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

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

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

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

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

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 ∆

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

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

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.

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

Output Error Method

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

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

Simulation LMS adaptive FIR filter for equalization

Simulation

Simulation

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

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

Decision-Feedback Equalizer (Most popular adaptive IIR equalizer)

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*

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