The Impact of Finite Wordlength Implementation - Digital Signal Processing: An Introduction with MATLAB and Applications - page 194

To begin to answer this question, it is useful to recall the expression for the
Direct Form transfer function of an IIR filter, assuming infinite-precision

coefficients:

H ð z Þ¼ P M 1

i ¼ 0 b i z i
1 P N 1

i ¼ 1 a i z i :

ð 4 : 16 Þ

Now, the modified transfer function which arises as a result of quantized
coefficients is given by:

H ð z Þ¼ P M 1

i ¼ 0 b i z i
1 P N 1

i ¼ 1 a i z i :

ð 4 : 17 Þ

where a i ¼ a i þ Da i and b i ¼ b i þ Db i are the quantized coefficients. Note, the
locations of the zeros and poles are affected by the errors Db i and Da i , respec-
tively. Detailed analysis in Ref. [ 2 ] shows that direct-form implementations such
as those given in ( 4.17 ) exacerbate problems due to coefficient quantization. These
problems are particularly significant when the poles/zeros are tightly clustered.
Moreover, as the order of the IIR transfer function increases, the sensitivity to
coefficient quantization errors increases accordingly.
It has been found that one can significantly improve the problems due to
coefficient quantization by realizing the overall filter as a conglomerate of first-
and second-order filter sections. Note that one needs both first and second order
sections in general so as to avoid complex filter coefficients.
The realization of the filter can be achieved by either (i) doing a partial fraction
expansion of the overall transfer function, and then operating all of the first and
second order sections in parallel, or (ii) factorizing the overall transfer function
into first and second order sections and then operating all of these sections in
sequential cascade.
When an overall filter is implemented as a conglomerate of first and second
order sections, each pole or pair of poles (zeros) is realized independently of the
remaining poles (zeros), and thus the movement occuring in a certain pole or pair
of poles does not affect the others. Hence, cascade-form implementation is much
less sensitive to coefficient quantization errors than direct-form implementation.
Now, in considering the coefficient quantization sensitivity, one may also
wonder whether the type of structure used to implement the second-order section
itself would make any difference. The answer is 'yes'. To understand this more
fully, consider the following example.

Example 2 A second—order IIR filter has the transfer function:

1
1 a 1 z 1 þ a 2 z 2 :

H ð z Þ¼ ð 4 : 18 Þ