Our first example of Mason`s rule will be to determine the H(z) transfer function of our old friend, the Biquad-Infinite Impulse Response (IIR) filter shown in Figure 3(a). We will see the application of Mason`s amplification formula to the signal flow diagram through a few examples, but first, let`s understand the different terms associated with the Freemason`s amplification formula. For our purposes, Mason`s rule is a method of deriving the Z domain transfer function of a discrete network by identifying different transfer paths from the input node to the exit node of a discrete network, and the different feedback paths that common signal nodes may or may not share with these transfer paths. It sounds mysterious, but it`s not really too complicated. Let`s define our terminology of Mason`s rule, and then demonstrate this analytical technique with examples. Again, the procedure of the Masonic rule seems a bit complicated, but we can show how simple the process is. Here are some examples of varying complexity that illustrate the execution and usefulness of the Masonic rule in the analysis of discrete networks. Mason`s rule is also particularly useful for deriving the Z domain transfer function from a discrete network that has internal feedback loops embedded in external feedback loops (nested loops). Here`s the good news: if we are able to draw the block diagram of a discrete network, then applying the Masonic rule will give us the Z-Domain H(z) transfer capability of that network. Once we have H(z), we can use all the algebraic tools and software at our disposal to determine the behavior of the frequency domain and the stability of the network. Here we describe Mason`s rule, accompanied by several examples, in the hope that this robust analysis technique will be useful to the reader in his future DSP network analysis efforts. As described above, q {displaystyle q} is a sum of cyclic products, each of which usually falls into an ideal (e.g.

strictly causal operators). The fractions of this form give a subring R ( 1 + ⟨ L i ⟩ ) − 1 {displaystyle R(1+langle L_{i}rangle )^{-1}} of the field of the rational function. This observation is transferred to the noncommutative case,[3] even though the Masonic rule itself is to be replaced by the Riegle rule. After going through this last example, step 7 at the end of section II is well deserved. For each forward trajectory in a signal flow diagram, there is an associated determinant, represented by Δi(z). If a diagram P = 3 forward paths (called paths P1(z), P2(z) and P3(z)), then there exists a determinant Δ1(z), a Δ2(z) and a determinant Δ3(z). The index variable i is simply the index that identifies each forward trajectory and its determinants. The determinant Δi(z) is the determinant of the flow diagram of the signal that does not touch the i-th path forward. For example, to find Δ1(z), we remove the forward trajectory P1(z) in a signal flow diagram (and all branches touching the P1(z) trajectory) and use equation (1) above for the remaining signal flow paths.

If there are no loops left after removing the path before P1(z), then Δ1(z) = 1. This example applies Mason`s rule to a DC polarization distance network that contains nested loops. Starting with the network in Figure 4(a), let`s convert it to the signal flow diagram in Figure 4(b). Notice how the subtraction in Figure 4(a) is replaced by a gain symbol of minus one in the signal flow diagram. With the technique of reducing functional diagrams, we have seen that to simplify a system for analysis purposes, we must apply certain reduction rules appropriately. And after simplifying the system, its global C(s)/R(s) transfer function is determined. Using Figure 2 as an example, the following definitions are established: In this example, there are two non-touch loops, [b,c,b] and [f,e,f], so the sum of the non-contact loop gains of the products is z-1/2 times z-1/3 = z-2/6. So let`s use equation (1) to define the primary determinant Δ(z) of the network as follows: Mason`s rule can be expressed in a simple matrix form. For example, suppose that T {displaystyle mathbf {T} } is the transient matrix of the graph, where t n m = [ T ] n m {displaystyle t_{nm}=left[mathbf {T} right]_{nm}} is the sum of the transmission of branches from node m to node n. Then the gain from node m to node n of the graph is equal to u n m = [ U ] n m {displaystyle u_{nm}=left[mathbf {U} right]_{nm}} , where This blog discusses a valuable analytical method (well known to our analog control system engineering brothers) for obtaining the Z-domain transmission function equations of digital signal processing (DSP) networks. This method, called Mason`s rule (sometimes called Mason`s gain formula), was developed by Samuel Mason in the early 1950s to analyze interconnected analog systems.[1-3] Here we describe Mason`s rule and present some examples that show the advantages of this network analysis technique.

However, Mason`s rule characterizes the transfer functions of interconnected systems in a way that is both algebraic and combinatorial, allowing for general statements and other calculations in algebraic systems theory.