American Institute of Mathematical Sciences

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April  2016, 12(2): 595-607. doi: 10.3934/jimo.2016.12.595

A global optimal zero-forcing Beamformer design with signed power-of-two coefficients

Bin Li 1, , Hai Huyen Dam 1, and  Antonio Cantoni 2,

1. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia, Australia
2. 

School of Electrical, Electronic and Computer Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

Received  November 2014 Revised  March 2015 Published  June 2015

In this paper, we investigate a zero-forcing beamformer design with signed power-of-two coefficients for rural applications. In this design, the minimum user information rate is taken as the performance measure, while a practical system design constraint, the per-antenna power constraint, is imposed. The problem is formulated as a constrained zero-one integer programming problem. Based on a transform between two different integer spaces, the problem is transformed into an equivalent constrained integer programming problem. A global optimal two-stage design is proposed for solving the problem. In the first stage, a polynomial time quantization method is applied to obtain an initial design. In the second stage, an auxiliary function method is used to find the global optimal design. For illustration, numerical examples under several different scenarios are studied and the results are compared with those obtained by an existing method. Furthermore, the impact of the mutual interference terms in the performance measure is also studied.
Keywords: signed power-of-two (SPT), integer programming, zero-forcing beamforming (ZFBF), auxiliary function method, MIMO system, per-antenna power constraint (PAPC)..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Bin Li, Hai Huyen Dam, Antonio Cantoni. A global optimal zero-forcing Beamformer design with signed power-of-two coefficients. Journal of Industrial and Management Optimization, 2016, 12 (2) : 595-607. doi: 10.3934/jimo.2016.12.595
References:
[1]

G. Caire and S. Shamai, On the achievable throughput of multiatenna Gaussian broadcast channel, IEEE Trans. Inf. Theory., 49 (2003), 1691-1706. doi: 10.1109/TIT.2003.813523.

[2]

H. H. Dam and A. Cantoni, Interior point method for optimum zero-forcing beamforming with per-antenna power constraints and optimal step size, Signal Process., 106 (2015), 10-14. doi: 10.1016/j.sigpro.2014.06.028.

[3]

H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, FIR variable digital filter with signed power-of-two coefficients, IEEE Trans. Circuits Syst., 54 (2007), 1348-1356. doi: 10.1109/TCSI.2007.897775.

[4]

Z. G. Feng and K. L. Teo, A discrete filled function method for the design of FIR filters with signed-powers-of-two coefficients, IEEE Trans. Signal Process., 56 (2008), 134-138. doi: 10.1109/TSP.2007.901164.

[5]

K. Karakayali, R. Yates, G. Foschini and R. Valenzuela, Optimal zero-forcing beamforming with per-antenna power constraints, IEEE International Symposium on Information Theory, Nice, France, (2007), 101-105.

[6]

S. R. Lee, J. S. Kim, S. H. Moon, H. B. Kong and I. Lee, Zero-forcing beamforming in multiuser MISO downlink systems under per-antenna power constraint and equal-rate metric, IEEE Trans. Wireless Commun., 12 (2013), 228-236. doi: 10.1109/TWC.2012.120312.120332.

[7]

B. Li, H. H. Dam, A. Cantoni and K. L. Teo, A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints, Optim. Lett., 8 (2014), 1829-1843. doi: 10.1007/s11590-013-0673-y.

[8]

B. Li, H. H. Dam, A. Cantoni and K. L. Teo, A first-order optimal zero-forcing beamformer design for multiuser MIMO systems via a regularized dual accelerated gradient method, IEEE Commun. Lett., 19 (2015), 195-198. doi: 10.1109/LCOMM.2014.2381245.

[9]

B. Li, H. H. Dam, A. Cantoni and K. L. Teo, Some interesting properties for zero-forcing beamforming under per-antenna power constraints in rural areas, J. Glob. Optim., 10.1007/s10898-014-0237-4. doi: 10.1007/s10898-014-0237-4.

[10]

B. Li, C. Z. Wu, H. H. Dam, A. Cantoni and K. L. Teo, A parallel low complexity zero-forcing beamformer design for multiuser MIMO systems via a regularized dual decomposition method, IEEE Trans. Signal Process., under review.

[11]

H. H. Dam, A. Cantoni and B. Li, A fast low complexity method for optimal zero-forcing beamformer MU-MIMO system, IEEE Signal Process. Lett., 22 (2015), 1443-1447.

[12]

B. Li, H. H. Dam, K. L. Teo and A. Cantoni, A Low Complexity Optimization Algorithm for Zero-Forcing Precoding under Per-antenna Power Constraints, The 40th IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, (2015), to appear.

[13]

B. Li, H. H. Dam, K. L. Teo and A. Cantoni, A Survey on Zero-Forcing Beamformer Design under Per-antenna Power Constraints for Multiuser MIMO Systems, 2015 IEEE International Conference on Digital Signal Processing, Singapore, (2015), to appear.

[14]

D. Li, J. Song and Y. C. Lim, A polynomial-time alogorithm for designing digital filters with power-of-two coefficients, Proc. IEEE Int. Symp. Circuits Syst., (1993), 84-87.

[15]

Y. C. Lim, Design of discrete-coefficient-value linear phase FIR filters with optimum normalized peak ripple magnitude, IEEE Trans. Circuits Syst., 37 (1990), 1480-1486. doi: 10.1109/31.101268.

[16]

Y. C. Lim and S. R. Parker, FIR filter design over a discrete powers-of-two coefficients space, IEEE Trans. Acoust. Speech Signal Process., 31 (1983), 583-591.

[17]

H. Lin, Y. Wang and X. Wang, An auxiliary function method for global minimization in integer programming, Math. Probl. Eng., 2011 (2011), 1-13. doi: 10.1155/2011/402437.

[18]

Y. Liu, An exterior point linear programming method based on inclusive nornal cone, Journal of Industrial and Management Optimization, 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825.

[19]

J. Quan, Z. Wu and G. Li, Global optimality conditions for some classes of polynomial integer programming problems, Journal of Industrial and Management Optimization, 7 (2011), 67-78. doi: 10.3934/jimo.2011.7.67.

[20]

A. Wiesel, Y. C. Eldar and S. Shamai, Linear precoding via conic optimizaiton for fixed MIMO receivers, IEEE Trans. Signal Process., 54 (2006), 161-176.

[21]

A. Wiesel, Y. C. Eldar and S. Shamai, Zero-forcing precoding and generalized inverses, IEEE Trans. Signal Process., 56 (2008), 4409-4418. doi: 10.1109/TSP.2008.924638.

[22]

C. J. Yu, K. L. Teo and H. H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Process., 95 (2014), 32-42. doi: 10.1016/j.sigpro.2013.08.005.

show all references

References:
[1]

G. Caire and S. Shamai, On the achievable throughput of multiatenna Gaussian broadcast channel, IEEE Trans. Inf. Theory., 49 (2003), 1691-1706. doi: 10.1109/TIT.2003.813523.

[2]

H. H. Dam and A. Cantoni, Interior point method for optimum zero-forcing beamforming with per-antenna power constraints and optimal step size, Signal Process., 106 (2015), 10-14. doi: 10.1016/j.sigpro.2014.06.028.

[3]

H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, FIR variable digital filter with signed power-of-two coefficients, IEEE Trans. Circuits Syst., 54 (2007), 1348-1356. doi: 10.1109/TCSI.2007.897775.

[4]

Z. G. Feng and K. L. Teo, A discrete filled function method for the design of FIR filters with signed-powers-of-two coefficients, IEEE Trans. Signal Process., 56 (2008), 134-138. doi: 10.1109/TSP.2007.901164.

[5]

K. Karakayali, R. Yates, G. Foschini and R. Valenzuela, Optimal zero-forcing beamforming with per-antenna power constraints, IEEE International Symposium on Information Theory, Nice, France, (2007), 101-105.

[6]

S. R. Lee, J. S. Kim, S. H. Moon, H. B. Kong and I. Lee, Zero-forcing beamforming in multiuser MISO downlink systems under per-antenna power constraint and equal-rate metric, IEEE Trans. Wireless Commun., 12 (2013), 228-236. doi: 10.1109/TWC.2012.120312.120332.

[7]

B. Li, H. H. Dam, A. Cantoni and K. L. Teo, A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints, Optim. Lett., 8 (2014), 1829-1843. doi: 10.1007/s11590-013-0673-y.

[8]

B. Li, H. H. Dam, A. Cantoni and K. L. Teo, A first-order optimal zero-forcing beamformer design for multiuser MIMO systems via a regularized dual accelerated gradient method, IEEE Commun. Lett., 19 (2015), 195-198. doi: 10.1109/LCOMM.2014.2381245.

[9]

B. Li, H. H. Dam, A. Cantoni and K. L. Teo, Some interesting properties for zero-forcing beamforming under per-antenna power constraints in rural areas, J. Glob. Optim., 10.1007/s10898-014-0237-4. doi: 10.1007/s10898-014-0237-4.

[10]

B. Li, C. Z. Wu, H. H. Dam, A. Cantoni and K. L. Teo, A parallel low complexity zero-forcing beamformer design for multiuser MIMO systems via a regularized dual decomposition method, IEEE Trans. Signal Process., under review.

[11]

H. H. Dam, A. Cantoni and B. Li, A fast low complexity method for optimal zero-forcing beamformer MU-MIMO system, IEEE Signal Process. Lett., 22 (2015), 1443-1447.

[12]

B. Li, H. H. Dam, K. L. Teo and A. Cantoni, A Low Complexity Optimization Algorithm for Zero-Forcing Precoding under Per-antenna Power Constraints, The 40th IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, (2015), to appear.

[13]

B. Li, H. H. Dam, K. L. Teo and A. Cantoni, A Survey on Zero-Forcing Beamformer Design under Per-antenna Power Constraints for Multiuser MIMO Systems, 2015 IEEE International Conference on Digital Signal Processing, Singapore, (2015), to appear.

[14]

D. Li, J. Song and Y. C. Lim, A polynomial-time alogorithm for designing digital filters with power-of-two coefficients, Proc. IEEE Int. Symp. Circuits Syst., (1993), 84-87.

[15]

Y. C. Lim, Design of discrete-coefficient-value linear phase FIR filters with optimum normalized peak ripple magnitude, IEEE Trans. Circuits Syst., 37 (1990), 1480-1486. doi: 10.1109/31.101268.

[16]

Y. C. Lim and S. R. Parker, FIR filter design over a discrete powers-of-two coefficients space, IEEE Trans. Acoust. Speech Signal Process., 31 (1983), 583-591.

[17]

H. Lin, Y. Wang and X. Wang, An auxiliary function method for global minimization in integer programming, Math. Probl. Eng., 2011 (2011), 1-13. doi: 10.1155/2011/402437.

[18]

Y. Liu, An exterior point linear programming method based on inclusive nornal cone, Journal of Industrial and Management Optimization, 6 (2010), 825-846. doi: 10.3934/jimo.2010.6.825.

[19]

J. Quan, Z. Wu and G. Li, Global optimality conditions for some classes of polynomial integer programming problems, Journal of Industrial and Management Optimization, 7 (2011), 67-78. doi: 10.3934/jimo.2011.7.67.

[20]

A. Wiesel, Y. C. Eldar and S. Shamai, Linear precoding via conic optimizaiton for fixed MIMO receivers, IEEE Trans. Signal Process., 54 (2006), 161-176.

[21]

A. Wiesel, Y. C. Eldar and S. Shamai, Zero-forcing precoding and generalized inverses, IEEE Trans. Signal Process., 56 (2008), 4409-4418. doi: 10.1109/TSP.2008.924638.

[22]

C. J. Yu, K. L. Teo and H. H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Process., 95 (2014), 32-42. doi: 10.1016/j.sigpro.2013.08.005.

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