Advance Digital Signal Process

Final Tutorial

Introduction of Weiner Filter

學號: R98943133

姓名: 馮紹惟老師: 丁建均教授

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ContentsAbstract 3

I. Introduction of Wiener Filter 4

Weiner FilterII. The Linear Optimal Filtering Problem 6

a. Linear Estimation with Mean-Square Error Criterion 7

III. The Real-Valued Case 9a. Minimization of performance function 10

b. Error performance surface for FIR filtering 12

c. Explicit form of J (w) 12

d. Canonical form of J (w) 13

IV. Principle of Orthogonality 15a. See the Wiener-Hopf Equation by Principle of Orthogonality

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b. More Of Principle of Orthogonality 17

V. Normalized Performance Function 19

VI. The Complex-Valued Case 20

VII. Application 22a. Modelling 22

b. Inverse Modelling 23

VIII. Implementation Issues 28

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Summary 30

Reference 31Abstract

The Wiener filter was introduced by Norbert Wiener in the 1940's and published in 1949 in signal processing. A major contribution was the use of a statistical model for the estimated signal (the Bayesian approach!). And the Wiener filter solves the signal estimation problem for stationary signals. Because the theory behind this filter assumes that the inputs are stationary, a wiener filter is not an adaptive filter. So the filter is optimal in the sense of the MMSE. The wiener filter’s main purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal.

As we shall see, the Kalman filter solves the corresponding filtering problem in greater generality, for non-stationary signals. We shall focus here on the discrete-time version of the Wiener filter.

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I. Introduction of wiener functionWiener filters are a class of optimum linear filters which involve linear estimation

of a desired signal sequence from another related sequence. In the statistical approach to the solution of the linear filtering problem, we assume the availability of certain statistical parameters (e.g. mean and correlation functions) of the useful signal and unwanted additive noise. The goal of the Wiener filter is to filter out noise that has corrupted a signal. It is based on a statistical approach. The problem is to design a linear filter with the noisy data as input and the requirement of minimizing the effect of the noise at the filter output according to some statistical criterion. A useful approach to this filter-optimization problem is to minimize the mean-square value of the error signal that is defined as the difference between some desired response and the actual filter output. For stationary inputs, the resulting solution is commonly known as the Weiner filter.

The Weiner filter is inadequate for dealing with situations in which nonstationarity of the signal and/or noise is intrinsic to the problem. In such situations, the optimum filter has to be assumed a time-varying form. A highly successful solution to this more difficult problem is found in the Kalman filter.

Now we summarize some Wiener filters characteristics Assumption:

Signal and (additive) noise are stationary linear stochastic processes with known spectral characteristics or known autocorrelation and cross-correlation

Requirement:We want to find the linear MMSE estimate of sk based on (all or part of)

yk. So there are three versions of this problem:a. The causal filter:

b. The non-causal filter:

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c. The FIR filter:

And we consider in this tutorial the FIR case for simplicity.

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II. Wiener Filter － The Linear Optimal filtering problemThere is a signal model, showed in Fig. 1,

Fig. 1 A signal model

Where u(n) is the measured value of the desired signal d(n). And there are some examples of measurement process:

Additives noise problem u(n) = d(n) + v(n) Linear measurement u(n) = d(n)* h(n) + v(n) Simple delay u(n) = d(n −1) + v(n)

=> It becomes a prediction problem Interference problem u(n) = d(n) + i(n) + v(n)

And the filtering main problem is to find an estimation of d(n) by applying a linear filter on u(n)

Fig. 2 The filtering goal

When the filter is restricted to be a linear filter => it is a linear filtering problem. Then the filter is designed to optimize a performance index of the filtering process, such as

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minwk

¿ y (n )−d(n)∨¿¿

minwk

¿ y (n )−d(n)∨¿2 ¿

minwk

E ¿¿

maxwk

SNR of y (n )

⋯

Solving wopt is a linear optimal filtering problem

The whole process can indicate Fig. 3

Fig. 3

where the output is y (n )= d (n )=∑k=0

∞

wk¿ u (n−k ) for n=0 ,1 , 2 ,⋯

a. Linear Estimation with Mean-Square Error CriterionFig. 4 shows the block schematic of a linear discrete-time filter W ( z ) for

estimating a desired signal d (n ) based on an excitation u (n ). We assume that both u (n ) and d (n ) are random processes (discrete-time random signals). The filter output is y (n ) and e (n ) is the estimation error.

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Fig. 4

To find the optimum filter parameters, the cost function or performance function must be selected. In choosing a performance function the following points have to be considered：

1. The performance function must be mathematically tractable.2. The performance function should preferably have a single minimum so that

the optimum set of the filter parameters could be selected unambiguously.

The number of minima points for a performance function is closely related to the filter structure. The recursive (IIR) filters, in general, result in performance function that may have many minima, whereas the non-recursive (FIR) filters are guaranteed to have a single global minimum point if a proper performance function is used.

In Weiner filter, the performance function is chosen to be

J (w )=E¿

This is also called “mean-square error criterion”

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u(n)

III.Wiener Filter － The Real-Valued CaseFig. 5 shows a transversal filter (FIR) with tap weights W 0 , W 1 , …,W N −1.

Fig. 5Let

W =[W 0 ,W 1 ,…,W N−1 ]T

U (n)=[u0 ,u1 ,…,un−N+1 ]T

The output is

y (n )=∑i=0

N−1

wiu (n−i)=W T U (n )=UT (n )W ……………[1]

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u(n) u(n−1)

Thus we may write

e (n )=d (n )− y ( n )=d (n )−UT (n)W ……………[2]

The performance function, or cost function, is then given by

J (w )=E [|e (n )|2 ]

¿ E [ (d (n )−W T U (n)) ( d ( n )−UT (n)W )] ¿ E [d2(n)]−W T E [U ( n ) d (n ) ]−E [UT (n ) d (n ) ] W +W T E [U (n ) UT (n ) ]W … [3]

Now we define the Nx1 cross-correlation vector

P ≡ E [U (n ) d (n ) ]=[ p0 , p1, …, pN −1 ]T

and the NxN autocorrelation matrix

R ≡ E [U (n ) UT (n ) ]¿ [r 0,0 r0,1 ⋯ r0 , N −1

r 1,0 r1,1 ¿ ¿⋮ ¿¿ ⋮ ¿r N−1,0¿r 00¿⋯¿r N −1 , N −1¿]

Also we note that

E [d (n ) UT (n ) ]=PT

W T P=PT W

Thus we obtain

J (w )=E [d2(n)]−2W T P+W T R W ………… …[4 ]

Equation 4 is a quadratic function of the tap-weight vector W with a single global minimum. We note that R has to be a positive definite matrix in order to have a unique minimum point in the w-space.

a. Minimization of performance function

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To obtain the set of tap weights that minimize the performance function, we set

∂ J ( w )∂ wi

=0 , for i=0 ,1 ,⋯ , N−1

or

∇ J ( w )=0

where ∇is the gradient vector defined as the column vector

Δ=[ ∂∂ w0

, ∂∂w1

,⋯ , ∂∂ wN −1 ]

T

and zero vector 0 is defined as N-component vector

0=[ 0 , 0 ,⋯ , 0 ]T

Equation 4 can be expanded as

J (w )=E [d2(n)]−2∑l=0

N −1

p l wl+∑l=0

N−1

∑m=0

N −1

wl wm r lm

and ∑l=0

N−1

∑m=0

N −1

wl wm rlm can be expanded as

∑l=0

N−1

∑m=0

N −1

wl wm r lm= ∑l=0 ,l ≠i

N−1

∑m=0 ,m≠i

N−1

wl wmr lm+wi ∑m≠ 0 ,l ≠ i

N −1

wm rℑ+wi2 rii

Then we obtain

∂ J ( w )∂ wi

=−2 p i+∑l=0

N −1

W l(rli+ril) , for i=0 ,1 , 2,⋯ , N−1……………[5]

By setting ∂ J ( w )

∂ wi=0, we obtain

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∑l=0

N−1

W l(r li+r il)=2 pi

Note that

rli=E [ x ( n−l ) x (n−i ) ]=R xx(i−l)

The symmetry property of autocorrelation function of real-valued signal, we have the relation

rli=r il

Equation 5 then becomes

∑l=0

N−1

ril wl=pi

In matrix notation, we then obtain

R W op=P …………… [6]

where W op is the optimum tap-weight vector.Equation 6 is also known as the Wiener-Hopf equation, which has the solution

W op=R−1 P

assuming that R has inverse.

b. Error performance surface for FIR filteringBy J ≡ E [e (n ) x (n−i) ], it can be deduced that

J=function [ w∨d (n ) , u(n)]=function [¿data ], and J=function [ w ] is defined as the error performance surface over all possible weights w .

c. Explicit form of J (w)

J=E [(d (n )−∑k=0

M−1

wk¿ u(n−k ))(d¿ (n )−∑

i=0

M−1

w iu¿ (n−i))]

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¿σ d2−∑

k=0

M−1

wk¿ p (−k )−∑

k=0

M −1

wk p¿ (−k )+∑k∑

iwk

¿ w ir (i−k )

¿Quadratic function of w0 ,w1 ,⋯ , wM −1

¿ A surface∈(M +1 )−dimension space

d. Canonical form of J (w)J (w) can be written in matrix form as

J (w )=σ d2−wH p−pH w+wH R w

And by

(w−R−1 p)H R(w−R−1 p)

¿wH R w−pH R−1 R w−wH R R−1 p+ pH R−1 R R−1 p

¿wH R w−pH w−wH p+ pH R−1 p

The Hermitian property of R−1 has been used, ( R−1)H=R−1

Then J (w) can be put into a perfect square-form as

J (w )=σ d2−pH R−1 p+(w−R−1 p)H R (w−R−1 p)

The squared form can yield the minimum MSE explicitly as

J (w )=σ d2−pH R−1 p+(w−w 0)

H R(w−w 0)

⟹when w=w0(¿R−1 p) is found, we have the minimum value of Jmin is

Jmin=E [d2(n)]−W opT P

Jmin=E [d2(n)]−W opT R W op… ……… … [7 ]

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Equation 7 can also expressed as

Jmin=E [d2(n)]−PT R−1 P

And the squared form becomes

J (w )=J min+ (w−w0 )H R (w−w0 )=Jmin+ε H R ε

where ε tape error=w−w0

If R is written in its similarity form, then J (w ) can be put into a more informative form named Canonical Form.

Using the eigen-decomposition R=Q ⋀ QH , we have

J (w )=J min+ (w−w0 )H Q ⋀ QH (w−w0 )=J min+vH ⋀ v ……The Canonical Form

Where v≡QH ε is the transformed vector of ε in the eigenspace of R.In eigenspace the vk

' s in v=[v1 , v2 , …,vM ]T are the decoupled tape-errors, then we have

J (w )=J min+v H ⋀ v

¿ Jmin+∑k=1

M

λk v k¿ vk=J min+∑

k=1

M

λk∨vk∨¿2 ¿

where ¿ vk∨¿2 ¿ is the error power of each coefficients in the eigenspace and λk is the weighting (ie. relative importance) of each coefficient error.IV. Wiener Filter － Principle of Orthogonality

For linear filtering problem, the wiener solution can be generalized to be a principle as following steps.1. The MSE cost function (or performance function) is given by

J (w )=E [e2(n)]

By the chain rule,

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∂ J ( w )∂ wi

=E[2 e (n)∂ e (n)∂ w i ] for i=0 , 1 ,2 ,⋯ ,N−1

where

e (n)=d(n)− y (n)

Since d (n) is independent of w i , we get

∂ e(n)∂ wi

=−∂ y (n)

∂ wi=−u(n−i)

Then we obtain

∂ J ( w )∂ wi

=−2 E [e (n ) u(n−i)] for i=0 , 1 ,2 ,⋯ , N−1……… ……[8]

2. When the Wiener filter tap, which weights are set to their optimal values,

∂ J ( w )∂ wi

=0

Hence, if e0(n) is the estimation error when w i are set to their optimal values, then equation 8 becomes

E [e0 (n )u (n−i)]=0 for i=0 , 1 ,2 ,⋯ , N−1

For the ergodic random process,

EΩ [ x (n−i)e¿ (n ) ]=∑n

x (n−i)e¿ (n )=⟨ x (n−i ) , e (n ) ⟩

⟹ ⟨ x (n−i ) , e (n ) ⟩=0means x (n−i )⊥e (n )

On the other hands, the estimation error is orthogonal to input u(n−i) for all i (Note: u(n−i) and e (n) are treated as vectors !).

That is, the estimation error is uncorrelated with the filter tap inputs, x (n−i). This is known as “ the principles of orthogonality”.

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3. We can also show that the optimal filter output is also uncorrelated with the estimation error. That is

E [e0 (n ) y 0(n)]=0 for i=0 , 1, 2 ,⋯ , N−1

This result indicates that the optimized Weiner filter output and theestimation error are “orthogonal”.

The orthogonality principle is important in that it can be generalized to many complicated linear filtering or linear estimation problem.

a. See the Wiener-Hopf Equation by Principle of OrthogonalityWiener-Hopf equation is a special case of the orthogonality principle, when

d (n )=u ( n )=∑k=1

M

wk¿ u(n−k ) which is the linear prediction problem. So we can derived

Weiner-Hopf equation based on the principle of orthogonality

E [u (n−k ) e¿ ( n ) ]=0

⟹ E[u (n−k )(d¿ (n )−∑i=1

M

w iu¿(n−i))]=0

⟹ E[∑i=1

M

w iu (n−k ) u¿(n−i)]=E [u (n−k )d¿ (n )]

where ∑i=1

M

wi r (i−k )=E [u (n−k ) u¿ (n ) ]=r (−k ) for k=1 M

This is the Wiener-Hopf equation R w=r¿. For optimal FIR filtering problem, d (n ) needs not to be u (n ), then the Wiener-Hopf equation is R w=p with p being the cross correlation vector of u (n−k ) and d (n ).

b. More Of Principle of OrthogonalitySince J is a quadratic function of w i 's , i.e.,

J(w0 , w1 ,⋯ ,w0 w1 , w1 w2 ,⋯ ,w02 , w1

2 ,⋯)

it has a bowel shape in the hyperspace of (w0 , w1 ,⋯), which has an unique extreme

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point at w opt .∇k J=0 is also a sufficient condition for minimizing J

This is the principle of orthogonality.

For the 2D case w ¿ [ w0 ,w1 ]T , the error surface is showed in Fig. 6

Fig. 6 Error surface of 2D case

→ y (n)⊥e ( n )By

E [e¿ (n ) y (n)]=E [e¿ (n )∑k

w k¿u (n−k)]=∑k

w k¿ E [e¿ (n ) u(n−k )]=0

=>Both input and output of the filter are orthogonal to the estimation error e (n ).The vector space interpretation

By y0⊥e0 and e0=d− y0 , i.e., eopt (n)=d (n)− yopt (n) => d=e0+ y 0 ,i.e., d is decomposed into two orthogonal components,

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Fig. 7 Estimation value + Estimation error = Desired signal.

In geometric term, d = projection of d on e0⊥ . This is a reasonable result !

V. Normalized Performance Function

1. If the optimal filter tap weights are expressed by w0 ,l , l=0 , 1 ,2 ,⋯ , N−1. The

estimation error is then given by

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e0 (n )=d (n )−∑l=0

N −1

w0 , lu (n−l)

and then

d (n )=e0 (n )+ y0 (n )

E [d2(n)]=E [ e02(n) ]+E [ y0

2(n)]+2E [e0(n) y0(n)]

2. We may note that

E [e02(n)]=J min

and we obtain

Jmin=E [d2(n)]−E [ y02(n)]

3. Define ρ as the normalized performance function, and

Jmin ≥ 0→ E [ d2(n) ]≥ E [ y02(n)]

ρ= JE [d2(n)]

4. ρ=1 when y (n)=15. ρ reaches its minimum value, ρmin when the filter tap-weights are chosen to achieve the minimum mean-squared error. This gives

ρmin=1−E [ y0

2(n) ]E [d2(n)]

and we have 0< ρmin<1

VI. Wiener Filter － The Complex-Valued Case19

In many practical applications, the random signals are complex-valued. For example, the baseband signal of QPSK & QAM in data transmission systems. In the Wiener filter for processing complex-valued random signals, the tap-weights are assumed to be complex variables.The estimation error, e (n), is also complex-valued. We may write

J=E ¿

The tap-weight w i is expressed by

w i=w i , R+ jw i , I

The gradient of a function with respect to a complex variable w i=w i , R+ jw i , I is defined as

∇wC ≡ ∂

∂ wR+ j ∂

∂ w I……………[9]

The optimum tap-weights of the complex-valued Wiener filter will be obtained from the criterion:

∇w i

C J=0 for i=0 ,1 , 2 ,⋯ , N−1

That is, ∂ J

∂ wi , R=0 and

∂ J∂ wi , I

=0

Since J=E [e(n)e¿(n)], we have

∇w i

C J=E[e (n )∇w i

C e¿ ( n )+e¿ (n )∇wi

C e (n )]……………[10]

Noting that

e (n )=d (n )−∑k=0

N −1

wk u(n−k )

∇w i

C e (n )=−u(n−i)∇w i

C w i

∇w i

C e¿ (n )=−u¿ (n−i)∇wi

C wi¿

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Applying the definition 9, we obtain

∇w i

C wi=∂ wi

∂ w i, R+ j

∂ w i

∂ wi , I=1+ j ( j )=0

and

∇w i

C wi¿=

∂ w i¿

∂ wi , R+ j

∂ wi¿

∂ w i , I=1+ j (− j )=2

Thus, equation 10 becomes

∇w i

C j=−2 E[e ( n )u¿(n−i)]

The optimum filter tap-weights are obtained when ∇w i

C J=0. This givens

E [e0 (n )u¿ (n−i ) ]=0 for i=0 ,1 , 2 ,⋯ , N−1……………[11]

where e0 (n ) is the optimum estimation error.Equation 11 is the “principle of orthogonality” for the case of complex-valued

signals in wiener filter.

The Wiener-Hopf equation can be derived as follows:Define

u (n ) ≡[u (n )u (n−1 )⋯u(n−N+1)]T

and

w ( n )≡ [w 0¿ w1

¿⋯wN −1¿ ]T

We can also write

u (n ) ≡[u¿ (n ) u¿ (n−1 )⋯ u¿(n−N+1)]H

and

w ( n )≡ [w0 w1⋯wN −1]H

where H denotes complex-conjugate transpose or Hermitian.Noting that

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e (n )=d (n )−w0H u (n)……………[12]

and

E [e0 (n )u (n ) ]=0

from equation 12, we have

E [u (n )(d¿ (n )−uH (n)w0)]=0

and then

R w0=P ……………[13]

where

R=E[u(n)uH (n)]

and

P=E [u(n)d¿(n)]

Equation 13 is the Wiener-Hopf equation for the case of complex-valued signals. The minimum performance function is then expressed as

Jmin=E [d2(n)]−W oH RW o

Remarks:In the derivation of the above Wiener filter we have made assumption that it is

causal and finite impulse response, for both real-valued and complex-valued signals.

VII. Wiener Filter － Application22

a. ModellingConsider the modeling problem depicted in Fig. 8

Fig. 8 The model of Modeling

μ(n), V 0(n), V i(n) are assumed to be stationary, zero-mean and uncorrelated with one another. The input to Wiener filter is given by

x (n )=μ (n )+V i(n)

and the desired output is given by

d (n )=gn∗μ (n )+V 0(n)

where gn is the impulse response sample of the plant.The optimum unconstrained Wiener filter transfer function

W o ( z )=Rdx (z)R xx(z)

Note that

R xx ( k )=E [ x ( n ) x (n−k ) ]

¿ E [ (μ (n )+V i(n)) ( μ ( n−k )+V i(n−k)) ]

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¿ E[ μ (n ) μ (n−k )]+E [ μ (n )V i (n−k ) ]+E [V i (n ) μ (n−k ) ]+ E[V i(n)V i(n−k )]¿ Rμμ (k )+RV i V i

(k )…… ………[14 ]

Taking Z-transform on both sides of equation 14, we get

R xx ( z )=Rμμ (z )+RV iV i( z )

To calculate W o ( z )=Rdx (z)Rxx(z) , we must first find the expression for Rdx (z), We can

show that

Rdx (z )=Rd' μ ( z )

where d ' (n ) is the plant output when the additive noise V 0(n) is excluded from that.Moreover, we have

Rd ' μ ( z )=G( z)Rμμ ( z )

Thus

Rdx (z )=G(z) Rμμ ( z )

and we obtain

W o ( z )=Rμμ ( z )

Rμμ ( z )+RV i V i(z )

G( z)……………[15]

We note that W o ( z ) is equal to G(z ) only when RV i V i( z )is equal to zero. That is,

when V i(n) is zero for all values of n. The noise sequence V i(n) may be thought of as introduced by a transducer that is used to get samples of the plant input. Replacing z by e jw in equation 15, we obtain

W o ( e jw )= Rμμ ( e jw )Rμμ ( e jw )+RV iV i

(e jw )G(e jw)

Define

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K (e jw )≡ Rμμ ( e jw )Rμμ ( e jw )+RV i V i

( e jw )

We obtain

W o ( e jw )=K ( e jw ) G(e jw)

With some mathematic manipulation, we can find the minimum mean-square error, min Jmin, expressed by

Jmin=Rμ0 μ0(0 )+ 1

2 π ∫−π

π

(1−K ( e jw))Rμμ (e jw )¿G(e jw)∨¿2dw ¿

The best performance that one can expect from the unconstrained Wiener filter is

Jmin=Rμ0 μ0(0 )

and this happens when V i (n )=0.The Wiener filter attempts to estimate that part of the target signal d (n ) that is

correlated with its own input x (n ) and leaves the remaining part of d (n ) (i.e. V o (n ) ) unaffected. This is known as “ the principles of correlation cancellation “.

b. Inverse ModellingFig. 9 depicts a channel equalization scenario.

Fig. 9 The model of inverse modeling

s (n ) :data samplesH ( z ): Systemfunction of communicationchanneln(n) :additive channel noise which isuncorrelated with s (n)W ( z ): filter functionused ¿ process thereceived noisy signal samples

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n(n)

¿recover the originaldata samples .When the additive noise at the channel output is absent, the equalizer has the

following trivial solution:

W o ( z )= 1H ( z )

This implies that y (n )=s (n ) and thus e (n )=0 for all n.When the channel noise, n(n), is non-zero, the solution provided by equation in

modeling may not be optimal.

x (n )=hn∗s (n )+ν (n )

and

d (n )=s (n )

where hn is the impulse response of the channel, H ( z ). From equation in modeling, we obtain

R xx ( z )=R ss ( z ) ¿ H ( z )∨¿2 Rμμ ( z ) ¿

Also

Rdx (z )=R sx ( z )=H (z¿¿−1)R ss ( z ) ¿

With z 1, we may also write

Rdx (z ) ¿ H ¿(z )R ss ( z )

And then

W o ( z )=H ¿( z)R ss ( z )

Rss ( z )¿ H ( z )∨¿2+Rμμ ( z ) ……………[16]¿

This is the general solution to the equalization problem when there is no constraint on the equalizer length and, also, it may be let to be non-causal.Equation 16 can be rewritten as

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W o ( z )= 1

1+Rμμ ( z )

R ss ( z ) ¿ H ( z )∨¿2 ∙ 1H ( z )

¿

Let z=e jw and define the parameter

ρ(e jw)≡ Rss ( e jw) ¿ H ( e jw )∨¿2

Rμμ ( e jw )¿

where R ss (e jw )¿H ( e jw )∨¿2¿ and Rμμ (e jw )are the signal power spectral density and the noise power spectral density, respectively, at the channel output.We obtain

W o ( e jw )= ρ(e jw)1+ρ(e jw)

∙ 1H (e jw )

We note that n(e jw) is a non-negative quantity, since it is the signal-to-noise power spectral density ratio at the equalizer input.

Also, 0 ≤ ρ(e jw)1+ρ(e jw)

≤ 1

Cancellation of ISI and noise enhancementConsider the optimized equalizer with frequency response given by

W o ( e jw )= ρ(e jw)1+ρ(e jw)

∙ 1H (e jw )

In the frequency regions where the noise is almost absent, the value of is very large and hence

W o ( e jw ) ≈ 1H (e jw )

The ISI will be eliminated without any significant enhancement of noise. On the other hand, in the frequency regions where the noise level is high, the value of n(e jw) is not large and hence the equalizer does not approximate the channel inverse well. This is of course, to prevent noise enhancement.

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VIII. Wiener Filter － Implementation IssuesThe Wiener (or MSE) solution exists in correlation domain, which needs X Ω(n) to

find the ACF and CCF in the Wiener-Hopf equation. This is the original theory

developed by Wiener for the linear prediction case given if we have another chance.

Fig. 10 The family of wiener filter in adaptive operation

The existence of Wiener solution depends on the availability of the desired signal d (n ), the requirement of d (n ) may be avoided by using a model of d (n ) or making it to be a constraint (e.g., LCMV). Another requirement for the existence of Wiener solution is that the RP must be stationary to ensure the existence of ACF and CCF representation. Kalman (MV) filtering is the major theory developed for making the Wiener solution adapt to nonstationary signals.

For real time implementation, the requirement of signal statistics (ACF and CCF) must be avoided=> search solution using LMS or estimate solution blockwise using LS.

Another concern for real time implementation is the computation issues in finding ACF, CCF and the inverse of ACM based on x (n ) instead of X Ω(n). Recursive algorithms for finding Wiener solution were developed for the above cases.

The Levinson-Durbin algorithm is developed for linear prediction filtering.

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Complicated recursive algorithms are used in Kalman filtering and RLS. LMS is the simplest recursive algorithm. SVD is the major technique for solving the LS solution.

So we do some conclusion, existence and computation are two major problems in finding the Wiener solution. And the existence problem includes: R , p ,d (n )∧R−1, we can find the computation problem is primarily for finding R−1.

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SummaryIn this tutorial, we described the discrete-time version of Wiener filter theory,

which has evolved from the pioneering work of Norbert Wiener on linear optimum filters for continuous-time signals. The importance of Wiener filter lies in the fact that it provides a frame of reference for the linear filtering of stochastic signals, assuming wide-sense stationarity. And the Wiener filter’s main purpose is that the output of filter can close to the desired response by filter out the noise which interference signal.

The Wiener filter has two important properties:1. The principle of orthogonality:

The error signal (estimation error) produced by the Wiener filter is orthogonal to its tap inputs.

2. Statistical characterization of the error signal as white noise:This condition is attained when the filter length matches the order if the multiple regression model describing the generation of the observable data (i.e., the desired response).

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Reference Wiener, Norbert (1949). Extrapolation, Interpolation, and Smoothing of

Stationary Time Series. New York: Wiley. ISBN 0-262-73005-7. Simon Haykin ： Adaptive Filter Theory, 4rd Ed. Prentice-Hall , 2002. Paulo S.R. Diniz ： Adaptive Filtering, Kluwer , 1994 D.G. Manolakis “Statistical and adaptive signal processing” et. al. 2000 WIKIPEDIA, “Weiner Filter” 張文輝教授, “適應性訊號處理”課程講義 魏哲和教授, “Adaptive Signal Processing”課程講義 曹建和教授, “Adaptive Signal Processing”課程講義

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