Advertisement

Circuits, Systems, and Signal Processing

, Volume 38, Issue 1, pp 407–424 | Cite as

New Criterion for l2l Stability of Interfered Fixed-Point State-Space Digital Filters with Quantization/Overflow Nonlinearities

  • Pooja RaniEmail author
  • Priyanka Kokil
  • Haranath Kar
Article
  • 49 Downloads

Abstract

The problem of l2l stability and disturbance attenuation performance analysis of fixed-point state-space digital filters with external disturbance and finite wordlength nonlinearities is studied in this paper. The finite wordlength nonlinearities are composite nonlinearities representing concatenations of quantization and overflow correction employed in practice. Using sector-based characterization of the finite wordlength nonlinearities, a new l2l stability criterion for the reduction of the effects of external disturbance to a given level and to confirm the nonexistence of limit cycles in the absence of external interference is established. Numerical examples are given to illustrate the efficacy of the proposed criterion.

Keywords

Asymptotic stability Digital filter Finite wordlength effect Lyapunov stability theory l2l approach 

Notes

Acknowledgements

The authors thank the Editors and the anonymous reviewers for their constructive comments and suggestions to improve the manuscript.

References

  1. 1.
    N. Agarwal, H. Kar, An improved criterion for the global asymptotic stability of fixed-point state-space digital filters with combinations of quantization and overflow. Digit. Signal Process. 28, 136–143 (2014)CrossRefGoogle Scholar
  2. 2.
    C.K. Ahn, IOSS criterion for the absence of limit cycles in interfered digital filters employing saturation overflow arithmetic. Circuits Syst. Signal Process. 32(3), 1433–1441 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    C.K. Ahn, l 2l elimination of overflow oscillations in 2-D digital filters described by Roesser model with external interference. IEEE Trans. Circuits Syst. II 60(6), 361–365 (2013)CrossRefGoogle Scholar
  4. 4.
    C.K. Ahn, l 2l stability criterion for fixed-point state-space digital filters with saturation nonlinearity. Int. J. Electron. 100(9), 1309–1316 (2013)CrossRefGoogle Scholar
  5. 5.
    C.K. Ahn, New passivity criterion for limit cycle oscillation removal of interfered 2D digital filters in the Roesser form with saturation nonlinearity. Nonlinear Dyn. 78(1), 409–420 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C.K. Ahn, l 2l suppression of limit cycles in interfered two-dimensional digital filters: a Fornasini–Marchesini model case. IEEE Trans. Circuits Syst. II 61(8), 614–618 (2014)CrossRefGoogle Scholar
  7. 7.
    C.K. Ahn, H. Kar, Passivity and finite-gain performance for two-dimensional digital filters: the FM LSS model case. IEEE Trans. Circuits Syst. II 62(9), 871–875 (2015)CrossRefGoogle Scholar
  8. 8.
    C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters. Signal Process. 109, 148–153 (2015)CrossRefGoogle Scholar
  9. 9.
    C.K. Ahn, P. Shi, Generalized dissipativity analysis of digital filters with finite wordlength arithmetic. IEEE Trans. Circuits Syst. II 63(4), 386–390 (2016)CrossRefGoogle Scholar
  10. 10.
    C.K. Ahn, P. Shi, Hankel norm performance of digital filters associated with saturation. IEEE Trans. Circuits Syst. II 64(6), 720–724 (2017)CrossRefGoogle Scholar
  11. 11.
    C.K. Ahn, L. Wu, P. Shi, Stochastic stability analysis for 2-D Roesser systems with multiplicative noise. Automatica 69, 356–363 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    P.H. Bauer, E.I. Jury, A stability analysis of two-dimensional nonlinear digital state-space filters. IEEE Trans. Acoust. Speech Signal Process. 38(9), 1578–1586 (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    B.W. Bomar, On the design of second-order state-space digital filter sections. IEEE Trans. Circuits Syst. 36(4), 542–552 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Bose, Combined effects of overflow and quantization in fixed-point digital filters. Digit. Signal Process. 4(4), 239–244 (1994)CrossRefGoogle Scholar
  15. 15.
    T. Bose, D.P. Brown, Limit cycles due to roundoff in state-space digital filters. IEEE Trans. Acoust. Speech Signal Process. 38(8), 1460–1462 (1990)CrossRefGoogle Scholar
  16. 16.
    S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    M.O. Camponez, A.C.S. Simmer, M.S. Filho, Low-noise zero-input, overflow, and constant-input limit cycle-free implementation of state space digital filters. Digit. Signal Process. 17(1), 335–344 (2007)CrossRefGoogle Scholar
  18. 18.
    Y.-Z. Chang, Z.-R. Tsai, Efficient 3D registration using two-stage scheme and spatial filter. Int. J. Innov. Comput. Inf. Control 6(2), 641–654 (2010)Google Scholar
  19. 19.
    T.A.C.M. Claasen, W.F.G. Mecklenbräuker, J.B.H. Peek, Second-order digital filter with only one magnitude-truncation quantiser and having practically no limit cycles. Electron. Lett. 9(22), 531–532 (1973)CrossRefGoogle Scholar
  20. 20.
    P. Diksha, H. Kokil, Kar Criterion for the limit cycle free state-space digital filters with external disturbances and quantization/overflow nonlinearities. Eng. Comput. 33(1), 64–73 (2016)CrossRefGoogle Scholar
  21. 21.
    K.T. Erickson, A.N. Michel, Stability analysis of fixed-point digital filters using computer generated Lyapunov functions-Part I: direct form and coupled form filters. IEEE Trans. Circuits Syst. 32(2), 113–132 (1985)CrossRefzbMATHGoogle Scholar
  22. 22.
    P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox (The Mathworks Inc., Natick, 1995)Google Scholar
  23. 23.
    H. Gao, C. Wang, Robust L 2L filtering for uncertain systems with multiple time-varying state delays. IEEE Trans. Circuits Syst. I 50(4), 594–599 (2003)CrossRefzbMATHGoogle Scholar
  24. 24.
    H. Kar, An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digit. Signal Process. 17(3), 685–689 (2007)CrossRefGoogle Scholar
  25. 25.
    H. Kar, An improved version of modified Liu–Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic. Digit. Signal Process. 20(4), 977–981 (2010)CrossRefGoogle Scholar
  26. 26.
    H. Kar, Asymptotic stability of fixed-point state-space digital filters with combinations of quantization and overflow nonlinearities. Signal Process. 91(11), 2667–2670 (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    H. Kar, V. Singh, Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities. IEEE Trans. Signal Process. 49(5), 1097–1105 (2001)CrossRefGoogle Scholar
  28. 28.
    H. Kar, V. Singh, Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach. IEEE Trans. Circuits Syst. II 51(1), 40–42 (2004)CrossRefGoogle Scholar
  29. 29.
    P. Kokil, S.S. Shinde, An improved criterion for peak-to-peak realization of direct-form interfered digital filters employing saturation nonlinearities. COMPEL-Int. J. Comput. Math. Electr. Electron. Eng. 34(3), 996–1010 (2015)CrossRefGoogle Scholar
  30. 30.
    P. Kokil, S.S. Shinde, Asymptotic stability of fixed-point state-space digital filters with saturation arithmetic and external disturbance: an IOSS approach. Circuits Syst. Signal Process. 34(12), 3965–3977 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    P. Kokil, S.S. Shinde, A note on the induced l stability of fixed-point digital filters without overflow oscillations and instability due to finite wordlength effects. Circuits Syst. Signal Process. 36(3), 1288–1300 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    T.-E. Lee, J.-P. Su, K.-W. Yu, K.-H. Hsia, Multi-objective optimization using fuzzy logic for an alpha-beta filter. ICIC Express Lett. 3(4(A)), 1173–1178 (2009)Google Scholar
  33. 33.
    A. Lepschy, G.A. Mian, U. Viaro, A contribution to the stability analysis of second-order direct-form digital filters with magnitude truncation. IEEE Trans. Acoust. Speech Signal Process. 35(8), 1207–1210 (1987)CrossRefGoogle Scholar
  34. 34.
    T. Li, W.X. Zheng, New stability criterion for fixed-point state-space digital filters with generalized overflow arithmetic. IEEE Trans. Circuits Syst. II 59(7), 443–447 (2012)CrossRefGoogle Scholar
  35. 35.
    T. Li, Q. Zhao, J. Lam, Z. Feng, Multi-bound-dependent stability criterion for digital filters with overflow arithmetics and time delay. IEEE Trans. Circuits Syst. II 61(1), 31–35 (2014)CrossRefGoogle Scholar
  36. 36.
    D. Liu, A.N. Michel, Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters. IEEE Trans. Circuits Syst. I 39(10), 798–807 (1992)CrossRefzbMATHGoogle Scholar
  37. 37.
    M.S. Mahmoud, Resilient L 2L filtering of polytopic systems with state delays. IET Control Theory Appl. 1(1), 141–154 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    W.L. Mills, C.T. Mullis, R.A. Roberts, Digital filter realizations without overflow oscillations. IEEE Trans. Acoust. Speech Signal Process. 26(4), 334–338 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Monteiro, R.V. Leuken, Integrated Circuit and System Design: Power and Timing Modeling, Optimization and Simulation (Springer, Berlin, 2010)CrossRefGoogle Scholar
  40. 40.
    T. Ooba, Stability of discrete-time systems joined with a saturation operator on the state-space. IEEE Trans. Autom. Control 55(9), 2153–2155 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    R.M. Palhares, P.L.D. Peres, Robust filtering with guaranteed energy-to-peak performance-an LMI approach. Automatica 36(6), 851–858 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    P. Rani, P. Kokil, H. Kar, l 2l suppression of limit cycles in interfered digital filters with generalized overflow nonlinearities. Circuits Syst. Signal Process. 36(7), 2727–2741 (2017)CrossRefzbMATHGoogle Scholar
  43. 43.
    J.H.F. Ritzerfeld, Noise gain expressions for low noise second-order digital filter structures. IEEE Trans. Circuits Syst. II 52(4), 223–227 (2005)CrossRefGoogle Scholar
  44. 44.
    D. Schlichtharle, Digital Filters: Basics and Design (Springer, Berlin, 2000)CrossRefzbMATHGoogle Scholar
  45. 45.
    H. Shen, J. Wang, J.H. Park, Z.-G. Wu, Condition of the elimination of overflow oscillations in two-dimensional digital filters with external interference. IET Signal Process. 8(8), 885–890 (2014)CrossRefGoogle Scholar
  46. 46.
    T. Shen, X. Wang, Z. Yuan, Stability analysis for a class of digital filters with single saturation nonlinearity. Automatica 46(12), 2112–2115 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    T. Shen, Z. Yuan, Stability of fixed-point state-space digital filters using two’s complement arithmetic: further insight. Automatica 46(12), 2109–2111 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    T. Shen, Z. Yuan, X. Wang, A new stability criterion for fixed-point state-space digital filters using two’s complement arithmetic. Automatica 47(7), 1538–1541 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Y. Sugita, N. Aikawa, T. Yoshikawa, Design method of FIR digital filters with specified group delay errors using successive projection. Int. J. Innov. Comput. Inf. Control 4(2), 445–455 (2008)Google Scholar
  50. 50.
    Y. Tsividis, Mixed Analog-Digital VLSI Devices and Technology (World Scientific Publishing, Singapore, 2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electronics and Communication EngineeringMotilal Nehru National Institute of Technology AllahabadAllahabadIndia
  2. 2.Department of Electronics EngineeringIndian Institute of Information Technology Design and ManufacturingKancheepuram, ChennaiIndia

Personalised recommendations