Abstract
To improve the performance of the blind equalizer (BE) in impulsive noise environments, a robust maximum likelihood algorithm (RMLA) is proposed for the communication systems using quadrature amplitude modulation signals. A novel robust maximum likelihood cost function based on the constant modulus algorithm is constructed to effectively suppress the influence of impulsive noise and ensure the computational stability. Theoretical analysis is presented to illustrate the robustness and good computational stability of the proposed algorithm under the impulsive noise ambient. Moreover, it is proved that the weight vector of the proposed BE can converge stably by LaSalle invariance principle. Simulation results are provided to further confirm the robustness and stability of the proposed RMLA.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grants 61801363, 61271299 and 61501348, the Natural Science Foundation of Shaanxi Province under Grant 2017JM6039, the Basic Scientific Research Foundation of Xidian University under Grants 8002/20101166309 and 8002/20103166309.
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Appendix A
Appendix A
1.1 Proof of Proposition 1
To prove the function, \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \) is a Lyapunov function, \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \) must satisfy the three conditions mentioned in Definition 1. So we divide the proof into three parts to prove Proposition 1.
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1.
\( J_{\text{RMLA}} ({\mathbf{w}}) \) is constructed of basic functions, such as linear function \( y(n) = {\mathbf{w}}^{H} {\mathbf{x}}(n) \), absolute value function \( \left| {y(n)} \right| \), quadratic function \( (\left| {y(n)} \right| - R)^{2} \). Obviously, the function \( J_{\text{RMLA}} ({\mathbf{w}}) \) is continuous.
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2.
We first construct iteration direction \( {\mathbf{d}}_{n + 1} = {\mathbf{w}}_{n + 1} - {\mathbf{w}}_{n} = \left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \times R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} - {\mathbf{w}}_{n} \) and prove its descent property. It is well known that \( \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} ){\mathbf{d}}_{n + 1} < 0 \) implies that \( {\mathbf{d}}_{n + 1} \) is a descent direction when \( \nabla J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \ne 0 \). This is completely proved by
$$ \begin{aligned} & \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} ){\mathbf{d}}_{n + 1} \\ &\quad = \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} ) \times \left( {\left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \times R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} - {\mathbf{w}}_{n} } \right) \\ &\quad = \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} )\left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \\ &\quad \times \left( {R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} - \left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right){\mathbf{w}}_{n} } \right) \\ &\quad = - \nabla J_{\text{RMLA}}^{H} ({\mathbf{w}}_{n} )\left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \nabla J_{\text{RLMA}} ({\mathbf{w}}_{n} ) < 0 . \end{aligned} $$(A.1)Because \( \left( {\sum\nolimits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \) is positive definite.
Hence, the following equation holds
$$ J_{\text{RMLA}} ({\mathbf{w}}_{n} + \lambda {\mathbf{d}}_{n + 1} ) \le J_{\text{RMLA}} ({\mathbf{w}}_{n} ) $$(A.2)where parameter \( \lambda \) is positive step size that is small enough. When \( \lambda = 1 \), we have \( {\mathbf{w}}_{n + 1} = {\mathbf{w}}_{n} + {\mathbf{d}}_{n + 1} = \left( {\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right){\mathbf{x}}\left( k \right){\mathbf{x}}^{H} \left( k \right)}}{{Y_{n} \left( k \right) + C}}} } \right)^{ - 1} \times R\sum\limits_{k = 1}^{K} {\frac{{Y_{n} \left( k \right)\left| {y_{n} \left( k \right)} \right|^{ - 1} y_{n}^{*} \left( k \right){\mathbf{x}}\left( k \right)}}{{Y_{n} \left( k \right) + C}}} \) and \( J_{\text{MMA}} ({\mathbf{w}}_{k + 1} ) \le J_{\text{MMA}} ({\mathbf{w}}_{k} ) \).
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3.
For any finite constant \( \xi \) in the codomain of \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \), it is obvious that set \( \left\{ {{\mathbf{w}}_{n} \left| {J_{\text{RMLA}} ({\mathbf{w}}_{n} )} \right. < \xi } \right\} \) is bounded. Because \( J_{\text{RMLA}} ({\mathbf{w}}_{n} ) \ge \xi \) holds when \( {\mathbf{w}}_{n} \) is unbounded.
Through the analysis of above three points, we can know that \( J_{\text{MMA}} ({\mathbf{w}}_{k} ) \) is a Lyapunov function. Finally, we can give the conclusion that the discrete sequence \( {\mathbf{w}}_{k} \) converges to the invariance set \( \tilde{\varOmega } = \left\{ {{\mathbf{w}}_{k} \left| {J_{\text{MMA}} ({\mathbf{w}}_{k + 1} ) - J_{\text{MMA}} ({\mathbf{w}}_{k} ) = 0} \right.} \right\} \) according to LaSalle invariance principle. This completes the proof of Proposition 1.
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Li, J., Feng, DZ., Li, B. et al. A Robust Maximum Likelihood Algorithm for Blind Equalization of Communication Systems Impaired by Impulsive Noise. Circuits Syst Signal Process 38, 2387–2401 (2019). https://doi.org/10.1007/s00034-018-0966-6
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DOI: https://doi.org/10.1007/s00034-018-0966-6