t
mse
1
ml
min
,
8:3:6
where l
min
is the minimum eigenvalue of the R matrix. Because t
mse
is inverselypropor-
tional to m, we have a large t
mse
when m is small (i.e., the speed of convergence is slow). If
we use a large value of m, the time constant is small, which implies faster convergence.
The maximum time constant t
mse
1=ml
min
is a conservative estimate of filter per-
formance, since onlylarge eigenvalues will exert significant influence on the conver-
gence time. Since some of the projections maybe negligiblysmall, the adaptive filter
error convergence maybe controlled byfewer modes than the number of adaptive
filter weights. Consequently, the MSE often converges more rapidly than the upper
bound of (8.3.6) would suggest.
Because the upper bound of t
mse
is inverselyproportional to l
min
, a small l
min
can
result in a large time constant (i.e., a slow convergence rate). Unfortunately, if l
max
is
also verylarge, the selection of m will be limited by(8.3.1) such that onlya small m can
satisfythe stabilityconstraint. Therefore if l
max
is verylarge and l
min
is verysmall, from
(8.3.6), the time constant can be verylarge, resulting in veryslow convergence. As
previouslynoted, the fastest convergence of the dominant mode occurs for m 1=l
max
.
Substituting this smallest step size into (8.3.6) results in
t
mse
l
max
l
min
:
8:3:7
For stationaryinput and sufficientlysmall m, the speed of convergence of the algorithm
is dependent on the eigenvalue spread (the ratio of the maximum to minimum eigen-
values) of the matrix R.
As mentioned in the previous section, the eigenvalues l
max
and l
min
are verydifficult
to compute. However, there is an efficient wayto estimate the eigenvalue spread from
the spectral dynamic range. That is,
l
max
l
min
max jX!j
2
min jX!j
2
,
8:3:8
where X! is DTFT of x(n) and the maximum and minimum are calculated over the
frequencyrange 0 ! p. From (8.3.7) and (8.3.8), input signals with a flat (white)
spectrum have the fastest convergence speed.
8.3.3 Excess Mean-Square Error
The steepest-descent algorithm in (8.2.14) requires knowledge of the gradient rxn,
which must be estimated at each iteration. The estimated gradient r
^
xn produces the
gradient estimation noise. After the algorithm converges, i.e., w(n) is close to w
o
, the
true gradient rxn % 0. However, the gradient estimator r
^
xn T 0. As indicated by
the update of Equation (8.2.14), perturbing the gradient will cause the weight vector
wn 1 to move awayfrom the optimum solution w
o
. Thus the gradient estimation
PERFORMANCE ANALYSIS
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