电子科技大学：《现代数字信号处理理论与算法 Modern theory and algorithm of digital signal processing》课程教学资源（课件讲稿）03 Wiener Filter

CH3 Wiener Filter Wiener filter theory and least squares parameter estimation serve as a basis for the derivation of the basic adaptive filtering algorithms: LMS algorithm from Wiener filter theory RLS procedure from least squares estimation theory

CH3 Wiener Filter Wiener filter theory and least squares parameter estimation serve as a basis for the derivation of the basic adaptive filtering algorithms: LMS algorithm from Wiener filter theory RLS procedure from least squares estimation theory

Contents o S.1 Prototype adaptive filter Filter structure Quadratic cost functions o $2.Wiener filter theory MMSE criterion Wiener-Hopf equations Orthogonality principle o S3.Unrealizable Wiener filter 2020-01-18 2

2020-01-18 2 Contents  S.1 Prototype adaptive filter  Filter structure  Quadratic cost functions  S2. Wiener filter theory  MMSE criterion  Wiener-Hopf equations  Orthogonality principle  S3. Unrealizable Wiener filter

S1.Prototype adaptive filter o Filter structure o Quadratic cost functions 2020-01-18 3

2020-01-18 3 S1. Prototype adaptive filter  Filter structure  Quadratic cost functions

Basic processes Filtering filter input and filter coefficient adaptive adaptation. filter parameters filter filter structure filter output and cost function, have a profound effect on the operation of the error desired signal adaptive filter. 2020-01-18 4

2020-01-18 4 Basic processes Filtering and filter coefficient adaptation. filter structure and cost function, have a profound effect on the operation of the adaptive filter

(1).Filter structure o Transversal FIR filters tractable mathematics,fairly simple algorithms Unconditionally stable filters o Generalization to adaptive IIR filters is nontrivial stability problems non-unimodal optimization problems o Non-linear filters Volterra filters neural network type filters 2020-01-18 5

2020-01-18 5 (1). Filter structure  Transversal FIR filters  tractable mathematics, fairly simple algorithms  Unconditionally stable filters  Generalization to adaptive IIR filters is nontrivial  stability problems  non-unimodal optimization problems  Non-linear filters  Volterra filters  neural network type filters

Tapped-delay line filter o Components unit delay elements,multiply-add cells o Order of the filter The number of delay elements The length of the impulse response. o Tap weights:coefficients w; Define the shape of the filter response It is the weights that will be adjusted o Alternative structures lattice filters:more desirable properties 2020-01-18 6

2020-01-18 6 Tapped-delay line filter  Components  unit delay elements, multiply-add cells  Order of the filter  The number of delay elements  The length of the impulse response.  Tap weights: coefficients wi  Define the shape of the filter response  It is the weights that will be adjusted  Alternative structures  lattice filters: more desirable properties

Single input FIR filter formula M-1 o input-output relation d(n)=∑wiu(n-k) k=0 o Vector form d(n)=w"u(n) w=[w。w·ww-] u(n)=[u(n)u(n-1)...u(n-M+1)] o z-transform of the input-output relation D(z)=W(z)U(2) 2020-01-18 7

2020-01-18 7 Single input FIR filter formula  input-output relation  Vector form  z-transform of the input-output relation     0 1 1 ( ) ( ) ( 1) ( 1) T M T w w w n u n u n u n M       w u 1 * 0 ˆ ( ) ( ) M k k d n w u n k      ˆ ( ) ( ) H d n n  w u ˆ D z W z U z ( ) ( ) ( ) 

W(z) o The z-transformed sequence o Shorthand to refer to a filter with impulse response (or weights) w=[%W…ww-]Y o When we consider adaptation techniques, where the filter assumes a time-varying form,the above z-transform interpretation of the filter's input-output relation is no longer valid 2020-01-18 8

2020-01-18 8 W(z)  The z-transformed sequence  Shorthand to refer to a filter with impulse response (or weights)  When we consider adaptation techniques, where the filter assumes a time-varying form, the above z-transform interpretation of the filter’s input-output relation is no longer valid  0 1 1  T w  w w wM 

Multiple input:multi-channel FIR filtering o interference cancellation with several reference sensors o antenna array processing o Trivially generalized to the multi- input case 2020-01-18 9

2020-01-18 9 Multiple input: multi-channel FIR filtering  interference cancellation with several reference sensors  antenna array processing  Trivially generalized to the multi￾input case

Linear combiner versus FlR filter o Adaptive linear combiner Narrow band beam-forming:multi-input adaptive filter reduces to zero-order FIR filters (special cases of the multi-channel FIR filter) Single input FIR filter may be viewed as a special case of the linear combiner o Adaptation algorithms have very similar mathematical formulations o Adaptive FIR algorithms are a factor M more efficient than the linear combiner algorithms! 2020-01-18 10

2020-01-18 10 Linear combiner versus FIR filter  Adaptive linear combiner  Narrow band beam-forming: multi-input adaptive filter reduces to zero-order FIR filters (special cases of the multi-channel FIR filter)  Single input FIR filter may be viewed as a special case of the linear combiner  Adaptation algorithms have very similar mathematical formulations  Adaptive FIR algorithms are a factor M more efficient than the linear combiner algorithms!