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Adaptive filtering

Adaptive filtering

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In this chapter, we have presented in detail the adaptive filtering methods with a special glance and analysis on their practical implementation. These methods, due to their iterative structure, offer an interesting architecture for real-time applications. However, when it comes to effective use, several questions must be raised beforehand: these questions are related to the algorithm behavior itself (what kind of iterative algorithm should be used? Are the filter coefficients converging toward the Wiener-Hopf solution? How much far away from this optimal solution (i.e., asymptotic performance) is expected at convergence? What is the convergence rate?) and to their practical uses (What is the computational load and hardware implementations for such algorithms? How the fixed-point implementation affects the performance?). The questions related to the algorithm behavior have been assessed in the first part of this chapter. For a wide range of iterative algorithms, a presentation and a study have been performed: from the legacy and simple LMS algorithms to more complex (and rapid) algorithm such as RLS and APA algorithms. Besides, we have also proposed a thorough analysis of alternatives methods-the so-called nonlinear methods. They are particularly suited when signal to recover exhibits a specific nature (sparsity, strong covariance, etc.). The second part of this chapter has been focused on the algorithm comparison: depending on the mathematical model they follow, the algorithms have a different complexity and provide various convergence rate and asymptotic performance. We have thus focused on the computational complexity and on their implementation and cost to clearly state the differences between the methods. Finally, to explain the methodology used in a practical implementation, we have derived an example of real channel estimation (when the channel to estimate is time varying). The trade-off between the complexity and the performance (convergence rate and asymptotic performance) has been clearly stated. The influence of fixed-point analysis has finally been demonstrated showing the important freedom degrees implementation perspective has to face. These algorithms combined with the exposed tools and methodologies are precious to practical implementation and have been presented in this book to pave the way for digitally enhanced mixed signal systems.

Chapter Contents:

  • 9.1 Introduction
  • 9.2 Algorithm presentations
  • 9.2.1 Least mean square algorithm
  • 9.2.1.1 Principle and mathematical formulation
  • 9.2.1.2 Discussion
  • 9.2.1.3 Complexity assessment and step size tuning rules
  • 9.2.2 Affine projection algorithms
  • 9.2.2.1 Convergence rate of the APA
  • 9.2.2.2 Steady state
  • 9.2.3 Recursive least square
  • 9.2.3.1 Exponentially weighted least square criterion
  • 9.2.3.2 Recursive implementation
  • 9.2.3.3 RLS algorithm parameters
  • 9.2.3.4 Fast RLS algorithms
  • 9.2.4 Nonlinear algorithms
  • 9.2.4.1 Sign algorithms
  • 9.2.4.2 Stability and convergence rate of the SD-LMS, SE-LMS and SS-LMS
  • 9.2.4.3 Leaky LMS
  • 9.2.4.4 Least mean forth algorithm
  • 9.2.4.5 Proportionate NLMS
  • 9.2.4.6 Dual-sign algorithm
  • 9.3 Algorithm comparison
  • 9.3.1 Complexity comparison
  • 9.3.2 Implementation and cost
  • 9.3.3 Discussion
  • 9.4 Application
  • 9.4.1 Context and model
  • 9.4.2 Floating-point and fixed-point model
  • 9.5 Conclusion
  • References

Inspec keywords: adaptive filters; least mean squares methods; computational complexity; recursive filters; convergence of numerical methods; affine transforms; integral equations; nonlinear filters; iterative methods; channel estimation

Other keywords: APA algorithms; nonlinear methods; convergence rate; LMS algorithms; affine projection algorithms; mathematical model; computational load; adaptive filtering; fixed-point analysis; iterative algorithm; Wiener-Hopf solution; RLS algorithms; channel estimation; least mean square algorithm; recursive least square algorithm; computational complexity

Subjects: Interpolation and function approximation (numerical analysis); Signal processing theory; Integral transforms in numerical analysis; Integral transforms in numerical analysis; Filtering methods in signal processing; Integral equations (numerical analysis); Integral equations (numerical analysis); Interpolation and function approximation (numerical analysis)

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