adaptive filtering chapter3
TRANSCRIPT
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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)