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VLSI Digital Signal Processing Systems: Design and Implementation by Keshab K. Parhi

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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 z3. Thus the transfer function of the transformed filter H′(z) is related to the original transfer function H(z) as

image

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

image

image

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 ...

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