receiver structure for symbol-wise beamforming;

FIGURE 3 - shows a graph of the convergence behaviour of the algorithm as used in one aspect of the present invention for a randomly generated channel; FIGURE 4 - shows a graph showing the performance gap between subcarrier-wise and symbol-wise beamforming;

FIGURE 5 - shows a graph illustrating how the 10% outage capacity changes with increasing spatial correlation;

FIGURE 6 - shows a graph showing the outage capacity of subcarrier-wise and symbol-wise beamforming plotted vs. the mean DoA #> DoA of the interfering signal;

FIGURE 7a - shows 10% outage capacity vs. mean DoA φ DoA of the interfering signal for moderate spatial correlation; and

FIGURE 7b - shows 10% outage capacity vs. mean DoA φ OoA of the interfering signal for high spatial correlation; FIGURE 8 - shows an exemplary structure of a MIMO-OFDMA system using sub-carrier wise beamforming;

FIGURE 9 - shows an exemplary structure of a MIMO system using sub-carrierwise-group- beamforming; FIGURE 10 - shows simulation results of mutual information complementary cumulative distribution functions (CCDFs) for OFDMA and subcarrier-group-wise beamforming for a range of subcarrier group sizes N grp corresponding to 1, 2, 4, 8, and 64 subcarriers in each group;

FIGURE 11 - shows simulation results of mutual information CCDFs for the two assymetric beamforming cases in the absence of spatial correlation; FIGURE 12 - shows simulation results of mutual information CCDFs for the two assymetric beamforming cases in the case that the transmit side correlation was set to 0.9;

FIGURE 13 - shows simulation results of mutual information CCDFs for the two assymetric beamforming cases in the case that the receive side correlation was set to 0.9;

FIGURE 14 - shows an flowchart of a method for joint estimation of transmit and receive beamforming vectors 1400 according to an embodiment of the invention;

FIGURE 15 - shows an more detailed flowchart of a method 1400;

FIGURE 16 - shows a flowchart of a method for joint estimation of a pair of transmit and receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system; FIGURE 17 - shows a flowchart of a method for joint estimation of a transmit beamforming vector and a plurality of receive beamforming vectors for use in a orthogonal frequency division multiplexing

(OFDM) multiple input multiple output (MIMO) communication system; and

FIGURE 18 - shows a flowchart of a method for joint estimation of a plurality of transmit beamforming vectors and a receive beamforming vector for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system.

DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The following description focuses on embodiments of the invention applicable to subcarrier-based multiple input multiple output (MIMO) communication systems. A subcarrier based communication system refers to a communication system using a transmission technique which is characterised by block based Discrete Fourier Transfer (DFT) processing, and includes Orthogonal Frequency Division Multiplexing (OFDM), Orthogonal Frequency Division Multiple Access (OFDMA) and Single Carrier Frequency Division Multiple Access (SC-FDMA) systems. In order to illustrate the invention, an OFDM MIMO communication system is described but it will be appreciated that the invention is not limited to this application but may be applied to many other systems and transmission schemes. Figure 1 illustrates the general structure of a MIMO-OFDM system 100 using sub-carrier wise beamforming (BF) at the transmitter and receiver. The MIMO system comprises M 1x transmit antennas and M rx receive antennas. The employed modulation system in this embodiment is OFDM with N subcarriers, where N is > 1. Subcarrier-wise beamforming is performed at both ends of the MEvIO link.

Beamforming assumes that at least the receiver has knowledge of at the MIMO channel and the interference plus noise covariance matrix. Such information may be obtained through the use of pilot channels and other methods for deriving a channel matrix H, as is known in the art. It may be estimated at one end of the link, such as the receiver and the estimate provided to the other end over a feedback link, which for the present embodiment will be assumed to be a perfect (i.e. error free) link.

The transmitter receives an input data symbol stream to be transmitted. Typically such a stream will be obtained through encoding, bit interleaving and symbol mapping a data stream. The input symbol stream is multiplexed 110 across the subcarriers to obtain the symbols S n which are then multiplied

with the weight vector V n e C " for the transmit antenna array to generate the signalvector

S n = v n S' n 120. In the figure, this operation is symbolised by the block labelled V n . If P s is the

available transmit power such that = μ n P s and = N , where N corresponds to the number of subcarriers, then this generates a power constraint of v"v π < 1 which ensures that the available transmit power is not exceeded. The j' h elements of all resulting signal vectors subsequently undergo an N -point inverse discrete Fourier transform (IDFT), which realises the OFDM modulator 130 (the DFT and IDFT are assumed to be unitary transforms in order to render the average power invariant under DFT/IDFT). After demultiplexing, the last N^ symbols of the signal at the output of the IDFT block are duplicated and prepended as cyclic prefix (CP) to form the OFDM symbol of length N+N cp 140, before the stream is launched from the j' h transmit antenna 150.

At the receiver 160, the CP of the signal at the i th receive antenna is stripped off 170, before the signal is OFDM demodulated by means of an N -point DFT 180. Provided that the CP is at least as long as the channel impulse response, i.e. N cp ≥ L , the received signal vector on subcarrier n after removal of

the CP and OFDM demodulation can be represented as J{ = H n S n + W n , where W n e C " models the combined effect of noise and multiple-access interference, and H n is the discrete frequency response of the channel. The receiver further processes the received signal vector y n to yield

at its outputl92, where « B e C n denotes the receive antenna weight vector. Receive beamforming is represented by the block labelled U n in Figure 1 190. Joint transmit and receive beamforming reduces the MMO link to a single-input single-output (SISO) channel and the mutual information between the input and output of a SISO-OFDM link can be expressed as

1 = ^7∑>g 2 (l + SINR π ). where SINR n is the average received signal to interference plus noise ratio of the n* subcarrier.

Since the channels Η n are assumed to be perfectly known at the receiver, the reciever can compute the antenna weight vectors U n and V n that maximise SINR n (and thus also X ). The solution to this joint optimisation problem is known (K.-K. Wong et al., ""Adaptive antennas at the mobile and base stations in an OFDM/TDMA system," IEEE Trans. Commun., vol. 49, no. 1, pp. 195-206, Jan. 2001.) and the maximum value of SINR n is achieved when V n is chosen as the dominant eigenvector of the matrix Η^RTL 1 H n and the receive antenna weight vector as the Wiener filter U n = OC n R^ T-C n V n n n where OC n is a scaling factor that does not affect the value of SINR n and R w denotes the n interference-plus-noise covariance matrix. The resulting SINR n is given by P 1 times the largest eigenvalue of Η^R^L H n where P 5 is the available transmit power which is uniformly allocated over n the subcarriers. The corresponding transmit antenna weight vectors V n can then be communicated to the transmitter via a perfect feedback link.

It should be stressed that the computational requirements of subcarrier-wise beamforming are still very high. As can be seen in Figure 1, one separate IDFT (DFT) block is needed per transmit (receive) antenna element. Additionally, an eigenvalue decomposition as well as some further matrix-vector arithmetic are required per subcarrier to compute the optimal weights. Although the DFT can be efficiently implemented via the fast Fourier transform (FFT), the overall computational complexity is likely to be prohibitive for a practical implementation, particularly in low-cost low-power systems that facilitate very high data rates such as those for use in the 60GHz band.

The computational complexity of MMO-OFDM systems that perform beamforming at both ends of the link can be reduced by performing transmit beamforming after the IDFT and receive beamforming before the DFT. A block diagram of the resulting transmitter and receiver is depicted in Figure 2 and this scheme will be referred to as symbol-wise beamforming. Similar numbering convention is used as per figure 1, symbolwise trnasmit and receive beamforming weight modules denoted 220 and 290 respectively. The advantage of symbol-wise beamforming is that it requires only a single IDFT block at the transmitter. Similarly, only one instead of M n DFT blocks is needed at the receiver. Therefore, the overall saving of IDFT / DFT blocks is {M a + M n ) /2-fold. Further symbol-wise beamforming not only reduces the computational complexity but also reduces the required amount of feedback by a factor of N , since the receiver needs to send only the antenna weight vector v to the transmitter rather than N vectors V n . In addition, weightings may be carried out at the carrier frequency, thereby reducing the number of required up/ down conversions and synthesis / sampling operations at the transmitter/ receiver to one.

Mathematically, symbol-wise beamforming is equivalent to subcarrier-wise beamforming using the same set of weights U n = u and V n = v on all N subcarriers. The expression of the received signal according simplifies to:

Applying an IDFT to the previous expression yields the received signal at the input of the OFDM demodulator:

= Yu H H [ vs f [k -l] + u li w[k],

1=0 where s'[k] and w[k] (0 < k < N ) are related to the frequency domain quantities S' n and W n (0 < n < N ) via the DFT (Note that s'[k] and w[k] are uncorrelated in time as a result of S n and W n being uncorrelated across subcarriers). The wireless medium is modelled as a tapped delay line with L taps. It can be seen from the above equation that each tap of the effective channel is simply the I th matrix-valued channel H 1 filtered with the antenna weight vectors u and v . The weight vectors remain constant during the entire OFDM symbol, giving rise to the term symbol-wise beamforming.

Of course, using the same antenna weight vectors across the entire band will in general incur a performance loss due to the reduced degrees of freedom compared to subcarrier-wise beamforming. An independent maximisation of all SFNR n is not possible, which means that the weight vectors will need to be chosen such that they xv match" all subcarriers in the best possible way.

The approach to choosing optimal beamforming vectors in the subcarrier case was to maximise the mutual information. However the ability to choose beamforming vectors that maximise the mutual information in the subcarrier case was possible only because optimisation of the antenna weight vectors could be decoupled from power allocation via a two-step procedure. First, the antenna weight vectors of each subcarrier are chosen such that the corresponding received SINR is maximised. In a second step, a waterfilling algorithm then operates on the resulting SINRs to compute the optimal power allocation. Unfortunately such an approach is not possible in symbol-wise beamforming due to reduced degrees of freedom, which renders the independent maximisation of the per-subcarrier SINRs impossible. If the antenna weight vectors u and v are chosen such that the received SINR on a particular subcarrier is maximised, then SINRs of the remaining subcarriers will in general not be maximised at the same time. Furthermore, the optimal u and v will not only be a function of the subcarrier channels Η n but also of the power allocation.

The problem of estimating the symbolwise beamforming that maximise the mutual information is to find the optimal power allocation vector μ = [/I 0 ,.. -M N - \ ] T and the optimal beamforming vectors u and v which solve the constrained maximisation problem:

N-\ i maximise I(u,v,μ) = T-IOg 2 (l+// π SINR (u,v))

N-I subject to V n V n < 1 and ∑μ n ≤ N where X is the mutual information and SINR ep π is the SESfR for n=0 equal power allocation and the contraints ensure the available transmit power is not exceeded. Using the method of Lagrange multipliers it can be shown that optimisation of the above problem generates a first condition for maximum mutual information: However the task of finding v is infeasible as it occurs in all summation terms in both the numerator and the denominator and thus a solution to optimisation problem in closed form appears infeasible. Furthermore it can be shown that the objection function X is non concave ruling out the use of convex optimisation techniques.

Symbol wise beamforming was previously considered by Huang and Letaief. However as discussed above this approach maximised the average pairwise code word distance, and is unlikely to be effective in many typical transmission environments where the noise does not follow a AWGN distribution.

However it has been advantageously realised that estimation of symbol wise transmit and receive beamforming vectors can be performed through optimisation of Signal to Noise Plus Interference Ratio at the input of the OFDM demodulator (SINR pre ).

The average received SINR previous to OFDM demodulation may be defined as:

where R w denotes the interference-plus-noise covariance matrix of the channel, such as may be obtained by means of estimation.

Unforunately the joint design, or estimation of u and v is not trivial as the SINR pre expression above is not jointly concave in u and v . In fact, even for the simplest case with L = I , i.e. a frequency-flat channel, the expression is non-concave and thus convex optimisation techniques can not be applied.

However even though the framework of convex optimisation cannot be applied, an algorithm that carries out the maximisation iteratively can be derived based on the method of Lagrange multipliers. Let the Lagrangian corresponding to the constrained maximisation of SINR be defined as

£ = SINR pre - <T(v H v -l) where ζ is the Lagrangian multiplier associated with the constraint v"v π < 1 . Furthermore, let A and B be shorthand for

B(v) = P s ∑H ιVV «H?.

1=0

With these definitions, the equation for SINR pre can be expressed as Taking the gradient V v £ v and setting it equal to zero yields the eigenvalue problem

A(u)v = ζ'v where ζ' = (u H R w u) ζ is the eigenvalue and v the eigenvector. Similarly, from setting the gradient V u £ with respect to u to zero we find

B{v)u = γR w u, which is a generalised eigenproblem with eigenvalue γ and eigenvector u . Note that the generalised eigenvalue just equals the SINR, i.e. γ = SINR pre . It is important to note that the eigenproblems are coupled through the matrices A and B and thus finding a closed form solution appears infeasible.

However it has been realised that whilst the closed form solution is unobtainable, the optimal solution can be obtained by jointly estimating u and v using an iterative approach. That is an estimate of u may be obtained by solving the generalised eigenproblem for some initial vector v , and then substituting the resulting vector u into the corresponding eigenproblem and repeated performing this iterative procedure. In pseudo-code, this algorithm can be written as shown in Table 1. It is noted that any of the eigenvectors could be used, however it is preferable that the dominant eigenvector is chosen as this is likely to increase the rate of convergence of the algorithm.

Table 1 - Iterative Maximisation of SINR pre

0 Set q <- 0

1 Initialise v[q] repeat 2 u[q + 1] <— dominant generalised eigenvector of B(v[q]) and R w

3 v[q + 1] <— dominant eigenvector of A( u[q + 1] )

4 q <r- q + l until stopping criterion met

After initialising the iteration index q and v , Step 2 maximises SINR for the given vector v by choosing u as the dominant generalised eigenvector of the matrix pencil formed by B and R w . Similarly, SINR pre is then maximised for the given vector u in Step 3. It follows that the sequence of iterates is non-decreasing.

If one applies a power constraint that v"v π < 1 , then it can be shown that the SINR pre is bounded from above, and the proposed algorithm is guaranteed to converge. It is important to note though that the algorithm might converge to a local maximum instead of the global maximum. As this is an iterative solution, extensive Monte Carlo simulations have been performed (discussed below) which demonstrated that the event of converging to a local maximum is rare.

In the next section, we will consider different approaches for the implementation of the proposed algorithm. Note also that the in Table 1, the iterative process of step 1 could start with first estimating u instead of v .Estimation of eigenvectors including dominant eigenvectors may be performed using standard eigenvalue decomposition (EVD) techniques.

Since the vectors u and v are updated in each iteration, the matrices A and B also need to be recomputed in every iteration. In the following, the two matrices will be rewritten with the aid of the Kronecker product, exploiting that the channel realisation does not change between iterations. Let vec( X ) denote the operation that stacks the columns of a (pXq) matrix X into a vector of dimension (pqxϊ) . Furthermore, let the inverse operation be denoted as unvec(-) . By applying the matrix identity CXD=(D 1 <8>C)X , the matrices A and B can be expressed as

B(v) = UnVeC(HVeC(Vv")), A(α) = unvec(H vec(«« H )), where the dependency of the vectors u and v on the iteration index q has been omitted for brevity. Note that the (M n XM 11 2 ) matrix H is completely determined by the current channel realisation. Due to the quasi-static nature of the channel, H remains constant for at least one codeword and therefore, need not be recomputed in every iteration. Whether the algorithm complexity is lower for the direct implementation or for the implementation via the Kronecker product depends on the number of channel taps L , the array sizes M a and M n , as well as on the number of iterations. Thus, the implementation approach needs to be chosen on a case-by-case basis.

Furthermore, the channel matrices and the covariance of noise and interference are not perfectly known in practice but need to be estimated by the receiver or provided to the receiver. Depending on if the estimation is best carried out in the time or frequency domain, the algorithm for symbol-wise beamforming can be implemented in the more convenient domain. In order to see this, note that by replacing H 1 in the equation for SINR pn . by its DFT expression an equivalent expression can be obtained for the SINR before OFDM demodulation. This expression has the same form as earlier and therefore leads to an equivalent iterative algorithm (with H 1 replaced by H n I sN and

R w replaced by ∑" ' ^ / N ).

As discussed above, each iteration involves solving the generalised and standard eigenproblems to produce an estimate of the receive and transmit beamforming vectors u and v. Typically u and v will be selected as the dominant eigenvectors as this is likely to improve the rate of convergence. Thus in another approach, a power method may be used to estimate the dominant eigenvectors. Power methods are appealing due to the simplicity low computational complexity as only one matrix-vector multiplcation per iterationis required. Table 2 lists psuedo code for implementing a power method based approach to the estimation of « and v. This algorithm will be referred to as maxSINRsym. Table 2 - Iterative Maximisation of SINR using the power method (PM)

0 Set q <- 0 1 Initialise u[q] and v[q] repeat repeat

2.1 «tø + l] <- R w - l B(v[q]) u[q]

2.2 <7 <- <7 + l until Np M , u iterations reached repeat

3.1 v[q + l] <- A(u[q])v[q]

3.2 q <^ q + l until Np MiV iterations reached until stopping criterion met

The stopping critereon may be selected based upon the application and could be chosen based upon difference in SINRp n . between successive iterations as being is less than a predefined value such as 10 "6 . Other stopping criteron such as a maximum number of iterations may also be used.

In order to illustate the effectiveness of the proposed algorithms a system model will be developed and then used to test the comparative effectiveness of the approach in a varied of simulated environments. Firstly we let NP S be the average power available for the transmission of N symbols. Due to the actual channel realisation being unknown at the transmitter, the S n are optimally chosen as independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and covariance E{ | S n " | 2 } = P s , i.e. the available transmit power is allocated uniformly across the subcarriers.

The wireless medium is described as a tapped delay line with L taps, whose I th tap is matrix-valued and denoted by H 1 e C w tt . The (i, j)" 1 entry of H 1 corresponds to the channel gain between the / Λ transmit antenna and the /'* receive antenna at lag / . The channel is assumed to be quasi-static and remains fixed for the duration of one codeword, which is composed of several OFDM symbols, but may change between codewords. The spatial correlation between the antenna elements is assumed to obey the Kronecker model i.e. H 1 ~ CM ( 0 M xM , (TJR 1x , ® R n , I , where 0 pxq is the all-zero matrix with p rows and q columns, (T J 1 l i ■s the average power delay profile, ® denotes the Kronecker product, and R 1x l € C a a and R n t € C n n are the transmit and receive covariance, respectively. Furthermore, the channel taps H 1 are assumed to be mutually uncorrelated. Note that the

Kronecker correlation model was demonstrated to potentially suffer from deficiencies in accurately capturing the channel properties in certain situations (when using large numbers of antennas). However, it is still widely used in the literature due to its simplicity and tractability, and is therefore also adopted in this work. For the results presented, we assume R 01 1 and R n l to be fixed for all channel taps and consequently, drop the subscript Z . The covariance matrices R tx and R n are assumed to have Toeplitz structure with exponentially decaying entries

The level of spatial correlation is determined by the parameter p , which takes on values between zero and one. The spatially uncorrelated case corresponds to p = 0 , whereas p = 1 results in full correlation.

As presented earlier, the received signal vector on subcarrier n after removal of the CP and OFDM demodulation can be represented as: x = n n s n +w n , where M^ e C a models the combined effect of noise and interference, and is the discrete frequency response of the channel. Without loss of generality we assume that only a single interfering transmit terminal is present (the generalisation to several interferers is straightforward).

Let the multiple-access interference and noise affecting the n' h subcarrier be given by

W n = K^ n +Z K . where Z n is i.i.d. noise ~CΛ/jO M - A ^J 2 - M ) ' ^ n the interfering symbol, and Η^ Dt n the channel from the interferer to the receiver. In analogy to JST n , we define with h mt l being the I th of L 1n , channel taps. We assume h mt l ~ CAf(O M xP ^ n , ,/ ^i nt,rx ) ' where σ L , ι and R im n denote the power delay profile and the receive covariance, respectively. Since the interfering signal arrives at the same antenna array as the desired signal, it is reasonable to assume ^i nt rx — R n ■ T fle interfering symbols X n are assumed to be i.i.d. zero-mean Gaussian with variance

E{ I X n | 2 } = cr^ . The random variables S n , X n , and Z n shall be mutually uncorrelated. Under these assumptions, W£ is uncorrelated across subcarriers.

The average received SINR of the n' h subcarrier is given as

£(k H W> | ! ) u n "R w u, where R w denotes the interference-plus-noise covariance matrix: n

R w = E(W n W n " } = <?<„,„<„ + . For later use we define average signal-to-noise ratio (SNR) and signal-to-interference ratio (SIR) as

SNR = P s I and SIR = P s I σf nt , respectively.

The average received SINR previous to OFDM demodulation (symbolwise beamforming), is given by L-I

E{\ Yu H H lV s'[k-l] \ 2 } SINR, pre - ^

E{| « H «OT I 2 }

^ | « H H,v | 2 SU u»R w u ' where R w =Ε{w[k]w H [k] } . Using the assumptions made about W n earlier is easily verified.

In the following, we will investigate the performance of symbol-wise beamforming for different degrees of spatial correlation, frequency selectivity, and multiple-access interference. Primarily, subcarrier-wise beamforming is used as reference system but in some cases a comparison will also be made with full MMO, i.e. with a MIMO system that performs spatial multiplexing, rather than beamforming. Performance will be assessed in terms of outage capacity, which is defined as the maximum information rate X out p that is supported by 100(1- p)% of all channel realisations:

Pr(j ≤ T out J = p with p denoting the outage probability. Results were obtained for N =64 , M tt =M n =3 , and

SNR=IOdB , and unless stated otherwise, were generated from an ensemble of 104 channel realisations. Furthermore, we assume the average power delay profile to be uniform with

and a number of L=2 channel taps. We will also consider an exponential power delay profile with

L=12 taps. Initially, we also assume that no multiple-access interference is present, i.e. W n =Z n . The effect of multiple-access interference will be investigated below. Plots will be annotated by maxSINRsc and maxSINRsym, to distinguish the two SINR maximising beamforming schemes that operate on a subcarrier or OFDM symbol basis.

As mentioned earlier, the proposed algorithm may not find the global maximum. Figure 3 illustrates the convergence behaviour of the algorithm for a randomly generated channel by plotting SINR re resulting from the current iterates u[q] and v[q] . Eigenvalue decompostion was implemented using MATLAB's EVD function eig. Each line corresponds to one of 200 random initialisations v[0] and clearly, two distinct convergence levels can be observed. Figure 3 shows convergence behaviour for a fixed 3x3 two-tap equal-gain Rayleigh channel without spatial correlation ( /7=0 ) at SNR=IOdB and 200 random initialisations v[0] at 60GHz. Despite the potential convergence to local maxima becoming obvious when investigating individual channel realisations, it was established through extensive Monte Carlo simulations using 1000 initialisation per realisation that the effect on ensemble results is only minor. Always using the antenna weight vectors u and v corresponding to the highest detected convergence level vs. using a single random initialisation results in an almost identical distribution of the mutual information. Our observations suggest that local maxima occur only with low probability and in addition, are mostly rather close to the globally optimal solution. Moreover, we have observed the probability of converging to a local maximum to decrease with increasing spatial correlation and decreasing frequency selectivity. All results presented in the following were obtained for random initialisations.

It can also be seen from Figure 3 that for this particular channel, the algorithm converges fairly quickly and comes close to the final convergence level after only a few iterations. Again, simulations have demonstrated this to also be true for most other realisations. In order to allow for convergence, the proposed algorithm was iterated 30 times for all results presented in the following.

Whilst on the topic of convergence, it is noted that the power method estimation of u and v need not be performed until convergence. Rather iterations may be stopped after N PM u and N PM , V iterations respectively and N PM u and N PMiV may be identical. A scheduling question can arise relating to the choice of N PM , U and N PMiV . That is, is it more efficient to perform fewer inner iterations and more outer iterations, or vice versa. Simulations have shown that where N PM , U and N PM , V are identical a small value is preferable (ie fewer inner iterations and more outer iteration) as this on average this leads to faster overall convergence. For example for values of N PM u = N PM , V = N PM equal to 10, 5, 4, 3, 2 and 1 respectively the mean number of outer iterations was 2.95,3.08, 3.20, 3.47, 4.25 and 7.1 corresponding to mean total number of iterations of 59.0, 30.8, 25.6, 20.8, 17.0 and 14.2. These numbers depend upon the actual configuration and simulation conditions, but the general approach can be used to guide the choice of N PM u and N PM , V in a specific implentation. In general choosing N PM u and N PM v sufficiently large to allow for convergence of the PM to the dominant eigenvector of the individual eigenproblems appears unnecessary.

As described earlier, the performance gap between subcarrier-wise and symbol-wise beamforming can be expected to be small when the channel is highly correlated in space. The results in Figure 4 confirm this behaviour. Figure 4 shows distribution of mutual information for different levels of spatial correlation (/9=0,0.4,0.6,0.8,0.9,1 ) for 3x3 two-tap equal-gain Rayleigh channels at SNR=IOdB and results were generated from an ensemble of 10 4 channel realisations. The figure shows the complementary cumulative distribution function (CCDF) of the mutual information for symbol-wise and subcarrier-wise beamforming (solid and dashed lines, respectively) for different levels of spatial correlation. Mutual information is computed according to i N-I J = -J]IOg 2 (H-SINR n ) . Results are included for p = 0 , 0.4 , 0.6 , 0.8 , 0.9 , and 1 , and the

N n=Q direction of increase is indicated by the arrows. For low correlation, the CCDFs for both beamforming schemes are considerably steeper than for large p , indicating a high degree of spatial diversity. The performance loss of symbol-wise beamforming relative to subcarrier-wise beamforming for p = 0 is about 1 bps/ Hz at the 90% mark, which corresponds to an outage probability of 10%. With increasing correlation, the performance gap decreases. At the same time the CCDFs become less steep, indicating the reduced availability of spatial diversity. In the extreme case of a fully correlated channel ( p — 1 ), the curves of the two beamforming schemes coincide (thick black line). It should be pointed out that even though symbol-wise beamforming is in general inferior to subcarrier-wise beamforming, this scheme is not limited to array gain but can still exploit spatial diversity to some extent. As a result, it clearly outperforms SISO systems as will be illustrated by the following results.

Figure 5 illustrates how the 10% outage capacity changes with increasing spatial correlation parameter p for 3x3 two-tap equal-gain Rayleigh channels at SNR = 1OdB . Along with the curves for the two beamforming schemes, results are included for the SISO case and also for a MIMO system that does not perform beamforming. The two cases that the current channel realisation is known or unknown to the transmitter are considered. If it is known, the available transmit power can be optimally allocated using waterfilling (WF) and MIMO capacity can be computed. In the second case, an equal power (EP) allocation is optimal and MIMO capacity can be found.

As described previously, the gap between the beamforming schemes is largest for p = 0 and then continually decreases with growing p , until it disappears completely for full correlation. The outage capacity of the SISO link is significantly lower than that of both beamforming approaches for the entire p range. Whilst the SISO system is not affected by spatial correlation, MIMO with or without WF suffers significantly. This effect has been studied extensively in the literature and can be attributed to a decreasing number of channel eigenmodes being available for spatial multiplexing. In the limit p — > 1 , only a single non-zero eigenmode is left. The results in Figure 5 show that at p = 1 , subcarrier-wise and symbol-wise beamforming have virtually the same outage capacity as MIMO with WF. In fact, subcarrier-wise beamforming with WF was shown to be MDVIO capacity achieving for highly correlated channels. Whilst MIMO with WF avoids allocating power to eigenmodes that are too weak for data transmission, MIMO with EP is unable to do so. As a result, the MIMO outage capacity even drops below that of beamforming once a certain p threshold is exceeded.

In this section, we investigate how well symbol-wise beamforming can cope with multiple-access interference. The performance is compared to that of subcarrier-wise beamforming and a conventional phased array. In the case of phased array beamforming, we assume uniform linear arrays, whose elements are spaced half a wavelength apart and have omnidirectional radiation patterns. The general system structure is the same as for symbol-wise beamforming (see Figure 2) but the vectors u and v are restricted to v = [1 e w ... e j(M< * ~1)! "] T with phase shift ψ and u accordingly with an independent phase shift ξ , The best combination of ψ and ζ is found by means of an exhaustive search, limiting the phase shifts such that the beam direction can be steered with 7.5° resolution (this was found to be a good compromise between complexity and accuracy). For all possible combinations, SINR is evaluated and ψ and ξ corresponding to the maximum are picked.

Furthermore, the channels from the desired source to the receiver H 1 and from the interfering terminal to the receiver h mt t are both modelled with L = Z 1n , = 2 taps of equal strength as described in the introduction of this section. The Kronecker correlation model gives rise to a mean direction of arrival (DoA) of 0° , which corresponds to the broadside direction if a linear antenna array is used. Here, the mean DoA of H 1 is left unchanged, whilst it is changed to φ DoA and assuming a uniform linear array with half wavelength spacing and uniform radiation pattern.

Figure 6 shows the outage capacity of subcarrier-wise and symbol-wise beamforming plotted vs. the mean DoA φ OoA of the interfering signal for 3x3 two-tap equal-gain Rayleigh channels.. The SISO case is included for reference. In these simulations, the average SNR and SIR were fixed at

SNR = 1OdB and SIR = -5dB , respectively. Without spatial correlation ( /7 = 0 , dash-dotted lines), the impinging signals experience an angular spread across the entire range [0, 2π) . As a result, the mean DoA is irrelevant and the outage capacity for both beamforming schemes is constant. For p = 0.6 (solid lines) a performance degradation can be observed in the lower φ DoA range, which becomes even more prominent for high correlation ( p = 0.99 , dashed lines). Lower angular spread due to higher spatial correlation causes the signals to be more limited around their DoAs, and thus, a large fraction of the total signal power comes from this constant direction. This results in higher sensitivity of the receiver against the interference DoA. Particularly for p = 0.99 and when φ DoA = 0° , i.e. when desired and interfering signal arrive from the same direction, there is no way to avoid the interference without rejecting the desired signal power at the same time. The effect is that the outage capacities of both schemes almost revert to that of a SISO link. A SISO system has no means to mitigate the interference and therefore, suffers from the full amount of interference regardless of its DoA. Whilst the performance of subcarrier-wise beamforming deteriorates with increasing correlation (the same effect was observed in the results for spatial correlation), symbol-wise beamforming actually benefits from it provided that ^? DoA is sufficiently large. The curves for subcarrier-wise and symbol-wise beamforming move closer to each other when p grows, which is in line with the implications of Theorem 1 and the results presented previously (In fact, both curves coincide for p = 1 ).

Figure 7a shows the extent at which the two beamforming schemes can mitigate multiple-access interference for varying strengths of the interfering terminal. The 10% outage capacity is plotted as a function of the SIR for Figure 7a for moderate spatial correlation ( p = 0.6 ), and Figure 7b for high spatial correlation (/? = 0.99 ). Both Figures 7a and 7b 10% outage capacity vs. SIR for 3x3 two-tap equal-gain Rayleigh channels at SNR=IOdB and the mean direction of arrival for desired signal chosen from a uniform distribution between ±30° and the the mean direction of arrival for the interfering signal chosen from a uniform distribution between 60° and 120°. Note that the interference gets stronger from the right to the left. In both cases subcarrier-wise beamforming (maxSESTRsc) copes best and is able to maintain an almost constant outage capacity even for high interference levels. For moderate correlation, the relative performance loss of symbol-wise beamforming (maxSINRsym) increases with increasing interference power since the scheme cannot suppress the interference on a subcarrier-by-subcarrier basis. Nevertheless, this scheme (maxSINRsym) clearly outperforms the interference unware case maxSNRsym, a SISO system and also a communication system using conventional phased array beamforming at both link ends (Phased Array). The outage capacity of these two approaches deteriorates quickly with growing interference levels. The reduced flexibility of how to choose u and v significantly limits the ability of the phased array to mitigate interference and to exploit spatial diversity. Therefore, the gain of symbol-wise beamforming relative to the phased array is considerable, even for weak interference. Additionally, the reduced flexibility of how to choose u and v significantly limits the ability of the phased array to mitigate interference. From Figure 7b it can be seen that in the highly correlated case, symbol-wise beamforming performs close to the optimal beamforming approach and for positive values of SIR, the outage capacity loss is withing 0.5bps/Hz.. Its outage capacity diverges from that of subcarrier-wise beamforming only for strong interference.

These results illustrate that methods based upon jointly estimating a pair of transmit and receive beamfoπning vectors by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SINR pre ) are suitable for use in the presence of multiple access interfernce. Thus this approach provides an efficient method and system for for estimating and performing symbol wise beamforming in MIMO-OFDM systems.

Figure 14 provides an flowchart of a method for joint estimation of transmit and receive beamfoπning vectors 1400 according to an embodiment of the invention. The which comprises jointly estimating a pair of transmit and receive beamforming vectors for use over a set of subcarriers and a block of time- domain symbols in a subcarrier-based MIMO system by iteratively maximising an expression for the Signal to Interference Plus Noise Ratio in the MIMO channel at the input of the receive side demodulator (SDSfR pre ).

Figure 15 provides a flowchart of the method in which wherein maximisation of the expression for SINRpre is expressed as a generalised eigenproblem in which the receive beamforming vector is an eigenvector and a standard eigenproblem in which the transmit beamforming vector is an eigenvector. The step of jointly estimating can be further outlines as 1500 as comprising: initialising an estimate of at least one of the pair of transmit and receive beamforming vectors

1510; updating an estimate of the receive beamforming vector by solving the generalised eigenproblem using the current estimate of the transmit beamforming vector 1520; updating an estimate of the transmit beamforming vector by solving the standard eigenproblem using the current estimate of the receive beamforming vector 1530; and testing if a predefined stopping criterion is met 1540 and if the result is no 1542 then repeating the two updating steps otherwise if the stopping criterion is met 1544 then stopping 1550.

In another aspect the approach may be applied in other subcarrier-based multiple input multiple output (MIMO) communication systems such as OFDMA and SC-FDMA systems. Such systems utilise a set of subcarriers and use the beamforming vectors for a block of time-domain symbols (which in the context of OFDM corresponds to an OFDM symbol). In an orthogonal frequency-division multiple access (OFDMA) system, the wireless medium is shared among multiple users by assigning them disjoint subsets of subcarriers. For simplicity, identical transmit powers per user are assumed, which are evenly shared across each user's subcarriers. Furthermore, it is assumed that each user is assigned a contiguous group of subcarriers. A schematic of an OFDMA system is presented in Figure 8. In an OFDMA system each user terminal performs only a single IDFT regardless of the number of users. Each of the transmitted signals is characterised by a partial spectral occupancy, whereas the received signal occupies the entire frequency band and is processed by the receiver depicted in Figure 8. Therefore, most of the processing is carried out at the access point/base station. Numbering follows figure 1 with group transmit weights 820 and receive weights 890, 892 and demultpexing 892.

In another aspect, symbol wise beamforming may be extended to a system in which sets of weights are used for groups of carriers and will be referred to as subcarrier-group-wise beamforming. One such system is shown schematicaly in Figure 9. Numbering follows figure 1 with transmit multiplexing 910, group transmit weights 920, 922, group receive weights 990, 992 and receiver demultiplexing 994. First, the input symbol stream is assigned to N grp separate streams, each of which is multiplexed across N q adjacent subcarriers. The remaining subcarriers are filled with null symbols. After undergoing an IDFT, demultiplexing and CP insertion, the signals are passed on to N grp beamforming stages. The spectral occupancy of the signals at the input of these beamforming stages is indicated in Figure 8. Finally, transmission occurs from M Tx separate antennas, each of which is fed by the coherent sum of the N grp signals destined for this particular antenna. The receiver basically performs the inverse process as shown in Figure 9. Note that symbols which are received on subcarriers not part of the group of interest are simply discarded. This approach requires one DFT per group and is primarily attactive when using large antenna arrays. As in the OFDMA case, the orthogonality of subcarriers allows optimisation of beamforming weights for each group to be carried out seperately.

Symbolwise beamforming for single-user systems can be readily extended to the OFDMA case. As the groups are disjoint, the OFDMA case for a group corresponds to an OFDM case with a reduced number of subcarriers. The problem then becomes estimation of pairs of transmit and receive beamforming weights for each group. However as the groups are disjoint then joint estimation of the beamforming weights for each disjoint group can be performed seperately (or sequentially) for each of the groups forming the OFDMA system. Similar reasoning applies to the SC-FMDA case.

The perfomance of the methods in OFDMA and subcarrier-group-wise beamforming was investigated via computer simulations comprising 10 4 realisations of a spatially uncorrelated 3x3 MIMO channel modelled using a Kronnecker correlation mode) with 12 taps, and an exponential power delay profile (PDP) with delay spread parameter K σ = 2T S . This channel model was selected as the absence of spatial correlation and the comparably high level of frequency selectivity result in a large performance gap between symbol-wise and subcarrier-wise beamforming. Figure 10 presents simulation results of mutual information complementary cumulative distribution functions (CCDFs) for OFDMA and subcarrier-group-wise beamforming for a range of subcarrier group sizes N grp corresponding to 1, 2, 4, 8, and 64 subcarriers in each group. For reference the SISO case is also presented. Note that for the computer simulations, independent channel realisations where selected for each user to reflect the spatial separation of the user terminals. Figure 10 clearly shows a general trend of increasing performance with growing group size N grp in both the OFDMA and subcarrier-group-wise case.

In another aspect of the invention, an approach based on maximisation of mutual information using a power method is considered. As was stated above the method of Lagrange multipliers can be used to obtain a first condition for maximum mutual information. Whilst a closed form solution is not obtainable, it is noted that this condition does resemble an eigenproblem:

where uniform power allocation has been assumed (μ n = 1 ).However, unlike in a normal eigenproblem, the matrix C is a function of the vector v , and the eigenproblem is non-linear in its eigenvector v . As conventional eigendecomposition techniques cannot be applied to such problems, the modified power method (MPM) may be used. When compared to the original PM, the MPM incorporates an additional step of updating the matrix at the beginning of each iteration. Given that the matrix changes in each iteration, it is not possible to prove convergence of the MPM, however simulation results indicate that the MPM algorithm generally converges.

For the case where the interference-plus-noise covariance matrices are constant across all subcarriers,

R w = R w for all n, an optimality condition for ύ can be written as:

U

where Assuming an invertible R w this can be rewritten as: = γu which can be solved via a modified power method. This gives rise to an iterative algorithm for maximisation of the mutual information which is shown below in Table 3 which is called maxMIsym for maximisation of mutual information in the symbolwise case. Table 3 - Iterative Maximisation of Mutual Information using the modified power method (MPM)

0 Set q <- 0 1 Initialise u[q] and v[q] repeat repeat

2.1 Update D(u[q],v[q])

2.2 «tø + l] <- R^D{u[q\,v[q\)u[q\ 2.3 <? <- tf + l until Np M . u iterations reached repeat

3.1 Update C(«[q],v[q])

3.2 vtø + l] <- C(iιtø],vtø])vtø] 3.3 q <^ q + l until Np M ,v iterations reached until stopping criterion met

While maxMIsym maximises the mutual information, this algorithm comes at the expense of higher computational complexity than the SINR pre -maximising algorithm maxSINRsym. Not only is the computation of the matrices C and D much more involved than that of A and B , but they also need to be updated in every MPM iteration. As in the maxSINRpre case using the power method, simulations indicated that in general choosing N PM u and N PM,v sufficiently large to allow for convergence of the MPM to the dominant eigenvector of the individual eigenproblems appears unnecessary.

The above method is illustated in Figure 16. Figure 16 shows a flowchart of a method for joint estimation of a pair of transmit and receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MMO) communication system. At step 1600 the method comprises jointly estimating a pair of transmit and receive beamforming vectors for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol, in a OFDM MIMO system by maximising an expression for the mutual information (MI) in the MDvIO channel, wherein the expression for MI is a function of at least the transmit and receive beamforming vectors, a MIMO channel response and a interference plus noise covariance matrix, and the maximisation is performed assuming a constant interference plus noise covariance matrix for all subcarriers and uniform power allocation, and the joint estimation is performed using a modified power method (MPM).

In another aspect of the invention the method is applied to the case of asymmetric beamforming in which one side performs subcarrier wise beamforming and the other side performs symbol wise beamforming. Typically the side with greater computational resouces would perform subcarrierwise beamforming. Two cases are considered - symbol wise transmit and subcarrier wise receive beamforming and the opposite case of symbol wise receive and subcarrier wise transmit beamforming. The approach is to dervive an optimisation problem in terms of the mutual information and then through the method of Lagrange multipliers obtain two optimisation equations. As will be presented closed form solutions are obtainable for one, but not both equations, and a modified power method is used to obtain an iterative solution to the second equation.

The optimisation problem for symbol wise transmit and subcarrier wise receive beamforming may be expresed as:

max iirmniissee X(u n ,v) v> π ≤ l Using the method of lagrange multipliers a generalised eigenproblem can be derived for the optimal receive subcarrier wise beamforming vectors U n where U n «= R^ H n V (see Andre Pollok, "Multi- Antenna Techniques for Millimetre-Wave Radios", PhD Thesis, University of South Australia, 2009 for more details). The transmit beamforming vector may also be expressed as

Whilst conventional EVD methods cannot be applied due to the non-linearity in v , the dominant eigenvector of Qv) can be computed numerically using the modified power method. When compared to the original PM, the MPM incorporates an additional step of updating the matrix at the beginning of each iteration. The mutual information of the considered beamforming system is therefore maximised by the following procedure: in a first step, the optimal vector v is computed via the MPM. Using the resulting solution, the optimal receive beamforming vectors U n are then found from U n °c R^ ΗJ> . This solution will subsequently be referred to as maxMI-symTx-scRx. The optimisation problem for subcarrier wise transmit and symbol wise receive beamforming may be expresed as:

maximise * >o v ≤ 1 Using the method of lagrange multipliers the closed form solution for the optimal transmit subcarrier

wise beamforming vectors V n is given by V n = . .. = . == (see Pollok 2009 for details).

The optimisation equation for the receive beamforming vector may also be expressed as r s ψ KM H K H u =

Nlog2 iά ύ H (R w +P s Η,ΗΪ)u

Solution for arbitrary R w appears difficult, however in the AWGN case with R w = O"ll Mn: simplies n n the above equation to a non linear eigenproblemD(ά)ά = γu where

γ is a real valued scalar. The non linear eigenproblem can be solved numerically for ύ using a modified power method.

The perfomance of the beamforming methods in assymetric MIMO systems was investigated via computer simulations comprising 10 4 realisations of a spatially uncorrelated 3x3 MMO channel modelled using a Kronnecker correlation mode) with 12 taps, and an exponential power delay profile (PDP) with delay spread parameter K σ = 2T S at SNR=IOdB. Figure 11 shows simulation results of mutual information CCDFs for the two assymetric beamforming cases in the absence of spatial correlation ( p^ = p n = 0 ). In this case the multipath components are departing in / arriving from arbitrary directions, i.e DoDs and DoAs are distributed uniformly between zero and 360°. It can be seen that for these spatially uncorrelated channels, the curves for maxMI-scTx-symRx and maxMI- symTx-scRx coincide. Due to the absence of spatial correlation, the channel does not impose any preference on which of the two terminals should be equipped with a subcarrier-wise beamforming stage Transmit side correlation was performed using a uniform direction of departure (DoD) distribution in the range ±30° and receive side corelation was performed using a uniform direction of arrival (DoA) distribution in the range ±30°. Figure 12 shows simulation results of mutual information CCDFs for the two assymetric beamforming cases in the case that the transmit side correlation was set to 0.9 (P 0 , = 0.9, p n = 0).Figure 13 shows simulation results of mutual information CCDFs for the two assymetric beamforming cases in the case that the receive side correlation was set to 0.9

The results in Figure 12 show that for this channel, maxMI-symTx-scRx clearly outperforms maxMI- scTx-symRx: the larger degree of freedom of subcarrier-wise beamforming is more beneficial at the receiver, where MPCs arrive from arbitrary directions and where spatial correlation is absent. When the terminals swap their role, the channel conditions are reversed, i.e. now the spatial correlation at the receiving terminal is high and the angular spread low. Consequently, maxMI-scTx-symRx can be expected to outperform maxMI-symTx-scRx in this case. This behaviour is confirmed by the results in Figure 13.

In all cases simulations show that the performance of assymetric beamforming systems lie between the performance of symbol wise (maxSINRsym) and subcarrier wise beamforming (maxSINRsc) cases.

Methods for performing estimation of beamforming vectors for assymetric cases are shown in Figures 17 and 18.

Figure 17 shows a flowchart of a method for joint estimation of a transmit beamforming vector and a plurality of receive beamforming vectors for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system. At step 1700 the method comprises: jointly estimating a transmit beamforming vector and a plurality of receive beamforming vectors by maximizing an expression for the mutual information, wherein the transmit beamforming vector is estimated using a modified power method (MPM) and the plurality of receive beamforming vectors is estimated using a closed-form expression that is a function of the transmit beamforming vector and the transmit beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of receive beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers. Figure 18 shows a flowchart of a method for joint estimation of a plurality of transmit beamforming vectors and a receive beamforming vector for use in a orthogonal frequency division multiplexing (OFDM) multiple input multiple output (MIMO) communication system. At step 1800 the method comprises: jointly estimating a receive beamforming vector and a plurality of transmit beamforming vectors by maximising an expression for the mutual information, wherein the receive beamforming vector is estimated using a modified power method (MPM) and the plurality of transmit beamforming vectors is estimated using a closed-form expression that is a function of the receive beamforming vector and the receive beamforming vector is for use over a set of OFDM subcarriers and a block of time-domain symbols which correspond to one OFDM symbol and each of the plurality of transmit beamforming vectors is for use with one of the subcarriers in the set of OFDM subcarriers.

The various aspects and methods of the present invention may also be embodied in software stored as instructions on a computer readable medium, which can cause a computer or processor or other machine to carry out the various aspects of the invention described herein. Such media can include hard drives, removable memory storage, solid state memory device, Flash memory devices, MultiMedia Card, Secure Digital Card, Compact Discs (CDs) and Digital Video Discs (DVDs).

The various aspects and methods of the present invention may also be embodied in hardware. For a hardware implementation, the estimation of beamforming vectors may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, other electronic units designed to perform the functions described herein, or a combination thereof. These may be part of receivers, transmitters or other apparatus, and may be implemented by processors in communication with memory or other specific modules. In some cases processing may be distributed between transmit and receive sides or elements of a system and information may be exhanged between the sides in order to perform the method as would be apparent to the person skilled in the art. In this cases the apparatus may include a communication meands for providing this information to the other side. For example the joint estimation could be performed by the transmit side provided information was provided from the receive side to allow estimation of the channel matrix or matricies, and the interference plus noise matrix or matricies (depending on whether time or frequency domain implementations are used).

In practice, the channel and the covariance of noise and interference are not perfectly known but need to be estimated by the receiver. Depending on where the estimation of these quantities is carried out most conveniently, the algorithm for symbol-wise beamforming can be implemented in either the time or frequency domain. Depending on if the estimation of these quantities is more conveniently carried out in the time or frequency domain, the algorithm for symbol-wise beamforming can be implemented in the more suitable domain. As would be apparent to the person skilled in the art, SINR pre can either be expressed in terms of the time-domain or the frequency-domain quantities. Both expressions have the same form and thus, lead to an equivalent algorithm. To obtain the frequency-domain version of maxSINRsym, H 1 and R w simply need to be replaced by Ti n / VN and ∑ n=0 ^vv / N (and the summations need to be carried out over all subcarriers rather than all channel taps), the channel from the interferer to the receiver.

Furthermore, maxSINRsym may be deployed in a distributed fashion if the channel is reciprocal, although it is noted that the assumption of channel reciprocity is only realistic when interference is absent (spatially separated terminals are likely to be subject to different interference signals). A major advantage of this approach is that each terminal only needs to estimate a vector channel rather than the full channel matrix H 1 . In the initial phase, the first terminal uses an arbitrary beamforming vector v

. The second terminal estimates the effective vector channel b t (v) =H l v and computes the SINR pre -

maximising receive vector u as the dominant eigenvector of B(v) = P s ∑b,(v )bf ( v ) .

Z=O

Due to channel reciprocity, u is also used for transmission from the second terminal. Analogously, the first terminal then estimates the effective vector channel a t (u) =H H u and updates v based

L-I on A(u) = P s ^ j a l (u)a^(u) . This procedure corresponds to one maxSINRsym iteration and may be

7=0 repeated a number of times. Also note that whilst the aforementioned procedure depends on H 1 , it is independent of the AWGN variance σ w 2 and thus, can still be applied even if the two terminals have different noise figures.

It will be appreciated that a particular advantage of the present invention over the prior art is that it is able to be used effectively in the presence of multiple-access interference. For example, multiple- access interference arises when multiple transmitters are operating in the same spectrum without cooperation such that the signals launched from the other transmitters cause interference for the desired signal. Such application in the prior art has not been addressed by means of symbol-wise beamforming. A further advantage of the present invention is that knowledge of the antenna array geometries (i.e. orientation and location of the individual antenna elements) and antenna types is not required.

A further advantage of the present invention is that the methods described may be implemented in existing MIMO-OFDM systems without requring additional hardware, thereby being a cost-efficient implementation of beamforming in MIMO-OFDM systems. The methods may also be extended to rlated

Throughout the specification and the claims that follow, unless the context requires otherwise, the words "comprise" and "include" and variations such as "comprising" and "including" will be understood to imply the inclusion of a stated integer or group of integers, but not the exclusion of any other integer or group of integers.

The reference to any prior art in this specification is not, and should not be taken as, an acknowledgement of any form of suggestion that such prior art forms part of the common general knowledge.

It will be appreciated by those skilled in the art that the invention is not restricted in its use to the particular application described. Neither is the present invention restricted in its preferred embodiment with regard to the particular elements and/or features described or depicted herein. It will be appreciated that the invention is not limited to the embodiment or embodiments disclosed, but is capable of numerous rearrangements, modifications and substitutions without departing from the scope of the invention as set forth and defined by the following claims.