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A matrix-based IRLS algorithm for the least \({l}_{p}\)-norm design of 2-D FIR filters

  • Ruijie Zhao
  • Xiaoping Lai
  • Xiaoying Hong
  • Zhiping LinEmail author
Article
  • 194 Downloads

Abstract

Fast design of two-dimensional FIR filters in the least \({l}_{p}\)-norm sense is investigated in this brief. The design problem is first formulated in a matrix form and then solved by a matrix-based iterative reweighted least squares algorithm. The proposed algorithm includes two loops: one for updating the weighting function and the other for solving the weighted least squares (WLS) subproblems. These WLS subproblems are solved using an efficient matrix-based WLS algorithm, which is an iterative procedure with its initial iterative matrix being the solution matrix in the last iteration, resulting in a considerable CPU-time saving. Through analysis, the new algorithm is shown to have a lower complexity than existing methods. Three design examples are provided to illustrate the high computational efficiency and design precision of the proposed algorithm.

Keywords

2-D FIR filter Least \(l_p\)-norm design Matrix-based algorithm Iterative reweighted least squares algorithm 

References

  1. Adams, J. W., Member, S., & Sullivan, J. L. (1998). Peak-constrained least-squares optimization. IEEE Transactions on Signal Processing, 46(2), 306–321.CrossRefGoogle Scholar
  2. Aggarwal, A., Kumar, M., Rawat, T. K., & Upadhyay, D. K. (2016). Optimal design of 2-d FIR digital differentiator using \({L}_1\)-norm based cuckoo-search algorithm. Multidimensional Systems and Signal Processing Online First, 28, 1–19.Google Scholar
  3. Algazi, V. R., Suk, M., & Rim, C. S. (1986). Design of almost minimax FIR filters in one and two dimensional by WLS techniques. IEEE Transactions on Circuits and Systems, 33(6), 590–596.CrossRefGoogle Scholar
  4. Aravena, J. L., & Gu, G. (1996). Weighted least mean square design of 2-D FIR digital filters: The general case. IEEE Transactions on Signal Processing, 44(10), 2568–2578.CrossRefGoogle Scholar
  5. Barreto, J. A., & Burrus, C. S. (1994). Iterative reweighted least squares and the design of two-dimensional FIR digital filters. In Proceedings of 1st IEEE internaltional conference on image processing (pp 775–779). Austin.Google Scholar
  6. Burrus, C. S., Barreto, J. A., & Selesnick, I. W. (1994). Iterative reweighted least-squares design of FIR filters. IEEE Transactions on Signal Process, 42(11), 2926–2936.CrossRefGoogle Scholar
  7. Chalmers, B. L., Egger, A. G., & Taylor, G. D. (1983). Convex \({L}^p\) approximation. Journal of Approximation Theory, 37, 326–334.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Diniz, P. S. R., & Netto, S. L. (1999). On WLS-Chebyshev FIR digital filters. Journal of Circuits, Systems and Computers, 9(3–4), 155–168.CrossRefGoogle Scholar
  9. Fujisawa, K., Futakata, Y., Kojima, M., Matsuyama, S., Nakamura, S., Nakata, K., & Yamashita, M. (2005). SDPA-M (Semidefinite programming algorithm in MATLAB) users manual—Version 6.2.0. Dept. Math. Comput. Sci., Tokyo Institute of Technol., Tokyo, Japan.Google Scholar
  10. Hamamoto, K., Yoshida, T., & Aikawa, N.(2015). A design of linear phase band-pass FIR digital differentiators with flat passband and \({L}_p\) norm-based stopband characteristics. In 10th international conference on information, communications and signal processing (Vol. 2, pp. 3–6). SingaporeGoogle Scholar
  11. Hong, X. Y., Lai, X. P., & Zhao, R. J. (2013). Matrix-based algorithms for constrained least-squares and minimax designs of 2-d FIR filters. IEEE Transactions on Signal Processing, 64(14), 3620–3631.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hong, X. Y., Lai, X. P., & Zhao, R. J. (2016). A fast design algorithm for elliptic-error and phase-error constrained LS 2-D FIR filters. Multidimensional Systems and Signal Processing, 27(2), 477–491.CrossRefzbMATHGoogle Scholar
  13. Horn, R. A., & Johnson, C. R. (1994). Topics in matrix analysis. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  14. Hsieh, C. H., Kuo, C. M., Jou, Y. D., & Han, Y. L. (1997). Design of two-dimensional FIR digital filters by a two-dimensional WLS technique. IEEE Transactions on Circuits and Systems-II, 44(5), 348–358.CrossRefGoogle Scholar
  15. Karlovitz, L. A. (1970). Construction of nearest points in the \({L}^p\), \(p\) even, and \({L}^{\infty }\) norms. Journal of Approximation Theory, 3(2), 123–127.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kok, C. W., Siu, W. C., & Law, Y. M. (2008). Peak constrained two-dimensional quadrantally symmetric eigenfilter design without transition band specification. Signal Process, 88(6), 1565–1578.CrossRefzbMATHGoogle Scholar
  17. Lai, X. P. (2008). Online estimation of minimum sizes of 2-D FIR frequency-selective filters with magnitude constraints. IEEE Signal Processing Letters, 15(1), 135–138.Google Scholar
  18. Lawson, C. L. (1961). Contributions to the theory of linear least maximum approximations. PhD thesis, University of CaliforniaGoogle Scholar
  19. Lim, Y. C., Lee, J. H., Chen, C. K., & Yang, R. H. (1992). A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design. IEEE Transactions on Signal Processing, 40(3), 551–558.CrossRefGoogle Scholar
  20. Lu, W. S. (2002). A unified approach for the design of 2-D digital filters via semidefinite programming. IEEE Transactions on Circuits and Systems-I, 49(6), 814–826.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Lu, W. S., & Hinamoto, T. (2011). Two-dimensional digital filters with sparse coefficients. Multidimensional Systems and Signal Processing, 22(1), 173–189.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Mousa, W. (2012). Iterative design of one-dimensional efficient seismic \({L}_p\) infinite impulse response \(f-x\) digital filters. IET Signal Processing, 6(6), 541–545.MathSciNetCrossRefGoogle Scholar
  23. Rice, J. R., & Usow, K. H. (1968). The lawson algorithm and extensions. Mathematics of Computation, 22(101), 118–127.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Savin, C. E., Ahmad, M. O., & Swamy, M. N. S. (1999). \({L}_p\) norm design of stack filters. IEEE Transactions on Image Processing, 18(12), 1730–1743.CrossRefGoogle Scholar
  25. Tseng, C. C., & Lee, S. L. (2013). Designs of two-dimensional linear phase FIR filters using fractional derivative constraints. Signal Processing, 93(5), 1141–1151.CrossRefGoogle Scholar
  26. Vargas, R. A. (2012). Iterative design of \(l_p\) digital filters. PhD thesis, Rice UniversityGoogle Scholar
  27. Vargas, R. A., & Burrus, C. S. (2009). Iterative design of \({L}_p\) FIR and IIR digital filters. In 13th digital signal processing workshop and 5th IEEE signal processing education workshop (pp. 468–473). Marco, Ls1, FLGoogle Scholar
  28. Watson, G. A. (1988). Convex \({L}^p\) approximation. Journal of Approximation Theory, 55(1), 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Zhao, R. J., & Lai, X. P. (2011). A fast matrix iterative technique for the WLS design of 2-D quadrantally sysmmetic FIR filters. Multidimensional Systems and Signal Processing, 22(4), 303–317.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Zhao, R. J., & Lai, X. P. (2013). Efficient 2-D based algorithms for WLS design of 2-D FIR filters with arbitrary weighting functions. Multidimensional Systems and Signal Processing, 24(3), 417–434.MathSciNetCrossRefzbMATHGoogle Scholar
  31. Zhao, R. J., Lai, X. P., & Lin, Z. P. (2016). Weighted least squares design of 2-D FIR filters using a matrix-based generalized conjugate gradient method. Journal of the Franklin Institute, 353(8), 1759–1780.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Zhu, W. P., Ahmad, M. O., & Swamy, M. N. S. (1999). A least-square design approach for 2-D FIR filters with arbitrary frequency response. IEEE Transactions on Circuits and Systems-II, 46(8), 1027–1034.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mechanical, Electrical and Information EngineeringShandong UniversityWeihaiChina
  2. 2.School of Electrical and Electronic EngineeringNanyang Technological UniversitySingaporeSingapore
  3. 3.Institute of Information and ControlHangzhou Dianzi UniversityHangzhouChina

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