Signal-flow graph

A signal-flow graph or signal-flowgraph (SFG), invented by Claude Shannon, but often called a Mason graph after Samuel Jefferson Mason who coined the term, is a specialized flow graph, a directed graph in which nodes represent system variables, and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes. Thus, signal-flow graph theory builds on that of directed graphs (also called digraphs), which includes as well that of oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications.Wai-Kai Chen wrote: 'The concept of a signal-flow graph was originally worked out by Shannon in dealing with analog computers. The greatest credit for the formulation of signal-flow graphs is normally extended to Mason , . He showed how to use the signal-flow graph technique to solve some difficult electronic problems in a relatively simple manner. The term signal flow graph was used because of its original application to electronic problems and the association with electronic signals and flowcharts of the systems under study.'Robichaud et al. identify the domain of application of SFGs as follows:The following illustration and its meaning were introduced by Mason to illustrate basic concepts:In general, there are several ways of choosing the variables in a complex system. Corresponding to each choice, a system of equations can be written and each system of equations can be represented in a graph. This formulation of the equations becomes direct and automatic if one has at his disposal techniques which permit the drawing of a graph directly from the schematic diagram of the system under study. The structure of the graphs thus obtained is related in a simple manner to the topology of the schematic diagram, and it becomes unnecessary to consider the equations, even implicitly, to obtain the graph. In some cases, one has simply to imagine the flow graph in the schematic diagram and the desired answers can be obtained without even drawing the flow graph.Linear signal-flow graph methods only apply to linear time-invariant systems, as studied by their associated theory. When modeling a system of interest, the first step is often to determine the equations representing the system's operation without assigning causes and effects (this is called acausal modeling). A SFG is then derived from this system of equations.The reduction of a graph proceeds by the elimination of certain nodes to obtain a residual graph showing only the variables of interest. This elimination of nodes is called 'node absorption'. This method is close to the familiar process of successive eliminations of undesired variables in a system of equations. One can eliminate a variable by removing the corresponding node in the graph. If one reduces the graph sufficiently, it is possible to obtain the solution for any variable and this is the objective which will be kept in mind in this description of the different methods of reduction of the graph. In practice, however, the techniques of reduction will be used solely to transform the graph to a residual graph expressing some fundamental relationships. Complete solutions will be more easily obtained by application of Mason's rule.The graph itself programs the reduction process. Indeed a simple inspection of the graph readily suggests the different steps of the reduction which are carried out by elementary transformations, by loop elimination, or by the use of a reduction formula.Before describing the process of reduction...the correspondence between the graph and a system of linear equations ... must be generalized...The generalized graphs will represent some operational relationships between groups of variables...To each branch of the generalized graph is associated a matrix giving the relationships between the variables represented by the nodes at the extremities of that branch...The elementary transformations and the loop reduction permit the elimination of any node j of the graph by the reduction formula:. With the reduction formula, it is always possible to reduce a graph of any order... the final graph will be a cascade graph in which the variables of the sink nodes are explicitly expressed as functions of the sources. This is the only method for reducing the generalized graph since Mason's rule is obviously inapplicable.For some authors, a linear signal-flow graph is more constrained than a block diagram, in that the SFG rigorously describes linear algebraic equations represented by a directed graph.The term 'cause and effect' was applied by Mason to SFGs:Signal-flow graphs can be used for analysis, that is for understanding a model of an existing system, or for synthesis, that is for determining the properties of a design alternative.Shannon's formula is an analytic expression for calculating the gain of an interconnected set of amplifiers in an analog computer. During World War II, while investigating the functional operation of an analog computer, Claude Shannon developed his formula. Because of wartime restrictions, Shannon's work was not published at that time, and, in 1952, Mason rediscovered the same formula.The amplification of a signal V1 by an amplifier with gain a12 is described mathematically byThere is some confusion in literature about what a signal-flow graph is; Henry Paynter, inventor of bond graphs, writes: 'But much of the decline of signal-flow graphs is due in part to the mistaken notion that the branches must be linear and the nodes must be summative. Neither assumption was embraced by Mason, himself !'Mason introduced both nonlinear and linear flow graphs. To clarify this point, Mason wrote : 'A linear flow graph is one whose associated equations are linear.'