Signal-flow graph

A signal flow graph is a representation of the signal processing in a system by 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 the nodes and edges.

  • 5.1 summarize parallel edges
  • 5.2 Sequential summarize edges

Terms

Signal flow graphs are formally defined. Therefore, first a definition of terms.

  • A path is a connected sequence of links ( edges) between nodes in one direction. In the example, (X3 → X4 → X5) a path.
  • An input node has only outgoing paths. X1 is the input node.
  • An output node has only incoming paths. X6 is the starting node.
  • A forward path leads to the exit node. (X2 → X3 → X4) and (X3 → → X7 X6) are forward paths.
  • A reverse path leads towards the entrance node. (X5 X8 → → X2 ) is a backward path.
  • A feedback loop is when the start node and end node are the same. (X2 → X3 → X4 → X5 → X2 → X8 ) is a feedback loop.
  • A self-referential loop is a path to the right leads back from one node to the same node without passing through other nodes.

Figure 1 shows a general directed weighted graph in the mathematical sense. For signal flow graph it is only through the following agreements:

  • A node represents a signal
  • An edge is about her weight processing the signal dar. so creates a new signal.

Furthermore:

  • Are static signals.
  • Are continuous signals.
  • Are the Laplace transform.
  • Are discrete signals.
  • Is the Z-transform.
  • Are transcription factors.
  • Are continuous impulse response functions.
  • Are continuous transfer functions.
  • Are discrete impulse response functions.
  • Are discrete transfer functions.

Elements of a signal flow graph

The addition takes place in the destination node.

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

The convolution is a common transmission link.

The integrator is only available in continuous time systems.

The delay element is only available in discrete time systems.

Basic circuits

For signal flow graph of the same rules as for signal flow diagrams apply. The only difference is the graphical representation. On a representation of the relationships in the time domain has been omitted here because they are too confusing. The ratios are much simpler in the image area. With the basic circuits complex signal flow graphs can reshaped, thus simplifying become.

Series

Parallel connection

Feedback

Creating signal flow graph

From the differential equation

Consider the ordinary, linear, inhomogeneous differential equation with constant coefficients 4th order

We introduce the four state variables

One. Thus, the differential equation of 4th order in a system of four differential equations first order

And

With the output equation

Be transferred. So we need a series of four integrators in the forward path of the signal flow graph. The multiplication by the coefficients is carried out in leading to the summing node backward paths.

Of the transfer function

Consider the transfer function

After multiplying numerator and denominator by the transfer function has a form from which the required integrators are seen immediately.

In the numerator are the factors of the forward path and its denominator is the reverse path. Thus, the signal flow graph can be drawn directly.

From the signal flow diagram

By permutation of nodes and edges are obtained from the signal flow graph of the signal flow diagram and vice versa.

Modifications of signal flow graphs

I the same way as systems of linear equations can be formed, and the corresponding signal flow graph can be reshaped. The following sections describe the different methods are discussed.

Summarize parallel edges

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

For this purpose, the vectors of the combined edges have to be added in the signal flow graph.

Summarize Sequential edges

Three points, and is connected only by the two edges in such a way, such that, the central node can be removed from the display. The associative law, ie, it is applied:

Software

There are some programs for creating signal flow graph. In this paper, the program yEd of yWorks was used. It allows you to draw graphs with different node shapes (eg, circle, rectangle ) and edge (arc or polyline ). A simulation of the system is not possible, in contrast to the signal flow diagram use application Simulink.

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