Stochastic Gradient Adaptive Algorithms - A Rapid Introduction to Adaptive Filtering - page 68

8
πσ

1
L ,

0 <μ< (4.116)

x

which is the same result obtained in [ 14 ] and [ 40 ]. From the results in [ 14 ], where
this stability bound is calculated based on a full transient analysis, we know that
this bound is also a necessary condition for convergence. These results, and the
ones corresponding to the mean stability of the SDA, are valid if L x is a positive
definite matrix. If this is not the case (for example if some of the eigenvalues of L x
have negative real parts), it would be possible that no choice of will guarantee

the stability of the algorithm (in mean and MSD sense). This will be the case for
some class of input signals as it is shown in [ 44 ]. Moreover, certain signals might
lead to stability of the LMS algorithm but not of the SDA [ 13 ].

μ

4.5.3 Steady State Behavior for a Large Family of Algorithms

Once we have conditions for the mean and MSD stability of an adaptive filter algo-
rithm we could get some insight into its asymptotic or steady state behavior. The
approach we take in this section will allow us to obtain the asymptotic value of:

E

2

E e 2

w T ( n ) = ˜ ( n − 1 ) x ( n ) + v ( n )

E

2

w T

2
v

= ˜ ( n − 1 ) x ( n ) + σ

v

= ξ( ) + σ ,

n (4.117)

wherewe used the assumptions about the noise sequence, the fact that the a priori error
can be written as E η ) . w T

2

η( n ) = ˜ ( n − 1 ) x ( n ) , and that the EMSE is ξ( n ) = ( n

Note that if the value of lim n →∞ ξ( n ) is different from zero, the final MSE will

v , which corresponds to the optimal Wiener error (as discussed in
Sect. 2.4 ) . This can be interpreted as follows: once the adaptive filter has reached
its steady state, it will keep on moving with random fluctuations in a vicinity of
the optimal Wiener solution. The amplitude of these fluctuations is measured by
the EMSE or the final MSD. Note that if we consider the independence assumption
between the input vectors, we have:

be greater than σ

E

w T ξ( n ) = tr [ R x K ( n − 1 ) ] , with K ( n − 1 ) = w ˜ ( n − 1 ) ˜ ( n − 1 ) , (4.118)

Here, we used the result ( 4.50 ), given the fact that is a function of the past

values of the input vectors and the noise sequence. This means, that we can bound

w ˜ ( n − 1 )