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:
DirectformII realization of Eq.(G.7).
Note that this form has a delayfree path from input to output.

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
.
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