On Boolean Techniques for Non Touching Loops of Signal Flow Graphs

Abstract

Vertex disjoint (non touching) loops have to be processed to form all combinations of disjoint loops for Mason’s graph. Similarly Coates’ graph-based method needs all spanning disjoint loop sets. This is required to obtain the transfer function/solve equations of a system represented by Mason’s/Coates’ signal flow graph. A Boolean formula is presented in this note to obtain them. A Boolean function is formed comprising product of sums of pairs of touching loops for each connected component. It is then simplified using some rules of Boolean algebra. The resulting terms are “complemented” to derive all combinations of non touching loops for a Mason’s graph. Using an additional Boolean rule, the same formula gives all spanning disjoint loop sets of a Coates’ graph. The graph need not be connected.