A Modification of the Coates Gain Formula for the Analysis of Linear Systems

In this paper, the Coates flow-graph gain formula is modified by the introduction of the loop-set and the two-loop-set, baaed on the concepts of the tree and the two-tree in network topology. The analogy is evident by observing that the two-trees T _ij of a given graph G can be obtained by coalescing vertices i and j of G and thon removing the self-loops. In a similar manner, the two-loop-sets can be obtained from the flow graph by first modifying the graph and then obtaining the loop-sets of tho modified graph. Because of such modification, the derivation of tho gain formula is considerably simplified. Consequently, the evaluation of the terms in the formula becomes easier. Finally, a simple method of determining the number of terms in the formula is derived as a check against possible omission of terms due to carelessness.