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SIGNAL-FLOW GRAPHS AND APPLICATIONS A signal-¯ow graph is a graphical means of portraying the relationship among the variables of a set of linear algebraic equations. S. J. Mason originally introduced it to represent the cause-and-effect of linear systems. Associated terms are de®ned in this chapter along with the procedure to draw the signal-¯ow graph for a given set of algebraic equations. Further, signal-¯ow graphs of microwave networks are obtained in terms of their S-parameters and associated re¯ection coef®cients. The manipulation of signal-¯ow graphs is summarized to ®nd the desired transfer functions | Radio-Frequency and Microwave Communication Circuits Analysis and Design Devendra K. Misra Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-41253-8 Hardback 0-471-22435-9 Electronic 9 SIGNAL-FLOW GRAPHS AND APPLICATIONS A signal-flow graph is a graphical means of portraying the relationship among the variables of a set of linear algebraic equations. S. J. Mason originally introduced it to represent the cause-and-effect of linear systems. Associated terms are defined in this chapter along with the procedure to draw the signal-flow graph for a given set of algebraic equations. Further signal-flow graphs of nacrour vc networks tire obtained in terms of their S-parameters and associated reflection coefficients. The manipulation of signal-flow graphs is summarized to find the desired transfer functions. Finally the relations for transdriecr power gam available power gain and operating power gain are formulated in this chapter. Consider a linear network that has N input and output ports. It is described by a set of linear algebraic equations as follows N Vi E t 1 -N 9.1 J 1 This says that the effect at the zth port is a sum of gain times causes at its N ports. Hence represents the dependent variable effect while Ij are the independent variables cause . Nodes or junction points of the signal-flow graph represent these variables. The nodes are connected together by line segments called branches with an arrow on each directed toward the dependent node. Coefficient is the gain of a branch that connects the zth dependent node with th independent node. Signal can be transmitted through a branch only in the direction of the arrow. 354 SIGNAL-FLOW GRAPHS AND APPLICATIONS 355 Basic properties of the signal-flow graph can be summarized as follows A signal-flow graph can be used only when the given system is linear. A set of algebraic equations must be in the form of effects as functions of causes before its signal-flow graph can be drawn. A node is used to represent each variable.
