Abstract
High computational complexity is a major problem encountered in the optimal design of two-dimensional (2-D) finite impulse response (FIR) filters. In this paper, we present an iterative matrix solution with very low complexity to the weighted least square (WLS) design of 2-D quadrantally symmetric FIR filters with two-valued weighting functions. Firstly, a necessary and sufficient condition for the WLS design of 2-D quadrantally symmetric filters with general nonnegative weighting functions is obtained. Then, based on this optimality condition, a novel iterative algorithm is derived for the WLS design problem with a two-valued weighting function. Because the filter parameters are arranged in their natural 2-D form and the transition band is not sampled, the computation amount of the proposed algorithm is reduced significantly, especially for high-order filters. The exponential convergence of the algorithm is established, and its computational complexity is estimated. Design examples demonstrating the convergence rate and solution accuracy of the algorithm, as well as the relation between the iteration number of the algorithm and the size and transition-band width of the filter are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad M. O., Wang J. D. (1989) An analytic least mean square solution to the design problem of two-dimensional FIR filters with quadrantally symmetric or antisymmetric frequency response. IEEE Transactions Circuits and Systems 36(7): 968–979
Aldhaheri R. W. (2004) Design of 2-D linear phase digital filters using Schur decomposition and symmetries. Multidimensional Systems and Signal Processing 15(1): 65–81
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
Charalambous C. (1985) The performance of an algorithm of minimax design of two-dimensional linear phase FIR digital filters. IEEE Transactions on Circuits and Systems 32(12): 1016–1028
Gislason E., Johansen M., Ersboll B. K., Jacobsen S. K. (1993) Three different criteria for the design of two-dimensional zero phase FIR digital filters. IEEE Transactions on Signal Processing 41(10): 3070–3074
Gu G., Aravena J. L. (1994) Weighted least mean square design of 2-D FIR digital filters. IEEE Transactions on Signal Processing 42(11): 3178–3187
Hanna M. T. (1996) Weighted least squares design of two-dimensional zero-phase FIR filters in the continuous frequency domain. IEEE Transactions on Circuits and Systems-II 43(7): 534–537
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
Lai X. P. (2007) Design of smallest size two-dimensional linear-phase FIR filters with magnitude error constraint. Multidimensional Systems and Signal Processing 18(4): 341–349
Lai X. P., Cheng Y. (2007) A sequential constrained least-square approach to minimax design of 2-D FIR filters. IEEE Transactions on Circuits and Systems-II 54(11): 994–998
Lu H.-C., Yeh K.-H. (1997) Optimal design of 2-d FIR digital filters by scaling-free McClellan transformation using least-squares estimation. Signal Processing 58(3): 303–308
Lu, W. S., Hinamoto, T. (2006). A second-order cone programming approach for minimax design of 2-D FIR filters with low group delay. In Proceedings IEEE international symposium on circuits and systems, pp. 2521–2524.
Reddy H. C., Khoo I. H., Rajan P. K. (2003) 2-D symmetry: Theory and filter design applications. IEEE Circuits and Systems Magazine 3(3): 4–23
Zhu W. P., Ahmad O. M., Swamy M. N. S. (1997) A closed-form solution to the least-square design problem of 2-D linear-phase FIR filters. IEEE Transactions on Circuits and Systems-II 44(12): 1032–1039
Zhu W. P., Ahmad O. M., Swamy M. N. S. (1999a) 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
Zhu W. P., Ahmad O. M., Swamy M. N. S. (1999b) Realization of 2-D linear-phase FIR filters by using the singular-value decomposition. IEEE Transactions on Signal Processing 47(5): 1349–1358
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, R., Lai, X. A fast matrix iterative technique for the WLS design of 2-D quadrantally symmetic FIR filters. Multidim Syst Sign Process 22, 303–317 (2011). https://doi.org/10.1007/s11045-010-0128-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11045-010-0128-x