lecture_25 - MIT OpenCourseWare http:/ocw.mit.edu 2.161...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
e n f n y n - 1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 25 1 Reading: Class Handout: Introduction to Least-Squares Adaptive Filters Class Handout: Introduction to Recursive-Least-Squares (RLS) Adaptive Filters Proakis and Manolakis: Secs. 13.1 13.3 Adaptive Filtering In Lecture 24 we looked at the least-squares approach to FIR filter design. The filter coef- ficients b m were generated from a one-time set of experimental data, and then used subse- quently, with the assumption of stationarity. In other words, the design and utilization of the filter were decoupled. We now extend the design method to adaptive FIR filters , where the coefficients are continually adjusted on a step-by-step basis during the filtering operation. Unlike the static least-squares filters, which assume stationarity of the input, adaptive filters can track slowly changing statistics in the input waveform. The adaptive structure is shown in below. The adaptive filter is FIR of length M with coefficients b k , k =0 , 1 , 2 ,...,M 1. The input stream { f ( n ) } is passed through the filter to produce the sequence { y ( n ) } . At each time-step the filter coefficients are updated using an error e ( n )= d ( n ) y ( n ) where d ( n ) is the desired response (usually based of { f ( n ) } ). c a u s a l l i n e a r F I R f i l t e r H ( z ) A d a p t i v e L e a s t - S q u a r e s A l g o r i t h m d n + e r r o r f i l t e r c o e f f i c i e n t s The filter is not designed to handle a particular input. Because it is adaptive, it can adjust to a broadly defined task. 1 copyright ± c D.Rowell 2008 25–1
Background image of page 2
± f ( n ) k k 1.1 The Adaptive LMS Filter Algorithm 1.1.1 Simplified Derivation In the length M FIR adaptive filter the coefficients b k ( n ), k =1 , 2 ,...,M 1, at time step n are adjusted continuously to minimize a step-by-step squared-error performance index J ( n ): ² ³ 2 M 1 J ( n )= e 2 ( n )=( d ( n ) y ( n )) 2 = d ( n ) b ( k ) f ( n k ) k =0 J ( n ) is described by a quadratic surface in the b k ( n ), and therefore has a single minimum. At each iteration we seek to reduce J ( n ) using the “steepest descent” optimization method, that is we move each b k ( n ) an amount proportional to ∂J ( n ) /∂b ( k ). In other words at step n + 1 we modify the filter coefficients from the previous step: ( n ) b k ( n +1) = b k ( n ) Λ( n ) , k =0 , 1 , 2 ,...M 1 ∂b k ( n ) where Λ(n) is an empirically chosen parameter that defines the step size, and hence the rate of convergence. (In many applications Λ( n ) = Λ, a constant.) Then ( n ) ∂e 2 ( n ) ( n ) = =2 e ( n ) = 2 e ( n ) f ( n k ) k k k and the fixed-gain FIR adaptive Least-Mean-Square (LMS) filter algorithm is b k ( n = b k ( n )+Λ e ( n ) f ( n k ) , k , 1 , 2 1 or in matrix form b ( n = b ( n e ( n ) f ( n ) , where b ( n )=[ b 0 ( n ) b 1 ( n ) b 2 ( n ) ··· b M 1 ] T is a column vector of the filter coefficients, and f ( n f ( n ) f ( n 1) f ( n 2) f ( n ( M
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.

Page1 / 13

lecture_25 - MIT OpenCourseWare http:/ocw.mit.edu 2.161...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online