Signal flow graph

A signal flow graph is a representation of the signal processing in a system through a directed , weighted graph . The nodes of this graph are small processing units that process the incoming signals in a certain form and then send the result to all outgoing edges.

From the signal flow diagram , they differ in the importance of nodes and edges.

Terms

Fig. 1: Example signal flow graph

Signal flow graphs are formally defined. Therefore, first some definitions of terms.

• A path is a connected sequence of connections (edges) between nodes in one direction. In the example (X3 → X4 → X5) is a path.
• An ingress node has only outbound paths. X1 is the input node.
• An exit node has only inbound paths. X6 is the output node.
• A forward path leads towards the exit node. (X2 → X3 → X4) and (X3 → X7 → X6) are forward paths.
• A backward path leads towards the entry node. (X5 → X8 → X2) is a backward path.
• A feedback loop exists when the start node and the end node are the same. (X2 → X3 → X4 → X5 → X8 → X2) is a feedback loop.
• A self-referential loop is a path that leads from a node directly back to the same node without going through other nodes.

Figure 1 shows a general directional, weighted graph in the mathematical sense. It only becomes a signal flow graph with the following agreements:

• A node represents a signal.
• An edge represents the processing of the signal via its weight. It therefore generates a new signal.

The following also applies:

• ${\displaystyle Y,X_{1},X_{2}\,}$ are static signals.
• ${\displaystyle y(t),x_{1}(t),x_{2}(t)\,}$ are continuous signals.
• ${\displaystyle Y(s),X_{1}(s),X_{2}(s)\,}$are their Laplace transforms .
• ${\displaystyle y(n),x_{1}(n),x_{2}(n)\,}$ are discrete signals.
• ${\displaystyle Y(z),X_{1}(z),X_{2}(z)\,}$are their Z-transforms .

• ${\displaystyle G,G_{1},G_{2}\,}$ are transfer factors.
• ${\displaystyle G(t),G_{1}(t),G_{2}(t)\,}$are continuous impulse response functions .
• ${\displaystyle G(s),G_{1}(s),G_{2}(s)\,}$are continuous transfer functions .
• ${\displaystyle G(n),G_{1}(n),G_{2}(n)\,}$ are discrete impulse response functions.
• ${\displaystyle G(z),G_{1}(z),G_{2}(z)\,}$ are discrete transfer functions.

Elements of a signal flow graph

The addition takes place in the destination node.

${\displaystyle Y=X_{1}+X_{2}\,}$ ${\displaystyle y(t)=x_{1}(t)+x_{2}(t)\,}$ ${\displaystyle Y(s)=X_{1}(s)+X_{2}(s)\,}$ ${\displaystyle y(n)=x_{1}(n)+x_{2}(n)\,}$ ${\displaystyle Y(z)=X_{1}(z)+X_{2}(z)\,}$

The multiplication by a constant is used, inter alia, a differential equation for the processing of the coefficients.

${\displaystyle Y=a\cdot X}$ ${\displaystyle y(t)=a\cdot x(t)}$ ${\displaystyle Y(s)=a\cdot X(s)}$ ${\displaystyle y(n)=a\cdot x(n)}$ ${\displaystyle Y(z)=a\cdot X(z)}$

The convolution is a general link.

${\displaystyle y(t)=\int _{0}^{t}G(t-\tau )\cdot x(\tau )d\tau }$ ${\displaystyle Y(s)=G(s)\cdot X(s)}$ ${\displaystyle y(n)=\sum _{i=0}^{n}G(n-i)\cdot x(i)}$ ${\displaystyle Y(z)=G(z)\cdot X(z)\,}$

The integrator only exists in systems that are continuous over time.

${\displaystyle y(t)=\int _{0}^{t}x(\tau )d\tau }$ ${\displaystyle Y(s)={\frac {1}{s}}X(s)}$

The delay element is only available in time-discrete systems.

${\displaystyle y(n)=x(n-1)\,}$ ${\displaystyle Y(z)=z^{-1}\cdot X(z)}$

Basic circuits

The same rules apply to signal flow graphs as to signal flow diagrams. The only difference is the graphical representation. The relationships in the time domain were not shown here, as they are too confusing. The relationships are much simpler in the image area. With the basic circuits, complex signal flow graphs can be transformed and thus simplified.

Series connection

${\displaystyle Y=G_{1}\cdot G_{2}\cdot X}$ ${\displaystyle Y(s)=G_{1}(s)\cdot G_{2}(s)\cdot X(s)}$ ${\displaystyle Y(z)=G_{1}(z)\cdot G_{2}(z)\cdot X(z)}$

Parallel connection

${\displaystyle Y=(G_{1}+G_{2})\cdot X}$ ${\displaystyle Y(s)=(G_{1}(s)+G_{2}(s))\cdot X(s)}$ ${\displaystyle Y(z)=(G_{1}(z)+G_{2}(z))\cdot X(z)}$

Feedback

${\displaystyle Y={\frac {G_{1}}{1+G_{1}\cdot G_{2}}}\cdot X}$ ${\displaystyle Y(s)={\frac {G_{1}(s)}{1+G_{1}(s)\cdot G_{2}(s)}}\cdot X}$ ${\displaystyle Y(z)={\frac {G_{1}(z)}{1+G_{1}(z)\cdot G_{2}(z)}}\cdot X}$

Creation of signal flow graphs

From the differential equation

4th order differential equation

The usual, linear, inhomogeneous differential equation with constant coefficients of the 4th order is given

${\displaystyle {\frac {d^{4}\left({\frac {y}{b_{0}}}\right)}{dt^{4}}}+a_{3}{\frac {d^{3}\left({\frac {y}{b_{0}}}\right)}{dt^{3}}}+a_{2}{\frac {d^{2}\left({\frac {y}{b_{0}}}\right)}{dt^{2}}}+a_{1}{\frac {d\left({\frac {y}{b_{0}}}\right)}{dt}}+a_{0}\left({\frac {y}{b_{0}}}\right)=x(t)}$

We carry the 4 state variables

${\displaystyle x_{1}=\left({\frac {y}{b_{0}}}\right)}$

${\displaystyle x_{2}={\dot {x}}_{1}={\frac {d\left({\frac {y}{b_{0}}}\right)}{dt}}}$

${\displaystyle x_{3}={\dot {x}}_{2}={\frac {d^{2}\left({\frac {y}{b_{0}}}\right)}{dt^{2}}}}$

${\displaystyle x_{4}={\dot {x}}_{3}={\frac {d^{3}\left({\frac {y}{b_{0}}}\right)}{dt^{3}}}}$

a. The 4th order differential equation can thus be converted into a system of 4 1st order differential equations

${\displaystyle {\dot {x}}_{1}=x_{2},{\dot {x}}_{2}=x_{3},{\dot {x}}_{3}=x_{4}}$

and

${\displaystyle {\dot {x}}_{4}=-a_{0}x_{1}-a_{1}x_{2}-a_{2}x_{3}-a_{3}x_{4}+x}$

with the initial equation

${\displaystyle y=b_{0}x_{1}\,}$

be convicted. So we need a series connection of 4 integrators in the forward path of the signal flow graph. The multiplication with the coefficients takes place in the backward paths leading to the summation node. ${\displaystyle a_{i},i=0...3\,}$

From the transfer function

4th order transfer function

The transfer function is given

${\displaystyle G(s)={\frac {b_{3}s^{3}+b_{2}s^{2}+b_{1}s+b_{0}}{s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}}}}$.

After multiplying the numerator and denominator by , the transfer function has a form from which the required integrators can be seen immediately. ${\displaystyle s^{-4}\,}$

${\displaystyle G(s)={\frac {b_{3}s^{-1}+b_{2}s^{-2}+b_{1}s^{-3}+b_{0}s^{-4}}{1+a_{3}s^{-1}+a_{2}s^{-2}+a_{1}s^{-3}+a_{0}s^{-4}}}}$

The numerator contains the factors of the forward path and the denominator those of the reverse path. This allows the signal flow graph to be drawn directly.

From the signal flow plan

Signal flow diagram of a 4th order transfer function

By interchanging nodes and edges, the signal flow diagram is obtained from the signal flow graph and vice versa.

Modifications of signal flow graphs

In the same way as linear systems of equations can be transformed, the associated signal flow graph can also be transformed. Some rules are explained below.

Combine parallel edges

Different edges with the same source and the same sink can be combined into one edge. So the distributive law is applied:

${\displaystyle x_{1}*a+x_{1}*b=x_{1}*(a+b)}$.

To do this, the vectors of the combined edges must be added in the signal flow graph.

Edges with the same source and target point can be combined.

Combine sequential edges

If three points , and connected only by two edges in such a way so that is valid , then the central node can be removed from the display. So the associative law is applied: ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a\rightarrow b\rightarrow c}$ ${\displaystyle b}$

${\displaystyle (x_{1}*a)*b=x_{1}*(a*b)}$.

Individual evidence

1. ↑ Mason, Samuel J .: Feedback Theory - Some Properties of Signal Flow Graphs , Proceeding of the IRE , 1953, vol. 41, pp. 1144-1156
2. ↑ ^{a b} Strauss, Frieder: Basic Course in High Frequency Technology , Vieweg + Teubner Verlag, Wiesbaden 2012, pp. 172–175