SUBSPACE TRACKING FOR SIGNAL PROCESSING - Adaptive Signal Processing: Next Generation Solutions - page 251

We note that these asymptotic biases are similar to those obtained in batch estimation
derived from a Taylor series expansion [77, p. 68] with expression (4.77) of C u .

! u 1 þo

and E[ l ( k )] l 1 ¼ o

:

X

n

1
k

l 1 l i
2( l 1 l i ) 2

1
k

1
k

E[ w ( k )] u 1 ¼

i¼ 2

Finally, we see that in adaptive and batch estimation, the square of these biases are an
order of magnitude smaller that the variances in O ( m )or O ( k ).
This methodology has been applied to compare the theoretical asymptotic per-
formance of several adaptive algorithms for minor and principal component analysis
in [22, 26, 27]. For example, the asymptotic mean square error E( kW ( k ) W k

2

Fro )of
the estimate W ( k ) given by the WSA algorithm (4.57) is shown in Figure 4.1, where
the stepsize m is chosen to provide the same value for m Tr( C u ). We clearly see in this
figure that the value b 2 / b 1 ¼ 0.6 optimizes the asymptotic mean square error / speed
of convergence tradeoff.

10 1

WSA algorithm

10 0

(1)

(2)

(6)

(5)

(3)

(4)

10 −1

(0)

10 −2

0 500 1000 1500 2000 2500 3000 3500 4000

Iteration number

2
Fro ) averaging 100
independent runs for the WSA algorithm, for different values of parameter b 2 / b 1 ¼ 0.96
(1), 0.9 (2), 0.1 (3), 0.2 (4), 0.4 (5), and 0.6 (6) compared with mTr(C u ) (0) in the case n ¼ 4,
r ¼ 2, C x ¼ Diag(1.75, 1.5, 0.5, 0.25), where the entries of W(0) are chosen randomly
uniformly in [0, 1].

Figure 4.1 Learning curves of the mean square error E(kW(k)W k