Infinite Impulse Response Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 185

N 2

= Â 1

() = () + ()

y n Cx n y n

(5.23)

i

i

The quantization error associated with the coefficients of an IIR filter depends on
the amount of shift in the position of its transfer function's poles and zeros in the
complex plane. This implies that the shift in the position of a particular pole depends
on the positions of all the other poles. To minimize this dependency of poles, an N th-
order IIR filter is typically implemented as cascaded second-order sections.

5.2.6 Lattice Structure

The lattice structure is used in applications such as adaptive filtering and speech
processing.

All-Pole Lattice Structure
We discussed the lattice structure in the previous chapter, where we derived the
k -parameters for an FIR or “all-zero” filter (except for poles at z

0). Consider now
an all-pole lattice structure associated with an IIR filter. This system is the inverse
of the all-zero FIR lattice of Figure 4.3, with N poles (except for zeros at z

=

0). A
solution for this system can be developed from the results obtained with the FIR
lattice structure. We can solve (4.52) and (4.53) backwards, computing y i -1 ( n ) in
terms of y i ( n ), and so on. For example, (4.52) becomes

=

() = () - ( )

yn ynken
i

- 1

(5.24)

- 1 i i i - 1

and (4.53) is repeated here as

() = () + ( )

en ky n e n - 1

(5.25)

i i i - 1 i - 1

Equations (5.24) and (5.25) are represented by the i th section lattice structure in
Figure 5.8, which can be extended for a higher-order all-pole IIR lattice structure.
For example, given the transfer function of an IIR filter with all poles, the recipro-
cal would be the transfer function of an FIR filter with all zeros. We also want to
make sure that this IIR system is stable by having all the poles inside the unit circle.
It can be shown that this is so if | k i | 1,2,..., N . Therefore, we can test the

stability of a system by using the recursive equation (4.49) to find the k -parameters
and check that each | k i |

< 1, i =

< 1.

Exercise 5.1: All-Pole Lattice Structure

The lattice structure for an all-pole system can be found. Let the transfer function
be