Energy Conservation in Adaptive Filtering - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 28

2 R u . This term can be deduced from Equation

with the weighted term E w i − 1

1.65 by writing it for = R u , which leads to

2 h U Tr R u , 2

R u =

2
R u −

2
R u + µ

E w i E w i − 1 2 µ h G E w i − 1

2

with the weighted term E w i

R u . This term can in turn be deduced from

R u . Continuing in this fashion, for suc-

Equation 1.65 by writing it for =

cessive powers of R u , we arrive at

2 h U Tr R u . 2

R M − 1
u

2
R M − 1
u

2
R u

E w i = E w i − 1 − 2 µ h G E w i − 1 + µ

As before, this procedure terminates. To see this, let p ( x ) = det ( xI − R u )

denote the characteristic polynomial of R u , say,

x M p M − 1 x M − 1 p M − 2 x M − 2

p ( x ) = + + +···+ p 1 x + p 0 .

Then, since p ( R u ) = 0 in view of the Cayley-Hamilton theorem, we have

2
R M

2 2

R u

2
R M − 1
u

E w i =− p 0 E w i − p 1 E w i −···− p M − 1 E w i .

2

This result indicates that the weighted term E w i

R M is fully determined by

the prior weighted terms.
Putting these results together, we find that the transient behavior of Filter
1.52 is now described by a nonlinear M -dimensional state-space model of the
form

2 h U

W = FW

i

+ µ Y

, (1.73)

i − 1

where the M × 1 vectors {W

i ,

Y} are defined by

2

E w i

( )

Tr R u

2 R u

E w i

(

R u )
.

Tr

.

i

= W = , Y , (1.74)

R M − 1
u

2
R M − 2
u

Tr ( )

E w i

R u

2
R M − 1
u

Tr ( )

E w i

and the M × M coefficient matrix F is given by

1 − 2 µ h G

0 1 − 2 µ h G

0 0 1 − 2 µ h G

= F .

0

0 1 − 2 µ h G

2 µ p 0 h G 2 µ p 1 h G ... 2 µ p M − 2 h G 1 + 2 µ p M − 1 h G