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Optimum and Adaptive Complex-Valued Bilinear Filters

Bernhard Plaimer, Matthias Wagner, Oliver Lang, Mario Huemer

TL;DR

This work extends nonlinear system identification with bilinear models from real-valued to complex-valued signals by developing fully CV BL filters and mixed CV-RV structures. It presents two RV-based CV BL extensions (2R and 4R) and introduces fully CV filters including cbwf, cbls, cblms, cbnlms, and cbrls, along with convergence analyses for cbwf and cblms and a derivation for crbnlms. The paper demonstrates, through simulations on CV MISO and CV Hammerstein systems, that fully CV BL filters offer faster convergence and robust performance compared to RV-based approaches and CV Volterra baselines, especially for correlated CV signals and tracking scenarios. It also discusses computational complexity and provides practical initialization and statistical estimation guidance, highlighting the applicability of CV BL filters in communications and radar contexts. Overall, the results indicate that CV BL filtering is a powerful framework for efficient, accurate identification of CV nonlinear systems, with potential for real-time adaptive implementations.

Abstract

The identification of nonlinear systems is a frequent task in digital signal processing. Such nonlinear systems may be grouped into many sub-classes, whereby numerous nonlinear real-world systems can be approximated as bilinear (BL) models. Therefore, various optimum and adaptive BL filters have been introduced in recent years. Moreover, in many applications, such as communications and radar, complex-valued (CV) BL systems in combination with CV signals may occur. Hence, in this work, we investigate the extension of real-valued (RV) BL filters to CV BL filters. First, we derive CV BL filters by applying two or four RV BL filters, and compare them with respect to their computational complexity and performance. Second, we introduce novel fully CV BL filters, such as the CV BL Wiener filter (C-BWF), the CV BL least squares (C-BLS) filter, the CV BL least mean squares (C-BLMS) filter, the CV BL normalized least mean squares (C-BNLMS) filter, and the CV BL recursive least squares (C-BRLS) filter. Finally, these filters are applied to identify CV multiple-input-single-output (MISO) systems and CV Hammerstein models.

Optimum and Adaptive Complex-Valued Bilinear Filters

TL;DR

This work extends nonlinear system identification with bilinear models from real-valued to complex-valued signals by developing fully CV BL filters and mixed CV-RV structures. It presents two RV-based CV BL extensions (2R and 4R) and introduces fully CV filters including cbwf, cbls, cblms, cbnlms, and cbrls, along with convergence analyses for cbwf and cblms and a derivation for crbnlms. The paper demonstrates, through simulations on CV MISO and CV Hammerstein systems, that fully CV BL filters offer faster convergence and robust performance compared to RV-based approaches and CV Volterra baselines, especially for correlated CV signals and tracking scenarios. It also discusses computational complexity and provides practical initialization and statistical estimation guidance, highlighting the applicability of CV BL filters in communications and radar contexts. Overall, the results indicate that CV BL filtering is a powerful framework for efficient, accurate identification of CV nonlinear systems, with potential for real-time adaptive implementations.

Abstract

The identification of nonlinear systems is a frequent task in digital signal processing. Such nonlinear systems may be grouped into many sub-classes, whereby numerous nonlinear real-world systems can be approximated as bilinear (BL) models. Therefore, various optimum and adaptive BL filters have been introduced in recent years. Moreover, in many applications, such as communications and radar, complex-valued (CV) BL systems in combination with CV signals may occur. Hence, in this work, we investigate the extension of real-valued (RV) BL filters to CV BL filters. First, we derive CV BL filters by applying two or four RV BL filters, and compare them with respect to their computational complexity and performance. Second, we introduce novel fully CV BL filters, such as the CV BL Wiener filter (C-BWF), the CV BL least squares (C-BLS) filter, the CV BL least mean squares (C-BLMS) filter, the CV BL normalized least mean squares (C-BNLMS) filter, and the CV BL recursive least squares (C-BRLS) filter. Finally, these filters are applied to identify CV multiple-input-single-output (MISO) systems and CV Hammerstein models.
Paper Structure (22 sections, 156 equations, 12 figures, 4 tables, 5 algorithms)

This paper contains 22 sections, 156 equations, 12 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. COMPLEX-VALUED BILINEAR MODEL
  3. COMPLEX-VALUED BILINEAR FILTERS
  4. 2rblms filter
  5. 4rblms filter
  6. cbwf
  7. cbls filter
  8. cblms filter
  9. cbnlms filter
  10. cbrls filter
  11. Mixed cv-rv bl filter structure
  12. SIMULATION RESULTS
  13. Identification of a cv miso system
  14. Bilinear versus linear approach
  15. cbwf versus cbls filter
  16. ...and 7 more sections

Figures (12)

  • Figure 1: Block diagram of a cv miso system.
  • Figure 2: Block diagram of a cv Hammerstein system.
  • Figure 3: Block diagram of a system identification task using optimum or adaptive filters.
  • Figure 4: Block diagram of a $2\mathbb{R}$bl filter structure.
  • Figure 5: Block diagram of a $4\mathbb{R}$bl filter structure.
  • ...and 7 more figures