Direct Form II

The signal flow graph for the Direct-Form-II (DF-II) realization of the second-order IIR filter section is shown in Fig.9.2.

Figure 9.2: Direct-Form-II implementation of a 2nd-order digital filter.

The difference equation for the second-order DF-II structure can be written as

$$\begin{eqnarray*} v(n) &=& x(n) - a_1 v(n-1) - a_2 v(n-2)\\ y(n) &=& b_0 v(n) + b_1 v(n-1) + b_2 v(n-2) \end{eqnarray*}$$

which can be interpreted as a two-pole filter followed in series by a two-zero filter. This contrasts with the DF-I structure of the previous section (diagrammed in Fig.9.1) in which the two-zero FIR section precedes the two-pole recursive section in series. Since LTI filters in series commute (§6.7), we may reverse this ordering and implement an all-pole filter followed by an FIR filter in series. In other words, the zeros may come first, followed by the poles, without changing the transfer function. When this is done, it is easy to see that the delay elements in the two filter sections contain the same numbers (see Fig.5.1). As a result, a single delay line can be shared between the all-pole and all-zero (FIR) sections. This new combined structure is called ``direct form II'' [60, p. 153-155]. The second-order case is shown in Fig.9.2. It specifies exactly the same digital filter as shown in Fig.9.1 in the case of infinite-precision numerical computations.

In summary, the DF-II structure has the following properties:

1. It can be regarded as a two-pole filter section followed by a two-zero filter section.

2. It is canonical with respect to delay. This happens because delay elements associated with the two-pole and two-zero sections are shared.

3. In fixed-point arithmetic, internal overflow can occur at the delay-line input (output of the leftmost summer in Fig.9.2), unlike in the DF-I implementation, which can only overflow if the output overflows.

4. As with all direct-form filter structures, the poles and zeros are sensitive to round-off errors in the coefficients $a_i$ and $b_i$ , especially for high transfer-function orders. Lower sensitivity is obtained using series low-order sections (e.g., second order), or by using ladder or lattice filter structures [86].