Signal Flow Graphs and Data Flow Graphs - Signal Processing Systems - page 1256

represent the output signal only in terms of the input signal. Note that the variables
used in the signal flow graphs and the equations correspond to frequency-domain
representations of signals.

2.2.1 Mason's Gain Formula

First of all, a few useful terminologies in Mason's gain formula have to be defined
related to an SFG.

￿ Forward path: a path that connects a source node to a sink node in which no node

is traversed more than once.

￿ Loop: a closed path without crossing the same point more than once.

￿ Loop gain: the product of all the transfer functions in the loop.

￿ Non-touching or non-interacting loops: two loops are nontouching or noninter-

acting if they have no nodes in common.

In general, Mason's gain formula [ 4 ] is presented as below:

N
j = 1 M j Δ j
Δ

Y
X =

M = (1)

where

￿ M = transfer function or gain of the system

￿ Y = output node

￿ X = input node

￿ N = total number of forward paths between X and Y

￿ Δ = determinant of the graph = 1 − ∑ loop gains + ∑ non-touching loop gains

taken two at a time − ∑ non-touching loop gains taken three at a time + ...

￿ M j =gainofthe j th forward path between X and Y

￿ Δ j = 1-loops remaining after eliminating the j th forward path, i.e., eliminate the

loops touching the j th forward path from the graph. If none of the loops remains,
Δ j =1.

To illustrate the actual usage of Mason's gain formula, the transfer function for
the example SFG shown in Fig. 1 is derived by following the steps below:

1. Find the forward paths and their corresponding gains
Two forward paths exist in this SFG:
M 1 = G 4

2. Find the loops and their corresponding gains
There are four loops in this example:
Loop 1 = −

G 1 G 2 G 3 and M 2 =

G 1 H 1 ,

Loop 2 = − G 3 H 2 ,