Signal Flow Graphs and Data Flow Graphs - Signal Processing Systems - page 1258

By substituting both Y 1 and Y 3 into ( 4 ) , we obtain

Y 2 = X − YH 3 − Y 3 H 1 = X − YH 3 − G 1 H 1 Y 2 (5)

Consequently,

−

X YH 3

Y 2 = (6)

1 + G 1 H 1

Then by substituting ( 6 ) into( 3 ) ,

G 1 G 2 G 3 + G 4 X −

G 1 H 1 = (

YH 3 G 1 G 2 G 3 + G 4 )( X − YH 3 )

Y =

G 3 H 2 ·

+ + ( + )( + )

1 1 1 G 3 H 2 1 G 1 H 1

+ ( + )

G 1 G 2 G 3 G 4 G 1 G 2 G 3 G 4 H 3

= X − Y

( 1 + G 3 H 2 )( 1 + G 1 H 1 ) ( 1 + G 3 H 2 )( 1 + G 1 H 1 )

As a result,

( G 1 G 2 G 3 + G 4 ) H 3 G 1 G 2 G 3 + G 4

1 + Y = X

( 1 + G 3 H 2 )( 1 + G 1 H 1 ) ( 1 + G 3 H 2 )( 1 + G 1 H 1 )

Y
X =

G 1 G 2 G 3 + G 4

( + )( + )+( + )

1 G 3 H 2 1 G 1 H 1 G 1 G 2 G 3 G 4 H 3

Finally, the transfer function between the output node Y and input node X is

G 1 G 2 G 3 + G 4

M = (7)

1 + G 1 H 1 + G 3 H 2 + G 1 G 3 H 1 H 2 + G 4 H 3 + G 1 G 2 G 3 H 3

which is exactly the same as the derived transfer function using Mason's gain
formula in ( 2 ) .

3 Data Flow Graphs

In this section, the notation of data flow graph (DFG) is introduced and is followed
by an overview of the single-rate DFG and the multi-rate DFG. How to construct
an equivalent single-rate DFG from the multi-rate DFG is then explained in detail.
After that, the concepts of retiming and pipelining are briefly introduced to derive
equivalent DFGs.

3.1 Notation

In data flow graph representations, the nodes represent computations or functions
or subtasks and the directed edges represent data paths (communications between
nodes) and each edge has a nonnegative number of delays associated with it.