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

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

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

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

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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

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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) Least-Mean-Square (LMS) Algorithm ∆

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

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

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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.

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Adaptive IIR Filter Two approaches: 1)Output error method 2)Equation error method

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Output Error Method

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

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

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Simulation LMS adaptive FIR filter for equalization

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Simulation

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Simulation

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

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

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Decision-Feedback Equalizer (Most popular adaptive IIR equalizer)

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

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

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