Adaptive FiltersS.B.RabetIn the Name of GODClass Presentation For The Course :Custom Implementation of DSP SystemsUniversity of Tehran2010Pages 9 ~15 are copied from second reference [Overview of Adaptive Filters, Gner Arslan, from Adaptive Filter Theory, 4e by Simon Haykin, 2002 Prentice Hall Inc]All the materials are copy rights of their respective authors as listed in references.

*introductionLinear filters : the filter output is a linear function of the filter input Design methods:1 The classical approach

frequency-selective filters such as lowpass / bandpass / notch filters etc

2 Optimal filter design

Mostly based on minimizing the mean-square value of the error signal[1]

*Wiener filterwork of Wiener in 1942 and Kolmogorov in 1939

it is based on a priori

statistical information when such a priori

information is not available, which is usually the case, it is not possible to design a Wiener filter in the first place

[1]

*Adaptive filterthe signal and/or noise characteristics are often nonstationary and the statistical parameters vary with time

An adaptive filter has an adaptation algorithm, that is meant to monitor the environment and vary the filter transfer function accordingly

based in the actual signals received, attempts to find the optimum filter design

[1]

*Adaptive filterIn a stationary environment, the filter is expected to converge, to the Wiener filter

In a nonstationary environment,

the filter is expected to track time variations and vary its filter coefficients accordingly

[1]

*Adaptive filterThe basic operation now involves two

processes :

1. a filtering process, which produces an output signal in response to a given input signal.

2. an adaptation process, which aims to adjust the filter parameters (filter transfer function) to the (possibly time-varying) environment Often, the (avarage) square value of the error signal is used as the optimization criterion

[1]

*Adaptive filterBecause of complexity of the optimizing algorithms most adaptive filters are digital filters that perform digital signal processing

When processing

analog signals, the adaptive filter is then preceded by A/D and D/A convertors. [1]

*Adaptive filter

The generalization to adaptive IIR filters leads to stability problems

Its common to use a FIR digital filter with adjustable coefficients. [1]

*LMS AlgorithmMost popular adaptation algorithm is LMS Define cost function as mean-squared error

Based on the method of steepest descent Move towards the minimum on the error surface to get to minimum gradient of the error surface estimated at every iteration

[2]

*LMS Algorithm[2]

*Stability of LMSThe LMS algorithm is convergent in the mean square if and only if the step-size parameter satisfy

Here max is the largest eigenvalue of the correlation matrix of the input dataMore practical test for stability is

Larger values for step sizeIncreases adaptation rate (faster adaptation)Increases residual mean-squared error

[2]

*

*Applications of Adaptive Filters: IdentificationUsed to provide a linear model of an unknown plant

Applications: System identification

[2]

*Applications of Adaptive Filters: Inverse ModelingUsed to provide an inverse model of an unknown plant

Applications: Equalization (communications channels)

[2]

*Applications of Adaptive Filters: PredictionUsed to provide a prediction of the present value of a random signal

Applications: Linear predictive coding

[2]

*Applications of Adaptive Filters: Interference CancellationUsed to cancel unknown interference from a primary signal

Applications: Echo / Noise cancellation

hands-free carphone, aircraft headphones etc[2]

Example:Acoustic Echo Cancellation *[1]

A new workNovel Adaptive IIR Filter for Frequency Estimation and Tracking

In many applications we may want to estimate (track) the signals fundamental frequency as well as any harmonic frequencies

In this article, we present a novel adaptive harmonic IIR notch filter with a single adaptive coefficient to efficiently perform frequency estimation and trackingin a harmonic frequency environment*[3]

Structurefrequency estimation of a measured signal x(n)

V(n) is a white Gaussian noise

To estimate frequency in such a harmonic frequency

environment, a IIR notch filter presented for the case of M=3 (the fundamental and two harmonics)*[3]

Pole zero plot*

Parameter r is chosen to be close to, but less than, one to achieve narrowband notches and avoid any filter stability problems

[3]

Transfer function

the transfer function has only one adaptive coefficient

Our objective, then, is to minimize the power of the last subfilter output

*[3]

MSE

we could determine a frequency capture range based on the plotted MSE function

*[3]

Performance*[3]

References[1] INTRODUCTIONto ADAPTIVE SIGNAL PROCESSING Marc Moonen ,Department of Electrical Engineering ESAT/SISTA

K.U. Leuven, Leuven, Belgium[2] Overview of Adaptive Filters, Gner Arslan, from Adaptive Filter Theory, 4e by Simon Haykin, 2002 Prentice Hall Inc[3] Li Tan , Jean Jiang Novel Adaptive IIR Filter for Frequency Estimation and Tracking , IEEE SIGNAL PROCESSING MAGAZINE [186] NOVEMBER 2009

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