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.
This is a preview of subscription content, log in via an institution to check access.
References
Adams, J. W., Member, S., & Sullivan, J. L. (1998). Peak-constrained least-squares optimization. IEEE Transactions on Signal Processing, 46(2), 306–321.
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.
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.
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.
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.
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.
Chalmers, B. L., Egger, A. G., & Taylor, G. D. (1983). Convex \({L}^p\) approximation. Journal of Approximation Theory, 37, 326–334.
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.
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.
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). Singapore
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.
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.
Horn, R. A., & Johnson, C. R. (1994). Topics in matrix analysis. Cambridge: Cambridge University Press.
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.
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.
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.
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.
Lawson, C. L. (1961). Contributions to the theory of linear least maximum approximations. PhD thesis, University of California
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.
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.
Lu, W. S., & Hinamoto, T. (2011). Two-dimensional digital filters with sparse coefficients. Multidimensional Systems and Signal Processing, 22(1), 173–189.
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.
Rice, J. R., & Usow, K. H. (1968). The lawson algorithm and extensions. Mathematics of Computation, 22(101), 118–127.
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.
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.
Vargas, R. A. (2012). Iterative design of \(l_p\) digital filters. PhD thesis, Rice University
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, FL
Watson, G. A. (1988). Convex \({L}^p\) approximation. Journal of Approximation Theory, 55(1), 1–11.
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.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported partially by the National Nature Science Foundation of China under Grants 61573123 and 61304142, and partially by the Singapore Academic Research Fund (AcRF) Tier 1 under Project RG 31/16.
Rights and permissions
About this article
Cite this article
Zhao, R., Lai, X., Hong, X. et al. A matrix-based IRLS algorithm for the least \({l}_{p}\)-norm design of 2-D FIR filters. Multidim Syst Sign Process 30, 1–15 (2019). https://doi.org/10.1007/s11045-017-0543-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-017-0543-3