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

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.

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

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

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

fig_03_30 Parallel branches can be summed to form a single branch

fig_03_31 Series branches can be multiplied to form a single branch

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

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

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

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

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

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

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

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

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

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

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

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

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

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.:

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

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