Adaptive IIR Adaptive IIR FilterFilter

Terry LeeTerry LeeEE 491DEE 491D

May 13, 2005May 13, 2005

OutlineOutline

Linear Filters – FIR & IIRLinear Filters – FIR & IIRLeast-mean-square algorithmLeast-mean-square algorithmAdaptive IIR using:Adaptive IIR using:

• Output Error MethodOutput Error Method• Equation Error MethodEquation Error Method

SimulationsSimulationsApplicationsApplications

Linear FiltersLinear Filters

FIR Filter ~FIR Filter ~

Moving-Average (MA)Moving-Average (MA)

present and past present and past inputsinputs

IIR Filter ~IIR Filter ~

Autoregressive Autoregressive Moving-Average Moving-Average

(ARMA)(ARMA)present and past present and past

inputsinputs

and past outputsand past outputs

IIR FilterIIR Filter

Difference equation of ARMA modelDifference equation of ARMA model

y(n) = ∑ ai(n)u(n-i) + ∑ bi(n)y(n-i)

i=0 i=1

M N

Forward filter Backwards

filter

Least-Mean-Square (LMS) Least-Mean-Square (LMS) AlgorithmAlgorithm

Linear adaptive filtering algorithmLinear adaptive filtering algorithm Differs from steepest descentDiffers from steepest descent Widely used for its simplicityWidely used for its simplicity Consists of:Consists of:1) A filtering process1) A filtering process

((mainly FIR modelmainly FIR model))

2) An adaptive process2) An adaptive process

Following the steepest descent algorithm,Following the steepest descent algorithm,

with an unknown environment:with an unknown environment: Tap-input vector: u(n)Tap-input vector: u(n) Tap-weight vector: w(n)Tap-weight vector: w(n) Estimation error: e(n)Estimation error: e(n) Cost function: J(n)=[|e(n)|]Cost function: J(n)=[|e(n)|] Gradient vector: J(n)Gradient vector: J(n) Update tap-weight vector: Update tap-weight vector: ŵŵ(n+1) (n+1)

Least-Mean-Square (LMS) Least-Mean-Square (LMS) AlgorithmAlgorithm

∆∆

Parameters: Parameters: M = # of taps (length of M = # of taps (length of filter)filter)

μ = μ = step-size parameterstep-size parameter

Filter output is: Filter output is: y(n) = y(n) = ŵŵHH(n)(n)uu(n)(n)

Error signal is: Error signal is: e(n) = d(n) – y(n) e(n) = d(n) – y(n)

Tap-weight vector: Tap-weight vector: ŵŵ(n+1) = (n+1) = ŵŵ(n) + (n) + μμuu(n)e*(n)(n)e*(n)

Summary of (LMS) Summary of (LMS) AlgorithmAlgorithm

Important Factors of an Important Factors of an AlgorithmAlgorithm

Rate of convergenceRate of convergence MisadjustmentMisadjustment TrackingTracking RobustnessRobustness Computational RequirementsComputational Requirements StructureStructure

Adaptive IIR Filter Adaptive IIR Filter

Motivation:Motivation:

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

Adaptive IIR FilterAdaptive IIR Filter

Two approaches:Two approaches:

1)1) Output error methodOutput error method

2)2) Equation error methodEquation error method

Output Error MethodOutput Error Method

y(n) = ∑ ai(n)u(n-i) + ∑ bi(n)d(n-i)

Equation Error MethodEquation Error Method

i=0 i=1

M N

y replaced by d

Output Error and Equation Output Error and Equation ErrorError

IIR has problems!IIR has problems!possible instabilitypossible instabilityslow convergenceslow convergencelocal minimalocal minima

SimulationSimulation

LMS adaptive

FIR filter for equalization

SimulationSimulation

50 100 150 200 250 300

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

iterations

MS

E

step size = 0.02, 0.01, 0.05, 0.1, and, 0.2

data1

data2

data3data4

data5

LMS adaptive

FIR filter for equalization

SimulationSimulation

0 100 200 300 400 500 600 700 800 900 10000.5

1

1.5

2

2.5

3

iterations

MS

E

step size = 0.02, 0.01, 0.05, 0.1, and 0.2

data1

data2

data3data4

data5

LMS adaptive

FIR filter for equalization

Applications of IIRApplications of IIR

acoustic echo cancellationacoustic echo cancellation linear predictionlinear prediction adaptive notch filteringadaptive notch filtering adaptive differential pulse code adaptive differential pulse code

modulation modulation adaptive array processingadaptive array processing * channel equalization ** channel equalization *

Adaptive EqualizerAdaptive Equalizer

Telephone channelsTelephone channels Fading radio channelsFading radio channels Bandwidth-limited channelsBandwidth-limited channels Removes ISIRemoves ISI Recovers informationRecovers information

Decision-Feedback Decision-Feedback EqualizerEqualizer

(Most popular

adaptive IIR equalizer)

IIR vs. FIRIIR vs. FIR

IIR has slower convergence rateIIR has slower convergence rateIIR is UNSTABLEIIR is UNSTABLEIIR introduces more complex structuresIIR introduces more complex structures

TRADEOFF:TRADEOFF:IIR uses less coefficients than FIR IIR uses less coefficients than FIR

*computationally cheaper**computationally cheaper*

*able to implement more complex *able to implement more complex filters*filters*

Linear Filters – FIR & IIRLinear Filters – FIR & IIR Least-mean-square algorithmLeast-mean-square algorithm Adaptive IIR using:Adaptive IIR using:

∙ Output Error MethodOutput Error Method∙ Equation Error MethodEquation Error Method

SimulationsSimulations ApplicationsApplications

SummarySummary