In this section, we will refer to the direct canonic structure that is characterized by the following transfer function:
First, we introduce an intermediary output x1(k), so that:
The transfer function in equation (7.1) can then be decomposed into a product of two transfer functions:
All this occurs as if the filter Hz(z) has been obtained by cascading an FIR filter H1(z) and an IIR filter H2(z).
We end up with the most spontaneous realization, called the direct form structure 1.
We see that this structure requires N+M−2 delay cells. We can then ask the following question: can we share some cells and only use M-l delay cells, knowing that M>N? The answer is yes, leading us to the direct form structure 2.