For the example of the previous section, suppose we are given Eq.(G.14) in directform II (DFII), as shown in Fig.G.1. It is important that the filter representation be canonical with respect to delay, i.e., that the number of delay elements equals the order of the filter. Then the third step (writing down controller canonical form by inspection) may be replaced by the following more general procedure:

The statespace description of the difference equation in Eq.(G.7) is given by Eq.(G.16). We see that controller canonical form follows immediately from the directformII digital filter realization, which is fundamentally an allpole filter followed by an allzero (FIR) filter (see §9.1.2). By starting instead from the transposed directformII (TDFII) structure, the observer canonical form is obtained [28, p. 87]. This is because the zeros effectively precede the poles in a TDFII realization, so that they may introduce nulls in the input spectrum, but they cannot cancel output from the poles (e.g., from initial conditions). Since the other two digitalfilter direct forms (DFI and TDFIsee Chapter 9 for details) are not canonical with respect to delay, they are not used as a basis for deriving statespace models.