## 7.3. Structure of IIR filters

### 7.3.1. *Direct structures*

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 *x*_{1}(*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 *H*_{z}(*z*) has been obtained by cascading an FIR filter *H*_{1}(*z*) and an IIR filter *H*_{2}(*z*).

**Figure 7.2.** Putting filters *H*_{1} and *H*_{2} into cascade

We end up with the most spontaneous realization, called the direct form structure 1.

**Figure** 7.3. *Direct form structure 1 of an IIR filter*

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.

**Figure 7.4.** *Direct form ...*