VLSI Digital Signal Processing Systems: Design and Implementation

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 U_i → V_(i+w)%J, i = 0,1, ... , J − 1, is same as the number of delays on the edge U → V in the original DFG. Mathematically, this can be stated as

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 w_l delays in the original DFG, and let A be a node in l. The loop l can be denoted as the path A A with w_l delays. If the loop l is traversed p times (p ≥ 1), this results in the path A A ... A with pw_l delays. The corresponding unfolded path starting at the the node A_i, 0 ≤ i ≤ J − 1, in the J-unfolded DFG ...