[Paper Review] Random Euler Complex-Valued Nonlinear Filters

Jiashu Zhang, Sheng Zhang+1 more|arXiv (Cornell University)|Jan 2, 2018

Advanced Adaptive Filtering Techniques31 references3 citations

TL;DR

This paper proposes two low-complexity random Euler complex-valued nonlinear filters—Linear Random Euler Complex-Valued Filter (LRECF) and Widely-Linear Random Euler Complex-Valued Filter (WLRECF)—that leverage random Fourier features and complex reproducing kernel Hilbert spaces to achieve efficient nonlinear filtering. The key contribution is a theoretical analysis of transient and steady-state performance in non-stationary environments, including closed-form expressions for minimum mean square error and optimal step-size, validated through simulations in system identification and channel equalization.

ABSTRACT

Over the last decade, both the neural network and kernel adaptive filter have successfully been used for nonlinear signal processing. However, they suffer from high computational cost caused by their complex/growing network structures. In this paper, we propose two random Euler filters for complex-valued nonlinear filtering problem, i.e., linear random Euler complex-valued filter (LRECF) and its widely-linear version (WLRECF), which possess a simple and fixed network structure. The transient and steady-state performances are studied in a non-stationary environment. The analytical minimum mean square error (MSE) and optimum step-size are derived. Finally, numerical simulations on complex-valued nonlinear system identification and nonlinear channel equalization are presented to show the effectiveness of the proposed methods.

Motivation & Objective

• Address the high computational cost of traditional kernel adaptive filters and complex-valued neural networks in nonlinear signal processing.
• Overcome the growing network structure issue in kernel filters by introducing a fixed, low-complexity architecture.
• Develop a theoretical framework for transient and steady-state performance analysis under non-stationary conditions modeled by a random-walk process.
• Enable efficient processing of complex-valued nonlinear systems using random Fourier features and complex kernel methods.
• Demonstrate effectiveness in real-world applications such as nonlinear system identification and nonlinear channel equalization.

Proposed method

• Derive the Linear Random Euler Complex-Valued Filter (LRECF) using complexification of real reproducing kernel Hilbert spaces (RKHS) and Bochner’s theorem.
• Extend LRECF to the Widely-Linear Random Euler Complex-Valued Filter (WLRECF) by incorporating the widely-linear model and Wirtinger’s derivative for enhanced performance with noncircular signals.
• Map input signals into a finite-dimensional random feature space using random Fourier features to approximate complex kernel mappings efficiently.
• Employ the complex least mean square (CLMS) algorithm with fixed network structure to ensure low computational complexity.
• Utilize a random-walk model to represent non-stationary environments for theoretical performance analysis.
• Derive closed-form expressions for mean-square deviation (MSD) and mean square error (MSE) using stochastic analysis techniques.

Experimental results

Research questions

• RQ1 Can a fixed-structure filter achieve comparable performance to growing kernel filters in complex-valued nonlinear filtering with significantly reduced computational cost?
• RQ2 What is the optimal step-size for minimizing steady-state mean square error in a non-stationary environment for the proposed filters?
• RQ3 How do the transient and steady-state behaviors of the LRECF and WLRECF compare under varying noise and system dynamics?
• RQ4 To what extent does the widely-linear model improve performance over the linear model in complex-valued nonlinear filtering?
• RQ5 Can theoretical predictions of MSE and MSD accurately reflect the actual performance of the proposed filters in non-stationary settings?

Key findings

• The proposed LRECF and WLRECF achieve low computational complexity due to their fixed network structure, unlike growing kernel filters.
• Theoretical analysis shows that there exists an optimal step-size that minimizes the steady-state mean square error (MSE) in non-stationary environments.
• Theoretical MSE and MSD curves closely match simulation results, validating the accuracy of the derived analytical expressions.
• In nonlinear system identification, the WLRECF outperforms CLMS and CKLMS in convergence speed and steady-state error.
• For nonlinear channel equalization, the WLRECF achieves superior symbol classification performance, as evidenced by improved eye diagrams.
• The choice of parameters μ (step-size), D (number of random features), and σ² (variance of random weights) significantly affects convergence rate and steady-state error, requiring careful tuning.