Example-3
Let's analyze that intricate network in Figure 1 by first converting it to the signal flow diagram in Figure 5(a).
Figure 5. Signal flow diagrams for Figure 1: (a) complete signal flow diagram; (b) signal flow diagram with forward path P1 deleted; (c) signal flow diagram with forward path P2 deleted.
Upon first glance at the flow diagram in Figure 5(a), we might guess this network has two forward paths and two loops. Watch out! The network actually has four forward paths, three loops, and a pair of nontouching loops. (Careful scrutiny is needed when evaluating a signal flow diagram in preparation for Mason's Rule. Stated in different words, using Mason's Rule is like pinning a cloth diaper on a baby—the process requires caution because a small mistake can cause big problems.)
Examination of the Figure 5(a) flow diagram tells us we have P = 4 forward paths, and three loops. The forward path gains (with the path nodes indicated in brackets) and loop gains are:
When forward path P1(z) is deleted, the remaining paths are shown in Figure 5(b), with one remaining loop, so Δ1(z) = 1–z–1/3. Similarly, when forward path P2(z) is deleted, the remaining paths are shown in Figure 5(c), with one remaining loop, thus Δ2(z) = 1–z–1/2. Now when forward paths P3(z) and P4(z) are deleted we have no paths remaining. Thus our removed-path determinants are:
In this example there are two nontouching loops, [b,c,b] and [f,e,f], so the sum of products nontouching loop gains taken two at a time is z–1/2 times z–1/3 = z–2/6. So we now use Eq. (1) to define the network's primary Δ(z) determinant to be:
Using Eq. (3) to obtain the network's transfer function, we have
(Click to enlarge)
After plowing through this last example, Step 7 at the end of Section II is well-deserved.
V. MATLAB Code
For those readers having access of MATLAB software, there is a Mason's Rule MATLAB function, written by Rob Walton, available at:
http://www.mathworks.com/matlabcentral/fileexchange/22
Walton's slick routine requires the user to create a ".txt" file describing a network's signal paths. The ".txt" file for our above Example# 3 contains the following 10 lines of text (parameters are separated by tabs).
Conclusion
Mason's Rule is an analog system analysis technique formulated in the early 1950s (around the time the young truck driver Elvis Presley arrived on the pop music scene) by Samuel Mason. Here we described how to use Mason's Rule to aid in modern-day discrete (DSP) network analysis. Although a bit tedious for complicated networks, Mason's Rule provides a super-effective method for determining the z-domain transfer function of even the most complicated discrete networks. Happily, public domain Mason's Rule MATLAB code is available.
About the Author
Richard (Rick) Lyons is a consulting Systems Engineer and lecturer with Besser Associates in Mountain View, California. He is the author of "Understanding Digital Signal Processing 2/E" (Prentice-Hall, 2004), and Editor of, and contributor to, "Streamlining Digital Signal Processing, A Tricks of the Trade Guidebook" (IEEE Press/Wiley, 2007). Lyons is also an Associate Editor for the IEEE Signal Processing Magazine.
References
[1] S. Mason, "Feedback Theory: Some Properties of Signal Flow Graphs," Proc. of IRE, Vol. 41, pp. 1144-1156, Sept. 1953.
[2] S. Mason, "Feedback Theory: Further Properties of Signal Flow Graphs," Proc. of IRE, Vol. 44, pp. 1920-926, Sept. 1956.
[3] S. Mason and H. Zimmerman, Electronic Circuits, Signals, and Systems, John Wiley & Sons, New York, 1960.
