Abstract
In this work, we propose an adaptive set-membership (SM) reduced-rank filtering algorithm using the constrained constant modulus criterion for beamforming. We develop a stochastic gradient type algorithm based on the concept of SM techniques for adaptive beamforming. The filter weights are updated only if the bounded constraint cannot be satisfied. We also propose a scheme of time-varying bound and incorporate parameter dependence to characterize the environment for improving the tracking performance. A detailed analysis of the proposed algorithm in terms of computational complexity and stability is carried out. Simulation results verify the analytical results and show that the proposed adaptive SM reduced-rank beamforming algorithms with a dynamic bound achieve superior performance to previously reported methods at a reduced update rate.
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ULA is considered for the sake of simplifying the presentation, while generalization to other antenna configurations is straightforward.
For the proposed algorithm with the fixed bound scheme, the convergence factor is the bound \(\gamma \). For the proposed algorithm with the time-varying bound scheme, the convergence factors are the tuning coefficients \(\rho \) and \(g\). The relationships between the convergence factors and the step-size values are shown in expressions (20), (21) and (22).
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Acknowledgments
The authors would like to thank Prof. Rodrigo C. de Lamare for his insightful suggestions which have helped to improve the quality of this contribution.
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This work was supported by the National Natural Science Foundation of China under Grants 61101103 and 61471319, the Fundamental Research Funds for the Central Universities, and the Zhejiang Provincial Natural Science Foundation of China under Grant LY14F010013.
Appendix: Derivation for (20) and (21)
Appendix: Derivation for (20) and (21)
In this appendix, we derive the expressions in (20) and (21). Note that each constraint set comprises of two parallel hyperstrips in the parameter space. Based on the constraint \(e^2(i)>\gamma ^2(i)\), we consider the following two cases for update: (1) \(|y(i)|>\sqrt{1+\gamma (i)}\) and (2) \(|y(i)|<\sqrt{1-\gamma (i)}\). Firstly, let us focus on the variable step-size \(\mu _{p}(i)\). We obtain \(\mu _{p}(i)\) for computing \(\mathbf {p}(i+1)\) by projecting \(\mathbf {p}(i)\) onto \(\mathcal {H}_{p}(i)\), i.e., the set of all \(\mathbf {p}\) that satisfy:
For case 1) \(|\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}(i)|>\sqrt{1+\gamma (i)}\), \(\mathbf {p}(i)\) is closer to the hyperplanes defined by
than to the ones defined by
By substituting (18) into (31) and using the constraint \(\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {D}(\theta _{0})\mathbf {p}^{*}(i)=\nu \) we have
By making an arrangement to (33) we have the following expression:
For case 2) \(|\bar{\mathbf {w}}^{H}(i)\mathbf {T}\mathbf {R}^{'}(i)\mathbf {p}^{*}(i)|<\sqrt{1-\gamma (i)}\), \(\mathbf {p}(i)\) is closer to the hyperplanes defined by (32). By following the same approach we have:
Therefore, the expressions for \(\mu _{p}(i)\) can be summarized as (20).
Then, we derive the expression for the variable step-size \(\mu _{w}(i)\). By employing (19) we have the following expression
Similarly, by considering these two cases to update and making the arrangement to (36), we can obtain (21).
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Cai, Y., Qin, B. & Zhang, H. An improved adaptive constrained constant modulus reduced-rank algorithm with sparse updates for beamforming. Multidim Syst Sign Process 27, 321–340 (2016). https://doi.org/10.1007/s11045-014-0304-5
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DOI: https://doi.org/10.1007/s11045-014-0304-5