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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
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
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
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*