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

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5.3    PROPERTIES OF UNFOLDING

Some basic properties of unfolding are discussed in this section. These properties are useful when applications of unfolding are considered in Section 5.5.

Property 5.3.1  Unfolding preserves the number of delays in a DFG.

This property is based on the fact that the sum of the delays on the J unfolded edges UiV(i+w)%J, i = 0,1, ... , J − 1, is same as the number of delays on the edge UV in the original DFG. Mathematically, this can be stated as

image

The proof of this is left as an exercise (see Problem 3). This property can be observed in Figs. 5.2, 5.3, and 5.4.

It is interesting to observe what happens when a loop is unfolded. Let l be a loop with wl delays in the original DFG, and let A be a node in l. The loop l can be denoted as the path A image A with wl delays. If the loop l is traversed p times (p ≥ 1), this results in the path A image A image ... image A with pwl delays. The corresponding unfolded path starting at the the node Ai, 0 ≤ iJ − 1, in the J-unfolded DFG ...

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