Realisation of Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 116

In the current problem we choose the Hessian [6] as

" #

¼ X

k
i ¼ 1 g i g

r 2 J ðp k Þ

t
i

1

¼ H

k ;

ð 5 : 46 Þ

and the parameter is incremented [7] via

p k ¼ H k g k k ; ð 5 : 47 Þ

p k þ 1 ¼ p k þ p k : ð 5 : 48 Þ

A heuristic choice of the step size k was chosen as the predicted objective function

given by

k ¼ J k ¼ J k g k p

t

k 1 :

ð 5 : 49 Þ

5.7.1.3 State Adjustment

In state adjustment, the AR(2) process (5.38) which serves as a model has no input
added to that at convergence; this AR(2) process is an oscillator. It makes parameter
incrementation unstable under normal circumstances. It was found that state
adjustment has a stabilising effect and is given by

^ x k ¼ s k 1 ^ a 1 þ s k 2 ^ a 2 : ð 5 : 50 Þ

5.7.1.4 Frequency Estimation

Frequency is computed by estimating the parameters a 1 and a 2 and substituting

^ ^

them in (5.39). Note that is different from ! p .

5.7.1.5 Recursive Estimation

The above algorithm is conveniently represented recursively by using the matrix
inversion lemma [1] for obtaining H k . This algorithm works from arbitrary initial
conditions by using a forgetting factor [5]

k while computing H k . The multi-
resolution regularised expectation maximisation (MREM) algorithm is as follows:

t

p

k ¼

½

a 1 ; a 2 ; v

; ð 5 : 51 Þ

x k ¼ a 1 x k 1 þ a 2 x k 2 ;

ð 5 : 52 Þ

s k ¼

a 1 s k 1 þ a 2 s k 2 þ y k ;

ð 5 : 53 Þ

e k ¼ y k x k ; ð 5 : 54 Þ

^

J k ¼ e k

v ;

ð 5 : 55 Þ

t

g

k ¼ 2e k s k 1 ;

½ 2e k s k 2 ; 1 ; ð 5 : 56 Þ