Introduction to DSP - IIR filters: direct form II

The transposition theorem says that if we take a filter diagram and reverse all the elements - swapping the order of execution for every element, and reversing the direction of all the flow arrows - then the result is the same:

• if everything is turned back to front, it all works just the same

This means that the direct form 1 diagram can be obtained by transposition of the direct form 2 diagram:

For this reason, direct form 1 is often called transposed direct form 2.

Don't ask me why these terms seem to be as confusing as they possibly could be - I didn't make them up. I imagine mathematicians sit around at coffee break and come up with new ways to spread despondency amongst us lesser mortals. Here are the two main sources of confusion:

direct form 1

so called because it can be drawn direct from the filter equation

direct form 2

so called because it can be derived by changing the diagram of direct form 1

transposed

so called because it is obtained by transposition of direct form 2 - but really, this is just direct form 1

canonic

so called because it has the minimum number of delay elements - but really, it is just direct form 2