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.

Formula

The gain formula is as follows: where:

Definitions

Procedure to find the solution

Examples

Circuit containing two-port

The transfer function from Vin to V2 is desired. There is only one forward path: There are three loops:

Digital IIR biquad filter

Digital filters are often diagramed as signal flow graphs.

Servo

The signal flow graph has six loops. They are: There is one forward path: The forward path touches all the loops therefore the co-factor And the gain from input to output is

Equivalent matrix form

Mason's rule can be stated in a simple matrix form. Assume \mathbf{T} is the transient matrix of the graph where is the sum transmittance of branches from node m toward node n. Then, the gain from node m to node n of the graph is equal to, where and \mathbf{I} is the identity matrix. Mason's Rule is also particularly useful for deriving the z-domain transfer function of discrete networks that have inner feedback loops embedded within outer feedback loops (nested loops). If the discrete network can be drawn as a signal flow graph, then the application of Mason's Rule will give that network's z-domain H(z) transfer function.

Complexity and computational applications

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 n vertices, having an edge between every pair of vertices. There is a path form y_\text{in} to for each of the (n-2)! 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, q 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.