1. Signal Flow Graph nodes : variables branches : gains e.g. y = a ∙ x
e.g. y = 3x + 5z – 0.1y a x y
-0.1 x z 3 y 5

2. Note: y6, y7, y8 are gained up versions of y1.
The value of a node is equal to the sum of all signal coming into the node.
The incoming signal needs to be weighted by the branch gains.
Note: y6, y7, y8 are gained up versions of y1.

3. e.g. e.g. Note: One node is introduced after each summation + u r G1 y- G2 1 u G1 r y
-G2
e.g. N G1 + + R + x z y G2 G3 + + - - H1 N G1 1 G3 x z R y 1 G2
Note: One node is introduced after each summation
-H1 -1

4. An input node is a node with only out going arrows.
R, N and r are input nodes. 1 u G1 r y
-G2
An output node is a node with
only incoming arrows.
There are no output nodes in these two graphs. N G1 1 G3 x z R y 1 G2
-H1 -1

5. But a non-input node cannot be converted into an input node
Any node can be converted into an output node by drawing an additional branch with gain = 1. 1 u G1 y r y 1
-G2 N G1 1 G3 x z R y 1 G2 y 1
-H1 -1
But a non-input node cannot be converted
into an input node

6. fig_03_30
Parallel branches can be summed to form a single branch

7. fig_03_31 Series branches can be multiplied to form a single branch

8. fig_03_32 Feedback connections can be simplified into a single branch1 r y r y 1
Note: the internal node E is lost!

9. G1 G3 x z R y 1 G2 y 1
-H1 -1 G1 G3 x z’ 1 z R y 1 G2 y 1
-H1 -1 G1 R x z’ R y 1 G2 y 1 -1

10. x z’ R y 1 G2 y 1 -1 G1 R x z’ y G2 y 1 -1
Overall:

11. Mason’s Rule A forward path: a path from input to output
Forward path gain Mx: total product of gains along the path
A loop is a closed path in which you can start at any point, follow the arrows, and come back to the same point
A loop gain Li: total product of gains along a loop
Loop i and loop j are non-touching if they do not share any nodes or branches

12. The determinant Δ:
Δx: The determinant of the S.F.G. after removing the k-th forward path
Mason’s Rule:

13. e.g. R y + - G1 G2 G3 H1 x z N R y z G3 1
-H1 x G1 G2 -1 N

14. Get T.F. from N to y
1 forward path: N y
M = 1
2 loops: L1 = -H1G3
L2 = -G2G3
Δ1: remove nodes N, y, and branch N y
All loops broken:  Δ1 = 1

15. Get T.F. from R to y
2 f.p.: R x z y : M1=G2G3
R z y : M2=G1G3
2 loops: L1 = -G3H1
L2 = -G2G3

16. Δ1: remove M1 and compute Δ Δ1 = 1 Δ2: remove M2 and compute Δ Δ2 = 1
Overall:

20. x y b3 b2 b1
-a1
-a2
-a3 x2 x1 e x3 Σ x3 e y x x2 x1 b1 b3 b2
-a3
-a2
-a1 1 s

21. b1
Example: b2 x1 x e y Σ b3 Σ x2 x3
-a1
-a2
-a3
Forward paths:
Loops:

22. Determinant:
Δ1: If M1 is taken out, all loops are broken.
therefore Δ1 = 1
Δ2: If M2 is taken out, all loops are broken.
therefore Δ2 = 1
Δ3: Similarly, Δ3 = 1

23. L1 and L3 are non-touching
Forward path:
M1 = H1 H2 H3
M2 = H4
Loops:
L1 = H1 H5
L2 = H2 H6
L3 = H3 H7
L4 = H4 H7 H6 H5
L1 and L3 are non-touching

24. Δ1: If M1 is taken out, all loops are broken.
therefore Δ1 = 1
Δ2: If M2 is taken out, the loop in the middle (L2) is still there.
therefore Δ2 = 1 – L2 = 1 – H2H6
Total T.F.:

26. One forward path, two loops, no non-touching loops.Vc U + y + -
One forward path, two loops, no non-touching loops.

27. Two forward paths, three loops, no non-touching loops.+ U + + Y + - -
Two forward paths, three loops, no non-touching loops.