自适应滤波adaptive_filtering_chapter3

h(n + 1) − h 0 = (I - µR )[h(n) - h 0 ] (3.17)
Adaptive Tapped-delay-line Filters Using the Gradient Approach According to the definition of the coefficient error vector we have
c(n) = h(n) − h 0 (3.15)
where h0 is the solution of Rh 0 = p (3.16) Subtracting h0 from both sides of (3.14) and using the normal equation to eliminate p we get
which can be given in short-hand notation as
∇(n) = −2 E [e( n)u(n)] (3.7)
where
u(n) = [u (n), u (n - 1),..., u (n - M + 1)] (3.8)
T
Adaptive Tapped-delay-line Filters Using the Gradient Approach Now defining the M-by-1 coefficient vector
M ∂ε = −2 p (k − 1) + 2∑ h(m, n)r (m − k ) k = 1,2,...M ∂h(k , n) m =1
(3.4)
which is obtained by differentiating both sides of Eq. (3.3) with respect to h(k,n). This expression can be simplified to obtain
QQ T = I , QT Rc(n) = QT RIc(n) = QT RQQ T c(n) = Λv (n) Since
ε (n) = Pd − 2∑ h(k , n) p(k − 1) + ∑∑ h(k , n)h(m, n)r (m − k )
k =1 k =1 m =1 M M M
(3.3)
Pd = E[d 2 (n)] , p (k − 1) = E[d (n)u (n − k + 1)] and where the quantities
r (m − k ) = E[u (n − k + 1)u (n − m + 1)] are results of ensemble
averaging.
Adaptive Tapped-delay-line Filters Using the Gradient Approach The Method of Steepest Descent The dependence of ε(n) on the filter coefficient can be visualized as a bowl-shaped surface with an unique minimum. The adaptive process has 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 ∇(n) whose kth element is
∂ε (n) / ∂h(1, n) − 2 E [e(n)u (n)] ∂ε (n) / ∂h(2, n) − 2 E [e(n)u (n − 1)] = . ∇ ( n) = . . . ∂ε (n) / ∂h( M , n) − 2 E [e(n)u (n − M + 1)] (3.6)
Adaptive Tapped-delay-line Filters Using the Gradient Approach Assuming the values of h(1,n), h(2,n), . . . , h(M,n) are known, the value of the mean squared error is
Define the transformed coefficient error vector
v (n) = QT c(n) (3.21)
which implies
v (n + 1) = QT c(n + 1) (3.22)
Adaptive Tapped-delay-line Filters Using the Gradient Approach
Adaptive Filtering
CHAPTER 3
Adaptive Tapped-delay-line Filters Using the Gradient Approach
Adaptive Tapped-delay-line Filters Using the Gradient Approach In the case of known correlation expression the solution for the optimal coefficients of the tapped-delay-line filter was
y (n) Leabharlann ∑ h(k , n)u (n − k + 1)
k =1 M
(3.1)
the error signal is
e(n) = d (n) − y (n) (3.2)
The error signal e(n) is utilized by the adaptive process to generate corrections at each iteration to be applied to the tap coefficients in order to move closer to the optimum Wiener configuration.
Substituting Eq. (3.12) into Eq. (3.11) we get
h(n + 1) = h(n) + µE u(n)(d (n) - u T (n)h(n)) = h(n) + µE [u(n)(d (n)] − µE u(n)uT (n) h(n) (3.13)
[
[
]
]
which can be rewritten as
h 0 = R -1p (1.25)
Where R was the correlation matrix of the filter tap inputs and p was the cross-correlation between the input vector and a desired response. If the filter operates in an environment where R and p are unknown, we may use all the data collected up to and including time n to compute the ˆ ˆ estimates R(n) and p(n) 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.
∂ε = −2 E [e(n)u (n − k + 1)] k = 1,2,...M ∂h(k , n)
(3.5)
Adaptive Tapped-delay-line Filters Using the Gradient Approach The gradient vector is then written as
h( n + 1) = h( n) + µE [e( n)u( n)] (3.11)
Adaptive Tapped-delay-line Filters Using the Gradient Approach The error signal e(n) is defined as
e(n) = d (n) − uT (n)h(n) (3.12)
h(n + 1) = h(n) + µp - µRh (n) = (I - µR )h(n) + µp
(3.14)
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 parameter µ (to be chosen by designer) For the stability analysis define a coefficient error vector as
Adaptive Tapped-delay-line Filters Using the Gradient Approach