Realisation of Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 115

In this problem we shall simultaneously estimate the parameters and the state of the
system.

5.7.1.1 Parameter Estimation

In parameter estimation we consider the error e k ¼ y k

x k and the objective

function J k ¼ e k , where the parameter is also estimated along with

a 1 and

v v

a 2 . The choice of v is such that E ½ J k ¼ 0. For minimising the objective function J k ,
gradients are obtained by differentiating (5.38) with respect to

^

a 1 and we get

^

s k ¼ @

e k
@ ^ a 1

¼ @ x k
@ ^

a 1 @ x k 1
@ ^

a 2 @ x k 2
@ ^

¼ þ þ x k 1 : ð 5 : 40 Þ

^ ^ ^

a 1 a 1 a 1

Using (5.40) we obtain the gradient vector as

t

@ J k

@p k

¼ 2e k s k 1 ; ½ 2e k s k 2 ; 1

t ¼r J ðp k Þ ½

t
k

¼ g ; ð 5 : 41 Þ

where parameter p k is given as

t
k ¼

p ½ a 1 ; ^ a 2 ; ð 5 : 42 Þ

^ v

and the value s k is obtained by recasting (5.40):

s k ¼ a 1 s k 1 þ a 2 s k 2 þ x k : ð 5 : 43 Þ

^ ^ ^

We can approximate (5.43) as

s k ¼ a 1 s k 1 þ a 2 s k 2 þ y k : ð 5 : 44 Þ

^ ^

This is due to the fact that (5.43) represents a high-Q filter very close to
convergence, hence we can drive the filter using y k to obtain s k due to non-
availability of

x k .

^

5.7.1.2 Parameter Incrementation

The general incrementation procedure [2] for minimising the function J ðp k Þ is
given as

1 r J ðp k Þ

p k þ 1 ¼ p k þr 2 J ðp k Þ ½ k : ð 5 :

45 Þ