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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 .
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F I R f i l t e r H ( z ) " d e s i r e d " o u t p u t { g } n { f } { e } e r r o r l e a s t - s q u a r e s d e s i g n a l g o r i t h m n n { d } n + - i n p u t o u t p u t { g } n f i l t e r c o e f f i c i e n t s b k { e } n { f } n Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 24 1 Reading: Class Handout: MATLAB Examples of Least-Squares FIR Filter Design Proakis and Manolakis: Sec. 12.3 12.5 Stearns and Hush: Ch. 14 1 Least-Squares Filter Design We now look at a FIR filter design technique that is based on “experimental” data Given an input sequence { f n } , and a “desired” filtered output sequence { d n } , the task is to design a FIR filter M 1 H ( z ) = b k z k k =0 that will minimize the error { e n } = { d n }−{ g n } in some sense, where { g n } is the filter output. In particular, we will look at a filter design method that minimizes the mean-squared-error (MSE), where 2 MSE = E e n = E ( d n g n ) 2 2 = E d 2 n + E g n 2 E { d n g n } 1 copyright c D.Rowell 2008 24–1
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1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 24 1 Reading: Class Handout: MATLAB Examples of Least-Squares FIR Filter Design Proakis and Manolakis: Sec. 12.3 12.5 Stearns and Hush: Ch. 14 Least-Squares Filter Design We now look at a FIR filter design technique that is based on “experimental” data " d e s i r e d " o u t p u t { d n } i n p u t F I R f i l t e r { g n } + e r r o r { f n } H ( z ) - { e n } o u t p u t { g n } b k f i l t e r c o e f f i c i e n t s { f } n l e a s t - s q u a r e s { e n } d e s i g n a l g o r i t h m Given an input sequence { f n } , and a “desired” filtered output sequence { d n } , the task is to design a FIR filter M 1 H ( z ) = b k z k k =0 that will minimize the error { e n } = { d n }−{ g n } in some sense, where { g n } is the filter output. In particular, we will look at a filter design method that minimizes the mean-squared-error (MSE), where MSE = E e 2 n = E ( d n g n ) 2 = E d 2 + E g 2 2 E { d n g n } n n 1 copyright c D.Rowell 2008 24–1
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( ) and in terms of correlation functions MSE = φ dd (0) + φ gg (0) 2 φ dg (0) . For the FIR filter with coefficients b k , the output is { g n } = { f n } ⊗ { b n } and from the input/output properties of linear systems (Lec. 22) φ gg ( n ) = Z 1 H ( z ) H ( z 1 ) φ ff ( n ) M 1 M 1 = b m b n φ ff ( m n ) . m =0 n =0 Similarly, φ dg ( n ) = E { d m g n + m } M 1 = E d m b k f n + m k k =0 M 1 = b k E { d m f n + m k } k =0 M 1 = b k φ fd ( k n ) k =0 and M 1 φ dg (0) = b k φ fd ( k ) k =0 The MSE is therefore M 1 M 1 M 1 MSE = φ dd (0) + b m b n φ ff ( m n ) 2 b k φ fd ( k ) .
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