Signal Flow Graphs and Data Flow Graphs - Signal Processing Systems - page 1255

2 Signal Flow Graphs

In this section, the notation of signal flow graph (SFG) is first overviewed. Then
two useful approaches, i.e., the Mason's gain formula and the equation-solving, are
explained in detail to derive the corresponding transfer function for a given SFG.

2.1 Notation

Signal flow graphs have been used for the analysis, representation, and evaluation
of linear digital networks, especially digital filter structures. An SFG is a collection
of nodes and directed edges [ 3 ] , where the nodes represent computations or tasks
and a directed edge

denotes a branch originating from node j and terminating
at node k . With input signal at node j and output signal at node k , the edge

( j , k )

)
denotes a linear transformation from the signal at node j to the signal at node k .An
example of SFG is shown in Fig. 1 , where both nodes j and k represent summing
operations and the edge

( j , k

denotes a unit gain transformation.
Two types of nodes exist in SFG, source nodes and sink nodes. A source node
is a node with no entering edges, and is used to represent the injection of external
inputs into a graph. A sink node is a node with only entering edges, and is used to
extract outputs from a graph.

( j , k )

2.2 Transfer Function Derivation of SFG

For a given SFG, there are mainly two approaches to derive its corresponding
transfer function. One is the Mason's gain formula, which provides a step-by-
step method to obtain the transfer function. The other is the equation-solving
approach by labeling each intermediate signal, writing down the equation for that
signal with dependency on other signals, and then solving the multiple equations to

G4

k

j G1 G2 G3

X Y

Y1 Y2 Y3 Y4 Y5

−H2

−H1

−H3

Fig. 1 An example of signal flow graph