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Enhanced q-least Mean Square

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Abstract

In this work, a new class of stochastic gradient algorithm is developed based on q-calculus. Unlike the existing q-LMS algorithm, the proposed approach fully utilizes the concept of q-calculus by incorporating a time-varying q parameter. The proposed enhanced q-LMS (Eq-LMS) algorithm utilizes a novel, parameterless concept of error-correlation energy and normalization of signal to ensure high convergence, stability and low steady-state error. The proposed algorithm automatically adapts the learning rate with respect to the error. For evaluation purposes the system identification problem is considered. The necessary condition of convergence for the proposed algorithm is analyzed, and the validation of analytical findings and simulation results is discussed. Extensive experiments show better performance of the proposed Eq-LMS algorithm compared to the standard q-LMS approach.

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References

  1. T. Abbas, S. Khan, M. Sajid, A. Wahab, J.C. Ye, Topological sensitivity based far-field detection of elastic inclusions. Results Phys. 8, 442–460 (2018)

    Article  Google Scholar 

  2. T. Aboulnasr, K. Mayyas, A robust variable step-size LMS-type algorithm: analysis and simulations. IEEE Trans. Signal Process. 45(3), 631–639 (1997)

    Article  Google Scholar 

  3. J. Ahmad, M. Usman, S. Khan, I. Naseem, H.J. Syed, RVP-FLMS: a robust variable power fractional LMS algorithm, in IEEE International Conference on Control System Computing and Engineering (ICCSCE) (IEEE, 2016)

  4. J. Ahmad, S. Khan, M. Usman, I. Naseem, M. Moinuddin, FCLMS: fractional complex LMS algorithm for complex system identification, in 13th IEEE Colloquium on Signal Processing and its Applications (CSPA 2017) (IEEE, 2017)

  5. A. Ahmed, M. Moinuddin, U.M. Al-Saggaf, q-State space least mean family of algorithms circuits, systems, and signal processing. Circuits Syst. Signal Process. 37, 729 (2017)

    Article  MATH  Google Scholar 

  6. A.U. Al-Saggaf, M. Arif, U.M. Al-Saggaf, M. Moinuddin, The q-normalized least mean square algorithm, in 2016 6th International Conference on Intelligent and Advanced Systems (ICIAS), pp. 1–6 (2016 Aug)

  7. U.M. Al-Saggaf, M. Moinuddin, A. Zerguine, An efficient least mean squares algorithm based on q-gradient, in 2014 48th Asilomar Conference on Signals, Systems and Computers, pp. 891–894 (2014 Nov)

  8. U.M. Al-Saggaf, M. Moinuddin, M. Arif, A. Zerguine, The q-least mean squares algorithm. Signal Process. 111(Supplement C), 50–60 (2015)

    Article  Google Scholar 

  9. A.R.A.L. Ali, V. Gupta, R.P. Agarwal, A. Aral, V. Gupta, Applications of q-Calculus in Operator Theory (Springer, New York, 2013)

    MATH  Google Scholar 

  10. H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee, A. Wahab, Mathematical Methods in Elasticity Imaging (Princeton University Press, Princeton, 2015)

    Book  MATH  Google Scholar 

  11. M. Arif, I. Naseem, M. Moinuddin, U.M. Al-Saggaf, Design of an intelligent q-LMS algorithm for tracking a non-stationary channel. Arab. J. Sci. Eng. 43, 2793 (2017). https://doi.org/10.1007/s13369-017-2883-6

    Article  Google Scholar 

  12. M. Arif, I. Naseem, M. Moinuddin, S.S. Khan, M.M. Ammar, Adaptive noise cancellation using q-LMS, in 2017 International Conference on Innovations in Electrical Engineering and Computational Technologies (ICIEECT), pp. 1–4 (2017 April)

  13. G. Bangerezako, Variational q-calculus. J. Math. Anal. Appl. 289(2), 650–665 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Benesty, S.L. Gay, An improved PNLMS algorithm, in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), vol 2, II–1881 (IEEE, 2002)

  15. N.J. Bershad, F. Wen, H.C. So, Comments on fractional LMS algorithm. Signal Process. 133, 219–226 (2017)

    Article  Google Scholar 

  16. B. Chen, S. Zhao, P. Zhu, J.C. Principe, Quantized kernel least mean square algorithm. IEEE Trans. Neural Netw. Learn. Syst. 23(1), 22–32 (2012)

    Article  Google Scholar 

  17. Y. Chen, Y. Gu, A.O. Hero, Sparse LMS for system identification, in IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2009 (IEEE, 2009), pp. 3125–3128

  18. S. Ciochin, C. Paleologu, J. Benesty, An optimized NLMS algorithm for system identification. Signal Process. 118, 115–121 (2016)

    Article  Google Scholar 

  19. H. Dai, Y. Huang, J. Wang, L. Yang, Resource optimization in heterogeneous cloud radio access networks. IEEE Commun. Lett. 22(3), 494–497 (2018)

    Article  Google Scholar 

  20. P.S.R. Diniz, Adaptive Filtering: Algorithms and Practical Implementation, 01, 4th Edn. Springer (2013). https://doi.org/10.1007/978-1-4614-4106-9

  21. S.C. Douglas, A family of normalized LMS algorithms. IEEE Signal Process. Lett. 1(3), 49–51 (1994)

    Article  Google Scholar 

  22. T. Ernst, A Comprehensive Treatment of q-Calculus, 1st edn. (Springer, Basel, 2012)

    Book  MATH  Google Scholar 

  23. J.M. Górriz, J. Ramírez, S. Cruces-Alvarez, C.G. Puntonet, E.W. Lang, D. Erdogmus, A novel LMS algorithm applied to adaptive noise cancellation. IEEE Signal Process. Lett. 16(1), 34–37 (2009)

    Article  Google Scholar 

  24. V. Kac, P. Cheung, Quantum Calculus (Springer, New York, 2012)

    MATH  Google Scholar 

  25. A. Khalili, A. Rastegarnia, W.M. Bazzi, Z. Yang, Derivation and analysis of incremental augmented complex least mean square algorithm. IET Signal Process. 9(4), 312–319 (2015)

    Article  Google Scholar 

  26. A. Khalili, A. Rastegarnia, S. Sanei, Quantized augmented complex least-mean square algorithm: derivation and performance analysis. Signal Process. 121, 54–59 (2016)

    Article  Google Scholar 

  27. S. Khan, I. Naseem, R. Togneri, M. Bennamoun, A novel adaptive kernel for the RBF neural networks. Circuits Syst. Signal Process. 36, 1–15 (2016)

    MATH  Google Scholar 

  28. S. Khan, J. Ahmad, I. Naseem, M. Moinuddin, A novel fractional gradient-based learning algorithm for recurrent neural networks. Circuits Syst. Signal Process. 37, 1–20 (2017)

    MathSciNet  MATH  Google Scholar 

  29. S. Khan, N. Ahmed, M.A. Malik, I. Naseem, R. Togneri, M. Bennamoun, FLMF: fractional least mean fourth algorithm for channel estimation in non-gaussian environment, in International Conference on Information and Communications Technology Convergence 2017 (ICTC 2017) (Jeju Island, Korea, 2017)

  30. S. Khan, M. Usman, I. Naseem, R. Togneri, M. Bennamoun, A robust variable step size fractional least mean square (RVSS-FLMS) algorithm, in 13th IEEE Colloquium on Signal Processing and its Applications (CSPA 2017) (IEEE, 2017)

  31. S. Khan, M. Usman, I. Naseem, R. Togneri, M. Bennamoun, VP-FLMS: a novel variable power fractional LMS algorithm, in 2017 Ninth International Conference on Ubiquitous and Future Networks (ICUFN) (ICUFN 2017), Milan, Italy (2017 Jul)

  32. S. Khan, I. Naseem, M.A. Malik, R. Togneri, M. Bennamoun, A fractional gradient descent-based RBF neural network. Circuits Syst. Signal Process. 37(12), 5311–5332 (2018). https://doi.org/10.1007/s00034-018-0835-3

    Article  MathSciNet  Google Scholar 

  33. S. Khan, I. Naseem, A. Sadiq, J. Ahmad, M. Moinuddin, Comments on momentum fractional LMS for power signal parameter estimation (2018). arXiv preprint arXiv:1805.07640

  34. S. Khan, A. Wahab, I. Naseem, M. Moinuddin, Comments on design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization (2018). arXiv preprint arXiv:1802.09252

  35. R.H. Kwong, E.W. Johnston, A variable step size LMS algorithm. IEEE Trans. Signal Process. 40(7), 1633–1642 (1992)

    Article  MATH  Google Scholar 

  36. R.H. Kwong, E.W. Johnston, A variable step size LMS algorithm. IEEE Trans. Signal Process. 40(7), 1633–1642 (1992)

    Article  MATH  Google Scholar 

  37. C. Li, P. Liu, C. Zou, F. Sun, J.M. Cioffi, L. Yang, Spectral-efficient cellular communications with coexistent one- and two-hop transmissions. IEEE Trans. Veh. Technol. 65(8), 6765–6772 (2016)

    Article  Google Scholar 

  38. C. Li, H.J. Yang, F. Sun, J.M. Cioffi, L. Yang, Multiuser overhearing for cooperative two-way multiantenna relays. IEEE Trans. Veh. Technol. 65(5), 3796–3802 (2016)

    Article  Google Scholar 

  39. C. Li, S. Zhang, P. Liu, F. Sun, J.M. Cioffi, L. Yang, Overhearing protocol design exploiting intercell interference in cooperative green networks. IEEE Trans. Veh. Technol. 65(1), 441–446 (2016)

    Article  Google Scholar 

  40. W. Liu, P.P. Pokharel, J.C. Principe, The kernel least-mean-square algorithm. IEEE Trans. Signal Process. 56(2), 543–554 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Y. Liu, R. Ranganathan, M.T. Hunter, W.B. Mikhael, Complex adaptive LMS algorithm employing the conjugate gradient principle for channel estimation and equalization. Circuits Syst. Signal Process. 31(3), 1067–1087 (2012). https://doi.org/10.1007/s00034-011-9368-8

    Article  MathSciNet  Google Scholar 

  42. P.A.C. Lopes, J.A.B. Gerald, A close to optimal adaptive filter for sudden system changes. IEEE Signal Process. Lett. 24(11), 1734–1738 (2017)

    Article  Google Scholar 

  43. A.K. Maurya, Cascade–cascade least mean square (LMS) adaptive noise cancellation. Circuits Syst. Signal Process. 37, 3785 (2017). https://doi.org/10.1007/s00034-017-0731-2

    Article  Google Scholar 

  44. P.P. Pokharel, W. Liu, J.C. Principe, Kernel least mean square algorithm with constrained growth. Signal Process. 89(3), 257–265 (2009)

    Article  MATH  Google Scholar 

  45. N. Sireesha, K. Chithra, T. Sudhakar, Adaptive filtering based on least mean square algorithm, in 2013 Ocean Electronics (SYMPOL), pp. 42–48 (2013 Oct)

  46. A.I. Sulyman, A. Zerguine, Convergence and steady-state analysis of a variable step-size NLMS algorithm. Signal Process. 83(6), 1255–1273 (2003)

    Article  MATH  Google Scholar 

  47. J. Tariboon, S.K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013(1), 282 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. J. Tariboon, S.K. Ntouyas, P. Agarwal, New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Adv. Differ. Equ. 2015(1), 18 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Tarrab, A. Feuer, Convergence and performance analysis of the normalized LMS algorithm with uncorrelated gaussian data. IEEE Trans. Inf. Theory 34(4), 680–691 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  50. N.V. Thakor, Y.-S. Zhu, Applications of adaptive filtering to ECG analysis: noise cancellation and arrhythmia detection. IEEE Trans. Biomed. Eng. 38(8), 785–794 (1991)

    Article  Google Scholar 

  51. A. Wahab, S. Khan, “Comments on” fractional extreme value adaptive training method: fractional steepest descent approach (2018). arXiv preprint arXiv:1802.09211

  52. W. Wang, H. Zhao, Boxed-constraint least mean square algorithm and its performance analysis. Signal Process. 144(Supplement C), 201–213 (2018)

    Article  Google Scholar 

  53. B. Widrow, J. McCool, M. Ball, The complex LMS algorithm. Proc. IEEE 63(4), 719–720 (1975)

    Article  Google Scholar 

  54. Y. Zheng, S. Wang, J. Feng, C.K. Tse, A modified quantized kernel least mean square algorithm for prediction of chaotic time series. Digital Signal Process. 48(Supplement C), 130–136 (2016)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Engineering, Karachi Institute of Economics and Technology, Korangi Creek, Karachi, 75190, Pakistan

    Alishba Sadiq & Imran Naseem

  2. Faculty of Engineering Science and Technology, Iqra University, Defence View, Shaheed-e-Millat Road (Ext.), Karachi, 75500, Pakistan

    Shujaat Khan

  3. School of Electrical, Electronic and Computer Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Imran Naseem & Roberto Togneri

  4. School of Computer Science and Software Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Mohammed Bennamoun

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  1. Alishba Sadiq
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Correspondence to Imran Naseem.

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Sadiq, A., Khan, S., Naseem, I. et al. Enhanced q-least Mean Square. Circuits Syst Signal Process 38, 4817–4839 (2019). https://doi.org/10.1007/s00034-019-01091-4

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