When converting a transfer function to statespace form by hand, the step of pulling out the direct path, like we did in going from Eq.(G.13) to Eq.(G.14), can be bypassed [28, p. 87].

Figure G.2 gives the standard directformII structure for a secondorder IIR filter. Unlike Fig.G.1, it includes a direct path from the input to the output. The filter coefficients are all given directly by the transfer function, Eq.(G.13).
This form can be converted directly to statespace form by carefully observing all paths from the input and state variables to the output. For example, reaches the output through gain 2 on the right, but also via gain on the left and above. Therefore, its contribution to the output is , as obtained in the DFII realization with directpath pulled out shown in Fig.G.1. The state variable reaches the output with gain , again as we obtained before. Finally, it must also be observed that the gain of the direct path from input to output is .