The Impact of Finite Wordlength Implementation - Digital Signal Processing: An Introduction with MATLAB and Applications - page 202

1

c m ¼

k ¼ 0 j f m ð k Þj 2 :

ð 4 : 33 Þ

P 1

While the above structure is less strict than that in ( 4.32 ), overflows are still
possible [ 3 ].

4.5.2.2 Scaling of Cascade-Form IIR Filters

For the cascade implementation of a high-order IIR filter, scaling is required to
avoid overflow in individual second and first-order sections in addition to the final
output node. A systematic rule, known as the 'pole-zero pairing rule', has been
developed to minimize the output noise power. The idea behind this rule is that
grouping a pole with an adjacent zero tends to reduce the peak gain of the relevant
section (Fig. 4.12 ).
In brief, the pole-zero pairing rule states the following Porat [ 3 ]. First, in the
z-plane, the complex pole pair that is closest to the unit circle should be paired
with its nearest complex zero pair. This procedure of matching pole and zeros pairs
is repeated until all the poles and zeros are paired. Second, the second-order
sections developed from the pairing process are ordered according to their peak
magnitude response either in descending or ascending order.

1

0.8 1

0.6 2

0.4

3

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

Real z

Fig. 4.12 Pole-zero pairing and ordering rule