Mason's gain formula

Mason's gain formula (MGF) is a method for finding the transfer function of a linear signal flow graph (SFG). The formula was derived by Samuel Jefferson Mason, for whom it is named. MGF is an alternate method to finding the transfer function algebraically by labeling each signal, writing down the equation for how that signal depends on other signals, and then solving the multiple equations for the output signal in terms of the input signal. MGF provides a step by step method to obtain the transfer function from a SFG. Often, MGF can be determined by inspection of the SFG. The method can easily handle SFGs with many variables and loops including loops with inner loops. MGF comes up often in the context of control systems, microwave circuits and digital filters because these are often represented by SFGs.

Path: a continuous set of branches traversed in the direction that they indicate. Forward path: A path from an input node to an output node in which no node is touched more than once. Loop: A path that originates and ends on the same node in which no node is touched more than once. Path gain: the product of the gains of all the branches in the path. Loop gain: the product of the gains of all the branches in the loop.

Make a list of all forward paths, and their gains, and label these Gk. Make a list of all the loops and their gains, and label these Li (for i loops). Make a list of all pairs of non touching loops, and the products of their gains (LiLj). Make a list of all pairwise non touching loops taken three at a time (LiLjLk), then four at a time, and so forth, until there are no more. Compute the determinant Δ and cofactors Δk. Apply the formula.

Mason's Rule can grow factorially, because the enumeration of paths in a directed graph grows dramatically. To see this consider the complete directed graph on vertices, having an edge between every pair of vertices. There is a path form to for each of the permutations of the intermediate vertices. Thus Gaussian elimination is more efficient in the general case. Yet Mason's rule characterizes the transfer functions of interconnected systems in a way which is simultaneously algebraic and combinatorial, allowing for general statements and other computations in algebraic systems theory. While numerous inverses occur during Gaussian elimination, Mason's rule naturally collects these into a single quasi inverse. General form is

Where as described above, is a sum of cycle products, each of which typically falls into an ideal (for example, the strictly causal operators). Fractions of this form make a subring of the rational function field. This observation carries over to the noncommutative case, even though Mason's rule itself must then be replaced by Riegle's rule.