Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 48

mathematically given as

!

X

i ¼ k þ N 1

1
N

k ¼ u i : ð 3 : 6 Þ

i ¼ k

To arrive at the recursive form similar to (3.4), we rewrite k þ 1 as

!
þ u k u k

X

i ¼ k þ N

k þ 1 ¼

N u i

i ¼ k þ 1

!
u k

i ¼ k þ N

X

¼ u i

i ¼ k

!

X

i ¼ k þ N 1

¼ u i þ u k þ N u k

i ¼ k

¼ N k þ u k þ N u k : ð 3 : 7 Þ

Equation (3.7) can be written as

1
N

k ¼ k 1 þ ge k where g ¼ and e k ¼ u k u k N : ð 3 : 8 Þ

k is obtained by passing the signal ge k through an integrator
(3.8). This integrator can be written in operator notation or as a transfer function
(TF) model:

The moving mean

1 z N
1 z 1

1
N

k ¼ u k : ð 3 : 9 Þ

The filter in (3.9) has feedback and feedforward elements and is called an
autoregressive moving average (ARMA) filter. This filter is stable in spite of a
pole on the unit circle due to the pole-zero cancellation at z ¼ 1.

3.2.4 Linear Phase Filters
Consider a narrowband filter (NBF) given by (3.17). Let the impulse response of
this IIR filter be h k . We truncate the infinite series h k and use these values to
generate the coefficients of an MA filter in a specific way, given by

b i ¼ h N i for i ¼ 1toN ;

b N þ i ¼ h i þ 1 i ¼ 1toN 1

: ð 3 :

10 Þ for