Energy Conservation in Adaptive Filtering - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 10

Now it can be verified that matrices F of the form of Equation 1.19, for
arbitrary { A > 0 ,B ≥ 0 } , are stable for all values of µ in the range:

min

,

1
λ max (

1

0 <µ< ,

max λ( IR +

(1.27)

A − 1 B )

H ) ∈

where the second condition is in terms of the largest positive real eigenvalue
of the block matrix,

A ,

/ 2 − B / 2

H =

I M 2 0

when it exists. Because H is not symmetric, its eigenvalues may not be positive
or even real. If H does not have any real positive eigenvalue, then the upper
bound on A − 1 B

alone.*
Likewise, the mean-stability of the filter, as dictated by Equation 1.18, re-
quires the eigenvalues of

µ is determined by 1 /λ max ( )

( I − µ P ) to lie inside the unit circle or, equivalently,

µ< 2 /λ max ( P ). (1.28)

Combining Equations 1.27 and 1.28 we conclude that the filter is stable in the
mean and mean-square senses for step-sizes in the range

min 2
λ

1 1

µ< , ,

max λ( IR +

. (1.29)

( ) λ (

A − 1 B

)

P H ) ∈

max max

1.7 Steady-State Performance

Steady-state performance results can also be deduced from Equation 1.21.
Assuming the filter is operating in steady state, Recursion 1.21 gives in the
limit

E

2
σ
g 2 [ u i ]

u i

2
(

2 2

v

lim
i

E w i

)σ = µ

σ .

I − F

→∞

This expression allows us to evaluate the steady-state value of E w i

S

for

any weighting matrix S , by choosing σ such that

( I − F )σ = vec ( S ) ,

i.e.,

) − 1 vec

σ = ( I − F ( S ).

A − 1 B

* The condition involving in Equation 1.27 guarantees that all eigenvalues of F are

less than 1, while the condition involving H ensures that all eigenvalues of F are larger than

λ max ( )

−

1.