2.7 Signal-Flow Graphs (SFGs)

A signal-flow graph (SFG) may be regarded as a simplified version of a block diagram.

The SFG was introduced by S. J. Mason for the cause-and-effect representation of linear systems that are modeled by algebraic equations.

Signal-flow graph Node and Branch

Nodes are used to represent the variables.

The node are connected by line segments called branches, according to the cause-and-effect equations.

The branches have associated branch transfer functions and directions.

A signal can transmit through a branch only in the direction of the arrow.

X Y

G

Signal-flow graph Path and Path Transfer function

Path is any collection of continuous succession of branches traversed in the same direction.

X

Z

Y G1 G2

The product of the branch transfer functions encountered in traversing a path is called the path transfer function.

P=G1G2

Signal flow graph Forward Path & Forward-path Transfer Function

Forward path is a path that starts at an input node and ends at an output node and along which no node is traversed more than once.

G1 G2 G3

X Y

P=G1G2G3

Forward path transfer function is the path transfer function of a forward path.

Signal flow graph Loop & Loop Transfer Function

Loop is a path that originates and terminates on the same node and along which no other node is encountered more than once.

Loop transfer function is defined as the path transfer function of a loop.

G1 G2 G3

X Y

32GGL

Nontouching loops are two loops of the SFG if they dot not share one or more common nodes.

The Mason Rule

Li=transfer function(TF) of i th loop;

LiLj=product of transfer functions(TFs) of two

non-touching loops;

LiLjLk=product of transfer functions(TFs) of

three non-touching loops;

= 1 - (sum of the TFs of all individual loops) + (sum of products of TFs of all possible combinations of two nontouching loops) (sum of products of TFs of

all possible combinations of three nontouching loops) +

kjijii

N

k

kkin

out

LLLLLL

PsX

sYs

1

1

)(

)()(

1

The Mason Rule

N=total number of forward paths between and ;

Pk=transfer function of the kth forward path between and ;

= the cofacter of for that part of the graph which is non-touching with the kth forward path.

k

kjijii

N

k

kkin

out

LLLLLL

PsX

sYs

1

1

)(

)()(

1

outY

inX

outY inX

Example1

There are four self loops:

221 HGL

)(sR )(sY

There are four self loops:

221 HGL

)(sR )(sY

332 HGL

Example1

There are four self loops:

221 HGL 332 HGL

663 HGL

Example1

)(sR )(sY

There are four self loops:

221 HGL 332 HGL

663 HGL 774 HGL

Example1

)(sR )(sY

)(sR )(sY

There are four self loops:

221 HGL 332 HGL

663 HGL 774 HGL

Loops L1 and L2 do not touch L3 and L4. The determinant is

)()(1 423241314321 LLLLLLLLLLLL

Path 1: 43211 GGGGP

Example1

There are four self loops:

221 HGL 332 HGL

663 HGL 774 HGL

Loops L1 and L2 do not touch L3 and L4. The determinant is

)()(1 423241314321 LLLLLLLLLLLL

Path 1: 43211 GGGGP

Example1

Path 2: 87652 GGGGP

)(sR )(sY

)(sR )(sY

There are four self loops:

221 HGL 332 HGL

663 HGL 774 HGL

Loops L1 and L2 do not touch L3 and L4. The determinant is

)()(1 423241314321 LLLLLLLLLLLL

Path 1: 43211 GGGGP

Path 2: 87652 GGGGP

The cofactor for path 1 is )(1 431 LL

The cofactor for path 2 is )(1 212 LL

Example1

)(sR )(sY221 HGL 332 HGL

663 HGL 774 HGL

)(

)(1

42324131

4321

LLLLLLLL

LLLL

43211 GGGGP

87652 GGGGP

)(1 431 LL

)(1 212 LL

The transfer function is

2211)(

)(

)( PPsT

sR

sY

Example1

G1 G2 G3

H2

H1

H3

R C

1G 2G 3G

1H

2H

3H

1

R C

Example2

1G 2G 3G

1H

2H

3H

1

R C

33211 HGGGL 122 HGL 2323 HGGL 1214 HGGL

1212321233211 HGGHGGHGHGGG

3211 GGGP 11

121232123321

321

1 HGGHGGHGHGGG

GGG

R

C

Example2