18.3 THE LFSR DEPENDENCE GRAPH

Based on Algorithm 18.1, the dependence graph of Fig. 18.1 is obtained. The gray circles indicate valid operations such as indicated in line 9 of Algorithm 18.1. The white circles have zero q inputs, and hence only transfer the northeast input to the southwest output. In that sense, we note that the inputs a₉ to a₄ become effective only when they arrive at the top line with gray circles. Likewise, the desired outputs r₄ to r₀ could be obtained as the outputs of the bottom row with gray circles.

Figure 18.1 Dependence graph of the LFSR polynomial division algorithm for the case n = 9 and m = 5.

Detailed explanations of how a dependence graph could be obtained were presented in Chapters 10 and 11 and by the author elsewhere [86, 100, 116–118]. The dependence graph is two-dimensional (2-D) since we have two indices, i and j. If we consider only the gray circles, then the bounds on the indices are 0 ≤ i ≤ n − m = 4 and 0 ≤ j < m = 5. Each node in the algorithm performs the operations indicated in line 9 of Algorithm 18.1. If we extended the algorithm such that the inputs are fed from the same side and the outputs are obtained from the same side, then the bounds of the algorithm indices would be 0 ≤ i ≤ n + m and 0 ≤ j < m.

Coefficient q_n−m−i of Q, on line 11 of Algorithm 18.1, is obtained at iteration i. Hence, this coefficient is represented by the horizontal lnes in the figure. For example, the horizontal line i = 3 represents the coefficient q_n−m−i = q₁. Similarly, the line at i = 0 represents q₄.

Coefficient b_j of B, on line 9 of Algorithm 18.1, has an index dependency

(18.7)

where c is a specific value of the index. For example, b₁ would be represented by the vertical line j = 1 as shown in the figure.

The partial remainder coefficient r_j(i + 1), on line 9 of Algorithm 18.1, is obtained from b_j, q_n−m−i and r_j−1(i). This explains the diagonal lines representing the inputs and outputs for each node.