Digital Filters Design for Signal and Image Processing

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 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 H_z(z) has been obtained by cascading an FIR filter H₁(z) and an IIR filter H₂(z).

Figure 7.2. Putting filters H₁ and H₂ 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 ...