Adaptive%20IIR%20Filter

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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 Method
• Equation 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
• y(n) ? ai(n)u(n-i) ? bi(n)y(n-i)

M
N
i0
i1
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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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 w(n1)

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Summary of (LMS) Algorithm

• Parameters M of taps (length of
  filter)
• µ step-size parameter
• Filter output is y(n) wH(n)u(n)
• Error signal is e(n) d(n) y(n)
• Tap-weight vector w(n1) w(n) µu(n)e(n)

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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
• Output error method
• Equation error method

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Output Error Method
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Equation Error Method

• y(n) ? ai(n)u(n-i) ? bi(n)d(n-i)

M
N
i0
i1
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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
LMS adaptive FIR filter for equalization
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Simulation
LMS adaptive FIR filter for equalization
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Applications of IIR

• acoustic echo cancellation
• linear prediction
• adaptive notch filtering
• adaptive differential pulse code
• 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 is UNSTABLE
• IIR introduces more complex structures
• TRADEOFF
• IIR uses less coefficients than FIR
• computationally cheaper
• able to implement more complex filters

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Summary

• Linear Filters FIR IIR
• Least-mean-square algorithm
• Adaptive IIR using
• Output Error Method
• Equation Error Method
• Simulations
• Applications