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Adaptive Filters S.B.Rabet In the Name of GOD Class Presentation For The Course : Custom Implementation of DSP Systems University of Tehran 2010 Pages 9 ~15 are copied from second reference [ “Overview of Adaptive Filters”, Güner 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.

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2 introduction Linear 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]

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3 Wiener filter work 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]

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4 Adaptive filter the 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]

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5 Adaptive filter In 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]

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6 Adaptive filter The 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]

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7 Adaptive filter Because 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]

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8 Adaptive filter The generalization to adaptive IIR filters leads to stability problems It’s common to use a FIR digital filter with adjustable coefficients. [1]

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9 LMS Algorithm Most 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]

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10 LMS Algorithm [2]

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11 Stability of LMS The 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 data More practical test for stability is Larger values for step size –Increases adaptation rate (faster adaptation) –Increases residual mean-squared error [2]

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12 Applications of Adaptive Filters: Identification Used to provide a linear model of an unknown plant Applications: –System identification [2]

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13 Applications of Adaptive Filters: Inverse Modeling Used to provide an inverse model of an unknown plant Applications: –Equalization (communications channels) [2]

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14 Applications of Adaptive Filters: Prediction Used to provide a prediction of the present value of a random signal Applications: –Linear predictive coding [2]

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15 Applications of Adaptive Filters: Interference Cancellation Used to cancel unknown interference from a primary signal Applications: –Echo / Noise cancellation hands-free carphone, aircraft headphones etc [2]

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Example: Acoustic Echo Cancellation 16 [1]

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A new work Novel Adaptive IIR Filter for Frequency Estimation and Tracking In many applications we may want to estimate (track) the signal’s 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 tracking in a harmonic frequency environment 17 [3]

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Structure frequency 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) 18 [3]

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Pole zero plot 19 Parameter r is chosen to be close to, but less than, one to achieve narrowband notches and avoid any filter stability problems [3]

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Transfer function the transfer function has only one adaptive coefficient Our objective, then, is to minimize the power of the last subfilter output 20 [3]

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MSE we could determine a frequency capture range based on the plotted MSE function 21 [3]

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Performance 22 [3]

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References [1] “INTRODUCTIONto ADAPTIVE SIGNAL PROCESSING” Marc Moonen,Department of Electrical Engineering · ESAT/SISTA K.U. Leuven, Leuven, Belgium [2] “Overview of Adaptive Filters”, Güner 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 23

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Goals of Adaptive Signal Processing Design algorithms that learn from training data Algorithms must have good properties: attain good solutions, simple.

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