adaptive filtering chapter3

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    CHAPTER 3

    Adaptive Tapped-delay-line Filters

    Using the Gradient Approach

    Adaptive FilteringAdaptive Filtering

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    In the case of known correlation expression the solution for the optimalcoefficients of the tapped-delay-line filter was

    (1.25)pRh-1

    0 !

    Where Rwas the correlation matrix of the filter tap inputs and p wasthe cross-correlation between the input vector and a desired response.

    If the filter operates in an environment where Rand p are unknown, we

    may use all the data collected up to and including time n to compute the

    estimates and in order to solve the normal equations. When,however the tapped-delay-line filter contains a large number of

    coefficients this procedure is highly inefficient. A more efficient

    approach is to use an adaptive filter.

    )(nR )(np

    Adaptive Tapped-delay-line Filters Using the Gradient Approach

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    Figure 3.1 Block diagram of adaptive filter

    Processes:

    1) adaptive or training process

    2) filtering or operating process. (d(n)= desired response must be

    provided)

    Adaptive Tapped-delay-line Filters Using the Gradient Approach

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    Let y(n) denote the output of the tapped-delay-line filter at time n,

    as shown by the convolution sum

    !

    !k

    knunkhny1

    )1(),()((3.1)

    the errorsignalis

    The error signal e(n) is utilized by the adaptive process to generatecorrections at each iteration to be applied to the tap coefficients inorder to move closer to the optimum Wiener configuration.

    )()()( nyndne ! (3.2)

    Adaptive Tapped-delay-line Filters Using the Gradient Approach

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    ! ! !

    !M

    k

    M

    k

    M

    m

    d kmrnmhnkhkpnkhPn1 1 1

    )(),(),()1(),(2)(I(3.3)

    where the quantities , and

    are results of ensemble

    averaging.

    )]([2 ndEPd ! )]1()([)1( ! knundEkp

    )]1()1([)( ! mnuknuEkmr

    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    Assuming the values ofh(1,n), h(2,n), . . . , h(M,n) are known, thevalue of the mean squared error is

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The Method of Steepest Descent

    The dependence ofI(n)on the filter coefficient can be visualized as abowl-shaped surface with an unique minimum. The adaptive processhas the task of continually seeking the minimum point of this error

    performance surface.The optimization technique used is the method of steepest descent.

    First we compute the M-by-1 gradient vector whose kth

    elementis

    )(n

    !

    !!x

    xM

    m

    Mkkmrnmhkpnkh 1

    ,...2,1)(),(2)1(2),(

    I

    (3.4)

    which is obtained by differentiating both sides of Eq. (3.3) withrespect to h(k,n).

    This expression can be simplified to obtain

    ? A MkknuneEnkh

    ,...2,1)1()(2),(

    !!x

    xI

    (3.5)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The gradient vector is then written as

    ? A? A

    ? A

    !

    xx

    xx

    xx

    !

    )1()(2

    .

    .

    )1()(2

    )()(2

    ),(/)(

    .

    .

    ),2(/)(

    ),1(/)(

    )(

    MnuneE

    nuneE

    nuneE

    nMhn

    nhn

    nhn

    n

    I

    I

    I

    which can be given in short-hand notation as

    (3.6)

    ? A)()(2)( nneEn u! (3.7)

    ? ATM-nu-nunun 1)(1),...,(),()( !u (3.8)where

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    Now defining the M-by-1 coefficient vector

    ? ATnMhnhnhn ),(),...,(2,),(1,)( !h (3.9)The update equation for the coefficient vector according to the

    steepest-descent algorithm is defined as

    ? A)(21)(1)( nnn ! Qhh

    where the factor 1/2 has been introduced for convenience and Q is a

    positive scalar.

    Substituting Eq. (3.7) in Eq. (3.10) the update law becomes;

    ? A)()()(1)( nneEnn uhh Q!

    (3.10)

    (3.11)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The error signal e(n) is defined as

    )()()()( nnndne T hu!

    Substituting Eq. (3.12) into Eq. (3.11) we get

    (3.12)

    ? A? A ? A )()()()()(()())()()()(()(1)(

    nnnEndnEn

    nn-ndnEnnT

    T

    hh

    hhh

    QQ

    Q

    !

    !

    (3.13)

    which can be rewritten as

    phR-I

    Rh-phh

    QQ

    QQ

    !

    !

    )()(

    )()(1)(

    n

    nnn

    (3.14)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    Stability of the Steepest Descent Algorithm

    The stability performance of the algorithm depends on

    i) the correlation matrix R(determined by the process)

    ii) the step-size parameterQ (to be chosen by designer)

    For the stability analysis define a coefficient error vector as

    0)()( hhc ! nn

    where h0

    is the solution of pRh !0 (3.16)

    (3.15)

    Subtracting h0 from both sides of (3.14) and using the normal

    equation to eliminate p we get

    ? A00 -)()(1)( hhR-Ihh nn Q! (3.17)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    According to the definition of the coefficient error vector we have

    )()(1)( nn cR-Ic Q!

    We can represent Rin terms of its eigenvalue matrix 0

    0!RQQT

    Premultiplying (3.18) by QT we get

    (3.18)

    (3.19)

    )(-)()()(1)( nnnn TTTT RcQcQcR-IQcQ QQ !! (3.20)

    Define the transformed coefficient error vector

    )()( nn TcQv ! (3.21)

    which implies

    1)(1)( ! nn TcQv (3.22)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    Since )()()()( nnnnTTTT vcRQQQRIcQRcQ 0!!!

    can be written. Thus we obtain )()(1)( nn v-Iv 0! Q

    IQQ !T

    (3.23)

    which represents a system ofn uncoupled scalar valued first-order

    difference equations, the kth one being

    M1,2,...,knvnv kkk !! )()1(1)( QP- (3.24)

    with solution

    M1,2,...,kvnv kn

    kk !! (0))1()( QP- (3.25)

    For stability

    M1,2,...,kk ! 110 QP-

    Therefore the steepest-descent algorithm is stable if

    maxP

    Q2

    0 (3.27)

    (3.26)

    ,

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The solution for the original coefficient vectorh(n) can be

    reformulated as follows:Premultiply (3.21) by Q to obtain )()()( nnn

    T ccQQQv !!

    Using (3.15) eliminate c(n) and solve forh(n) to obtain

    )()(0

    nn Qvhh !

    which can be rewritten in the form

    k

    M

    k

    k nvn qhh !

    !1

    0 )()(

    (3.28)

    (3.29)

    where qis are normalized eigenvectors associated with theeigenvalues Pis of the matrix R.

    Thus the behaviour of the ith coefficient is found to be

    M1,2,...,ivqihni,h

    M

    k

    n

    kkki !! !10 )1((0))()( QP- (3.30)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The Mean Squared Error

    At time n, the value of the mean squared error is given as

    !

    !M

    k

    kk nvn1

    2

    min )()( PII(3.31)

    Substituting (3.25) in (3.31) we get

    !

    !

    M

    k

    k

    2n

    kk vn1

    2

    min )0()1()( QPPII -(3.32)

    When the steepest descent algorithm is convergent, that is the step-

    size parameterQ is chosen within the bounds ,thenirrespective of the initial conditions.

    maxPQ 2/0

    min)(lim II !gp

    nn

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The steepest-descent algorithm, although shown to converge to theoptimum Wiener solution, irrespective of the initial conditionsunfortunately it requires exact measurements of the gradient vector ateach iteration which is not possible in reality.There is need for an algorithm that derives estimates of the gradientvector from the limited number of available data.One such algorithm is the so-called least-mean-square (LMS)

    algorithm which uses instantaneous unbiased estimates of thegradient vector in the form:

    )()(2)( nnen u!

    In terms of the coefficients update mechanism the LMS algorithm isformulated as

    ? A )()()()(2

    1)(1)( nnennnn uhhh QQ !!

    (3.33)

    (3.34)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The error signal was defined as

    )()()()( nnndne T hu! (3.12)

    which leads to the following adaptation scheme.

    Figure 3.2 Multidimensional signal-flow graph representation of the LMS algorithm

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    Convergence of the Coefficient Vector in the LMS Algorithm

    The statistical analysis of the LMS algorithm is based on the following

    conditions:1. Each sample vector u(n) of the input signal is assumed to beuncorrelated with all previous sample vectors u(k) fork= 0, 1,..., n-1.2. Each sample vectoru(n) of the input signal is uncorrelated with all

    previous samples of the desired response d(k) fork= 0, 1,..., n-1.

    Then from Eqs. (3.34) and (3.12), we observe that the coefficient vectorh(n+1) at time n+1 depends only on three inputs:

    1. The previous sample vectors of the input signal, namely, u(n), u(n-1)

    ,...,u(0).2. The previous samples of the desired response, namely, d(n), d(n-1),...,d(0).3.The initial value h(0) of the coefficient vector.

    The coefficient vectorh(n+1) is independent of both u(n+1)and d(n+1).

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    The analysis is as follows: Eliminate e(n) by substituting Eq. (3.34)

    in Eq. (3.12) to get? A

    ? A )()()()()()()()()()(1)(

    ndnnnn

    nnndnnn

    T

    T

    uhuuI

    huuhh

    QQ

    Q

    !

    !

    (3.35)

    Using (3.15) to eliminate h(n) from the right-hand side of Eq.(3.35) to get

    0)()( hhc ! nn (3.15)

    ? A? A

    ? A ? A00

    0

    )()()()()()()(

    )()()()()(1)(

    huuuhcuuI

    uhcuuIh

    nnndnnnn

    ndnnnnn

    TT

    T

    !

    !

    QQ

    QQ

    (3.36)

    which can be rewritten as

    ? A ? A0)()()()()()()(1)( huuucuuIc nnndnnnnn

    TT ! QQ (3.37)

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    Adaptive Tapped-delay-line Filters Using the Gradient Approach

    Taking expectations of both sides of (3.37) and using independence

    ofc(n) from u(n) we get? A ? A ? A? A ? A ? A ? A

    ? A 00

    0

    )(

    )()()()()()()(

    )()()()()()()(1)(

    Rh-pcRI

    huu-ucuuI

    huuucuuIc

    QQ

    QQ

    QQ

    !

    !

    !

    nE

    nnEndnEnEnnE

    nnndnEnnnEnE

    TT

    TT

    (3.38)

    where ? A)()( ndnEup ! ? A)()( nnE TuuR!andHowever since the right-hand side of (3.38) is zero, reducing topRh !0

    ? A ? A)(1)( nEnE cRIc Q!(3.39)

    which is in the same mathematical form as (3.18).

    Thus, the LMS algorithm converges in the mean provided that the step-

    size parameterQ satisfies the condition

    maxP

    Q2

    0 (3.40)