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<p>The momentum-based complex least mean square (MCLMS) algorithm introduces an additional momentum term to improve the convergence speed and stability of the algorithm, with only a negligible increase in complexity. However, the standard MCLMS algorithm assumes that the signal is second-order circular (proper), meaning that its real and imaginary components are uncorrelated and have equal energy. Under this assumption, the information within the complementary statistics is lost. To address this issue, this work bridges momentum acceleration with improper signal processing by establishing the first theoretical framework for a noncircular MCLMS. Key innovations include: 1. A joint mean-square error (MSE) and complementary MSE (CMSE) analysis that incorporates the signal pseudocovariance; 2. An Approximate Uncorrelating Transform (AUT) that reduces the complexity of covariance diagonalization from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(L^{3})\)</EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O(L^{2})\)</EquationSource> </InlineEquation>; 3. Decoupled mean-square stability conditions that facilitate a separate analysis of the impact of the step size <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mu\)</EquationSource> </InlineEquation> and the momentum parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation>; 4. A theoretical proof that noncircularity enhances the convergence acceleration effect of momentum, with greater improvement under stronger noncircularity. Extensive simulations validate the theoretical analysis and demonstrate the superior convergence speed, robustness, and generalization capability of the proposed MCLMS over standard CLMS benchmarks.</p>

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Performance Analysis of Momentum-Based Complex LMS Adaptive Filter for Non-circular Signals

  • 期刊论文

  • 发表日期：

    2025年10月25日

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  • Wanting Shi,
  • Yili Xia,
  • Wenjiang Pei

摘要

The momentum-based complex least mean square (MCLMS) algorithm introduces an additional momentum term to improve the convergence speed and stability of the algorithm, with only a negligible increase in complexity. However, the standard MCLMS algorithm assumes that the signal is second-order circular (proper), meaning that its real and imaginary components are uncorrelated and have equal energy. Under this assumption, the information within the complementary statistics is lost. To address this issue, this work bridges momentum acceleration with improper signal processing by establishing the first theoretical framework for a noncircular MCLMS. Key innovations include: 1. A joint mean-square error (MSE) and complementary MSE (CMSE) analysis that incorporates the signal pseudocovariance; 2. An Approximate Uncorrelating Transform (AUT) that reduces the complexity of covariance diagonalization from \(O(L^{3})\) to \(O(L^{2})\) ; 3. Decoupled mean-square stability conditions that facilitate a separate analysis of the impact of the step size \(\mu\) and the momentum parameter \(\alpha\) ; 4. A theoretical proof that noncircularity enhances the convergence acceleration effect of momentum, with greater improvement under stronger noncircularity. Extensive simulations validate the theoretical analysis and demonstrate the superior convergence speed, robustness, and generalization capability of the proposed MCLMS over standard CLMS benchmarks.