## 11.7 SLOW-DOWN, RETIMING, AND PIPELINING

Many useful realizations contain nodes that are not connected to unit-delay branches. These nodes or variables thus do not appear in a state variable description and the scaling and noise computation methods cannot be applied directly. To overcome this difficulty, the SRP (slow-down and retiming/pipelining) transformation technique (see chapters 4 and 3) can be used as a preprocessing step.

Consider the filter in Fig. 11.19(b) which is obtained by applying slowdown transformation (*M* = 3) to the filter in Fig. 11.19(a). By 3 slow-down transformation, every *z*-variable in Fig. 11.19(a) is changed into *z*^{3}. Thus the transfer function of the transformed filter *H′*(*z*) is related to the original transfer function *H*(*z*) as

Thus, if the unit-sample response from the input to the internal node *x* in Fig. 11.19(a) is defined by

*Fig. 11.20* (a) A filter with a nonstate variable node on a feed-forward path. (b) Nonstate variable node is converted into state variable node by pipelining.

the unit-sample response from the input to the internal node *x′* in Fig. 11.19(b) is

Then,

Similarly, it can be shown that

The foregoing analysis shows that slow-down ...