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q-State Space Least Mean Family of Algorithms

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Abstract

It is generally known that model-based estimation algorithms (such as Kalman filter and its family) perform better than the non-model-based algorithms [such as least mean square (LMS), recursive least squares] due to extra information available in terms of system dynamics (which can be used to provide state space model of the system). However, the computational complexity of the model based algorithms is very high. On the other hand, the convergence performance of model based least mean type algorithms [such as state space least mean (SSLM) algorithms] is slower and highly dependent on the step-size choice. Thus, the larger step size can provide faster convergence but gives poor steady-state excess mean square error (EMSE). To meet this conflicting demand, we propose to employ the q-calculus to minimize the generalized least mean cost function. The main advantage of using the q-calculus is that it can provide a nonlinear correction term in the adaptation of the state estimate vector. Consequently, this results in an intelligent adaptation by providing both faster convergence in the initial phase of adaptation and a lower steady-state EMSE in the final phase. The developed algorithms are termed as q-state space least mean (q-SSLM) algorithms. The performance of the proposed q-state space least mean square (q-SSLMS) algorithm is also investigated both in terms of convergence in the mean and the mean square sense. The supremacy of the proposed algorithm is validated by performing several simulations and it is also contrasted with the performance of the well-known Kalman filter. Finally, the theoretical convergence analysis is also validated via simulations.

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Notes

  1. Accuracy of the algorithm depends mainly on the dynamics of the problem.

  2. The SSLM algorithm is contrasted with the KF in terms of optimality and computational complexity in the Remarks presented at the end of this section.

  3. The choice of matrix \(\mathbf{G}\) is discussed in the Remarks.

  4. The analysis for higher values of L is quite involved which can be a future arena to explore.

  5. For higher noise variance, the system simulation time resolution had to be increased for stable simulation of the system.

  6. Similar to example 1, for higher noise variance the system simulation time resolution had to be increased for stable simulation of the system.

  7. The full algorithm and its variables can be referred to in [7, 12].

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Acknowledgements

The authors acknowledge the support provided by King Abdulaziz University and the Centre of Excellence in Intelligent Engineering Systems to carry out this work.

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Authors and Affiliations

  1. Centre of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Jeddah, Saudi Arabia

    Arif Ahmed, Muhammad Moinuddin & Ubaid M. Al-Saggaf

  2. Electrical And Computer Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia

    Arif Ahmed, Muhammad Moinuddin & Ubaid M. Al-Saggaf

  3. School of Engineering and Computer Science, Victoria University of Wellington, Wellington, New Zealand

    Arif Ahmed

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  1. Arif Ahmed
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  2. Muhammad Moinuddin
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  3. Ubaid M. Al-Saggaf
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Correspondence to Arif Ahmed.

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Ahmed, A., Moinuddin, M. & Al-Saggaf, U.M. q-State Space Least Mean Family of Algorithms. Circuits Syst Signal Process 37, 729–751 (2018). https://doi.org/10.1007/s00034-017-0569-7

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