Introduction to DSP - IIR filters: parallel and cascade IIR structures

In the cascade form, the output of one section forms the input to the next:

In practice, the propagation of errors is crucial to the success of an IIR filter so the order of the sections in the cascade, and the selection of which filter coefficients to group in each section, is vital:

• sections with high gain are undesirable because they increase the need for scaling and so increase quantisation errors
• it is desirable to arrange sections to avoid excessive scaling

To reduce the gain of each section we note that:

• poles cause high gain (bumps in the frequency response)
• zeroes cause low gain (dips in the frequency response)
• the closer to the unit circle, the greater the effect

This suggests a way to group poles and zeroes in each section to avoid high gain sections:

Note that the pole closest to the unit circle will provide the highest gain because it is a large value close to the unit circle. This can best be countered by pairing it with the zero closest to it. Here is a recipe for grouping poles and zeroes to create sections which avoid high gain:

• pair the pole closest to the unit circle with the zero closest to it (note: not closest to the unit circle)
• do this for all the poles, working up in terms of their distance from the unit circle
• arrange the sections in order of how close their poles are to the unit circle

The question remains, whether to place the high gain sections first or last.

Recall that:

• poles are large values (high gain)
• the closer to the unit circle, the higher the gain
• poles cause bumps in the frequency response
• the closer to the unit circle, the sharper the bump (high Q)
• poles in the early stages affect the input to later stages
• poles at late stages have the last word

So, the section with the pole closest to the unit circle will have the highest gain but also the sharpest shape. As with so much else in digital filter design, we are faced with a compromise between conflicting desires:

• poles close to the unit circle in early stages cause high gain early on, so require more signal scaling and worse quantisation errors later on
• poles close to the unit circle in late stages cause significant noise shaping at a late stage