Lecture 7 Wiener filter

Lecture 7 Wiener filter

Stochastic Processes II José Biurrun Manresa 20.10.2010

Lecture 7 – Wiener filter

Introduction The process of extracting the information-carrying signal
 from the observed signal , where   
   and  is a noise process, is called filtering











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Lecture 7 – Wiener filter

Introduction • Typical filters are designed based on a frequency response approach

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Lecture 7 – Wiener filter

Introduction • Wiener filters, on the other hand, are based on a statistical approach • If the spectral properties of the signals involved are known, a linear time-invariant filter can be designed whose output would be as close as possible to the original signal

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Lecture 7 – Wiener filter

Problem formulation Given two stochastic processes, •
, the signal to be estimated •   , the observed signal Which • Have zero mean • Are jointly wide-sense stationary (WSS) • Have known autocorrelations   ,    and covariance     Estimate
 as a function of   5

Lecture 7 – Wiener filter

Problem formulation That is to say, 


             ∗  

In the frequency domain
    

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Lecture 7 – Wiener filter

Properties of stochastic processes A signal
 is WSS if ∀ ,  1. 
  = 
0 2.  ,      Two signals
 and  are jointly-WSS if ∀ ,  1.   ,   
       

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Lecture 7 – Wiener filter

Performance criterion What is a suitable criterion to estimate
?   
 


• Estimation error • Mean square error

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!

Aim: to minimize the mean square error, i.e., MMSE criterion " !   0 ∀ "   8

Lecture 7 – Wiener filter

Mean square error !

!

!

 

 


 


!


    

!



  
!      #        #  2
       

!

$



  & !     #    #  2      

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Lecture 7 – Wiener filter

Minimum mean square error " !  0 "   " & !     #    #  2      "    $  0  2  0  2 

'(

0

     2   0

2       2   0 

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Lecture 7 – Wiener filter

Wiener-Hopf equations 

        

In matrix form )  *  +  • These are called the Wiener-Hopf equations, a system of (N+1) linear equations (similar in formulation to the Yule-Walker equations) 11

Lecture 7 – Wiener filter

Wiener-Hopf equations )  *  +  • )  is positive semidefinite (Hermitian matrix with non negative eigenvalues) and non-singular (has an inverse) • Further, it is a Toeplitz matrix (constant along the diagonals) • There exist efficient algorithms (Levinson-Durbin and others) that utilize this structure to efficiently compute * • Solving for * will result in a (N+1) FIR filter 12

Lecture 7 – Wiener filter

Wiener filter transfer function 

        ↔ ∗         

Taking the Fourier transform on both sides   -   - 

-    ↔   - 

•  represents the transfer function of the Wiener filter in the frequency domain 13

Lecture 7 – Wiener filter

Special cases The noise term in   
   is white, zero-mean, and uncorrelated to the original signal. Then -   -   & ! And       
    
   
        

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Lecture 7 – Wiener filter

Special cases It can be deduced that -    -  Replacing in the Filter transfer function, we obtain -   -      -  -   & !

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Lecture 7 – Wiener filter

Special cases If noise is not white, then -   -      -  -   -  1 1     1 -  1  1 -/ -  The transfer function of the filter depends of the signal-to –noise ratio frequency by frequency 16