access icon free Stability analysis in Gram-Schmidt QR decomposition

© The Institution of Engineering and Technology
Received 01/03/2016, Accepted 06/05/2016, Published 09/05/2016

In this study, important aspects concerning the stability of the QR decomposition (QRD) through the modified Gram-Schmidt (GS) orthogonalisation procedure with application in multiple-input–multiple-output (MIMO) detection are investigated. In particular, the numerical stability of GS-QRD is analysed through the condition number, considering a matrix with Gaussian entries, which is a very special class of matrix, especially for telecommunication systems in general and for MIMO system in particular. The condition number is analysed in the average sense, aided by random processes theory, including in special the central limit theorem, random variable transformation and moment generating functions. An analytical bound for the condition number is found and corroborated by numerical simulations.

Inspec keywords: random processes; MIMO communication; numerical stability; matrix decomposition; signal detection

Other keywords: GS orthogonalisation procedure; multiple-input-multiple-output detection; moment generating functions; stability analysis; MIMO detection; modified Gram-Schmidt orthogonalisation procedure; analytical bound; numerical simulations; random variable transformation; Gram-Schmidt QR decomposition; GS-QRD numerical stability; telecommunication systems; random processes theory; central limit theorem; Gaussian entries

Subjects: Signal detection; Linear algebra (numerical analysis); Other topics in statistics; Radio links and equipment

References

    1. 1)
      • 11. Bohnke, R., Wubben, D., Kuhn, V., et al: ‘Reduced complexity MMSE detection for blast architectures’. Global Telecommunications Conf., 2003. GLOBECOM ’03, vol. 4, December 2003, pp. 2258–2262.
    2. 2)
      • 3. Kim, T.H.: ‘Low-complexity sorted QR decomposition for MIMO systems based on pairwise column symmetrization’, IEEE Trans. Wirel. Commun., 2014, 13, (3), pp. 1388–1396.
    3. 3)
      • 16. Kobayashi, R.T., Abrão, T.: ‘Ordered MMSE–SIC via sorted QR decomposition in ill conditioned large-scale MIMO channels’, Telecommun. Syst., 2015, pp. 1–12. Available at http://www.dx.doi.org/10.1007/s11235-015-0123-5.
    4. 4)
      • 5. Peer, N., Murin, Y., Dabora, R.: ‘Improved QRD-QLD algorithm for low complexity MIMO decoding’, IEEE Commun. Lett., 2014, 18, (10), pp. 1703–1706.
    5. 5)
      • 18. Craig, C.C.: ‘On the frequency function of xy’, Ann. Math. Stat., 1936, 7, (1), pp. 1–15.
    6. 6)
      • 8. Nagatomi, K., Higuchi, K., Kawai, H.: ‘Complexity reduced MLD based on QR decomposition in OFDM MIMO multiplexing with frequency domain spreading and code multiplexing’. WCNC'09 – IEEE Conf. on Wireless Communications & Networking Conf., Piscataway, NJ, USA, 2009, pp. 676–681.
    7. 7)
      • 2. Lai, R.-H., Chen, C.-M., Ting, P.-A., et al: ‘A modified sorted-QR decomposition algorithm for parallel processing in MIMO detection’. 2009 IEEE Int. Symp. Circuits and Systems, 2009, pp. 1405–1408.
    8. 8)
      • 17. Cox, A.J., Higham, N.J.: ‘Stability of householder QR factorization for weighted least square problems’. Numerical Analysis 1997, Proc. of the 17th Dundee Conf., Pitman Research Notes Mathematics, 1998, vol. 380, p. 57.
    9. 9)
      • 12. Song, Y., Liu, C., Lu, F.: ‘Lattice reduction-ordered successive interference cancellation detection algorithm for multiple-input-multiple-output system’, IET Signal Process.., 2015, 9, (7), pp. 553–561.
    10. 10)
      • 15. Björck, Å.: ‘Numerics of Gram-Schmidt orthogonalization’, Linear Algebra Appl., 1994, 197–198, pp. 297–316.
    11. 11)
      • 13. Daniel, J.W., Gragg, W.B., Kaufman, L., et al: ‘Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization’, Math. Comput., 1976, 30, (136), pp. 772–795. Available at http://www.dx.doi.org/10.2307/2005398.
    12. 12)
      • 6. Mondal, S., Salama, K.N., Ali, W.H.: ‘A novel approach for K-best MIMO detection and its VLSI implementation’. IEEE Int. Symp. on Circuits and Systems, 2008, pp. 936–939.
    13. 13)
      • 1. Awasthi, A., Guttal, R., Al-dhahir, N., et al: ‘Complex QR decomposition using fast plane rotations for MIMO applications’, IEEE Commun. Lett., 2014, 18, (10), pp. 1743–1746.
    14. 14)
      • 9. Kim, J.S., Moon, S.H., Lee, I.: ‘A new reduced complexity ML detection scheme for MIMO systems’, IEEE Trans. Commun., 2010, 58, (4), pp. 1302–1310.
    15. 15)
      • 7. Han, S., Cui, T., Tellambura, C.: ‘Improved k-best sphere detection for uncoded and coded MIMO systems’, IEEE Wirel. Commun. Lett., 2012, 1, (5), pp. 472–475.
    16. 16)
      • 14. Gander, W.: ‘Algorithms for the QR-decomposition’. Eidgenoessische Technische Hochschule CH-8092 Zuerich, Technical Report, April1980. Available at https://www.inf.ethz.ch/personal/gander/papers/qrneu.pdf.
    17. 17)
      • 10. Wolniansky, P.W., Foschini, G.J., Golden, G.D., et al: ‘V-blast: an architecture for realizing very high data rates over the rich-scattering wireless channel’. 1998 URSI Int. Symp. on Signals, Systems, Electronics, 1998. ISSSE 98, September 1998, pp. 295–300.
    18. 18)
      • 4. Zimmermann, E., Rave, W., Fettweis, G.: ‘On the complexity of sphere decoding’, IEEE Trans. Signal Process., 2004, 53, (4), pp. 1474–1484.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-spr.2016.0123
Loading

Related content

content/journals/10.1049/iet-spr.2016.0123
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading