lecture2forelectronics and communication.ppt
- 1. Overview of Adaptive Signal Processing by Hajira Fathima Assistant Professor & I/c Head of ECE Department Maulana Azad National Urdu University Hyderabad Content and Figures are from Adaptive Filter Theory, 4e by Simon Haykin, ©2002 Prentice Hall Inc.
- 2. The Filtering Problem • Filters may be used for three information-processing tasks – Filtering – Smoothing – Prediction • Given an optimality criteria we often can design optimal filters – Requires a priori information about the environment – Example: Under certain conditions the so called Wiener filter is optimal in the mean-squared sense • Adaptive filters are self-designing using a recursive algorithm – Useful if complete knowledge of environment is not available a priori
- 3. Applications of Adaptive Filters: Identification • Used to provide a linear model of an unknown plant • Parameters – u=input of adaptive filter=input to plant – y=output of adaptive filter – d=desired response=output of plant – e=d-y=estimation error • Applications: – System identification
- 4. Applications of Adaptive Filters: Inverse Modeling • Used to provide an inverse model of an unknown plant • Parameters – u=input of adaptive filter=output to plant – y=output of adaptive filter – d=desired response=delayed system input – e=d-y=estimation error • Applications: – Equalization
- 5. Applications of Adaptive Filters: Prediction • Used to provide a prediction of the present value of a random signal • Parameters – u=input of adaptive filter=delayed version of random signal – y=output of adaptive filter – d=desired response=random signal – e=d-y=estimation error=system output • Applications: – Linear predictive coding
- 6. Applications of Adaptive Filters: Interference Cancellation • Used to cancel unknown interference from a primary signal • Parameters – u=input of adaptive filter=reference signal – y=output of adaptive filter – d=desired response=primary signal – e=d-y=estimation error=system output • Applications: – Echo cancellation
- 7. Stochastic Gradient Approach • Most commonly used type of Adaptive Filters • Define cost function as mean-squared error • Difference between filter output and desired response • Based on the method of steepest descent – Move towards the minimum on the error surface to get to minimum – Requires the gradient of the error surface to be known • Most popular adaptation algorithm is LMS – Derived from steepest descent – Doesn’t require gradient to be know: it is estimated at every iteration • Least-Mean-Square (LMS) Algorithm signal error vector input tap parameter rate - learning vector weight - tap of value old vector weigth - tap of value update
- 8. Least-Mean-Square (LMS) Algorithm • The LMS Algorithm consists of two basic processes – Filtering process • Calculate the output of FIR filter by convolving input and taps • Calculate estimation error by comparing the output to desired signal – Adaptation process • Adjust tap weights based on the estimation error
- 9. LMS Algorithm Steps • Filter output • Estimation error • Tap-weight adaptation 1 M 0 k * k n w k n u n y n y n d n e n e k n u n w 1 n w * k k
- 10. 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 • Demos – http://www.eas.asu.edu/~dsp/grad/anand/java/ANC/ANC.html – http://www.eas.asu.edu/~dsp/grad/anand/java/AdaptiveFilter/Zero.html max 2 0 power signal input 2 0
- 11. Questions ?