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Adaptive IIR Filter: Theory, Algorithms, and Applications

Explore the concept of adaptive IIR filters, including least-mean-square algorithm, output and equation error methods, simulations, and applications in signal processing. Learn about the differences between IIR and FIR filters and their respective advantages and challenges.

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Adaptive IIR Filter: Theory, Algorithms, and Applications

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  1. Adaptive IIR Filter Terry Lee EE 491D May 13, 2005

  2. Outline • Linear Filters – FIR & IIR • Least-mean-square algorithm • Adaptive IIR using: • Output Error Method • Equation Error Method • Simulations • Applications

  3. Linear Filters IIR Filter ~ Autoregressive Moving-Average (ARMA) present and past inputs and past outputs FIR Filter ~ Moving-Average (MA) present and past inputs

  4. IIR Filter Difference equation of ARMA model y(n) = ∑ ai(n)u(n-i) + ∑ bi(n)y(n-i) M N i=0 i=1 Forward filter Backwards filter

  5. Least-Mean-Square (LMS) Algorithm • Linear adaptive filtering algorithm • Differs from steepest descent • Widely used for its simplicity • Consists of: 1) A filtering process (mainly FIR model) 2) An adaptive process

  6. Least-Mean-Square (LMS) Algorithm Following the steepest descent algorithm, with an unknown environment: • Tap-input vector: u(n) • Tap-weight vector: w(n) • Estimation error: e(n) • Cost function: J(n)=[|e(n)|] • Gradient vector: J(n) • Update tap-weight vector: ŵ(n+1) ∆

  7. Summary of (LMS) Algorithm Parameters: M = # of taps (length of filter) μ = step-size parameter Filter output is: y(n) = ŵH(n)u(n) Error signal is: e(n) = d(n) – y(n) Tap-weight vector: ŵ(n+1) = ŵ(n) + μu(n)e*(n)

  8. Important Factors of an Algorithm • Rate of convergence • Misadjustment • Tracking • Robustness • Computational Requirements • Structure

  9. Adaptive IIR Filter Motivation: To build the adaptive process around a linear IIR filter with fewer number of adjustable coefficients than an FIR filter to achieve a desired response.

  10. Adaptive IIR Filter Two approaches: • Output error method • Equation error method

  11. Output Error Method

  12. y replaced by d Equation Error Method y(n) = ∑ ai(n)u(n-i) + ∑ bi(n)d(n-i) M N i=0 i=1

  13. Output Error and Equation Error IIR has problems! • possible instability • slow convergence • local minima

  14. Simulation LMS adaptive FIR filter for equalization

  15. Simulation LMS adaptive FIR filter for equalization

  16. Simulation LMS adaptive FIR filter for equalization

  17. Applications of IIR • acoustic echo cancellation • linear prediction • adaptive notch filtering • adaptive differential pulse code modulation • adaptive array processing • * channel equalization *

  18. Adaptive Equalizer • Telephone channels • Fading radio channels • Bandwidth-limited channels • Removes ISI • Recovers information

  19. Decision-Feedback Equalizer (Most popular adaptive IIR equalizer)

  20. IIR vs. FIR • IIR has slower convergence rate • IIR is UNSTABLE • IIR introduces more complex structures TRADEOFF: • IIR uses less coefficients than FIR *computationally cheaper* *able to implement more complex filters*

  21. Summary • Linear Filters – FIR & IIR • Least-mean-square algorithm • Adaptive IIR using: • Output Error Method • Equation Error Method • Simulations • Applications

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