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Represent the DSP network as a signal-flow graph
Start with the block diagram and translate it into a directed graph. Treat each signal location as a node and each directed connection as a branch labeled with its gain. A unit delay is represented by z−1; constant multipliers retain their values. At a subtraction, include a branch with gain −1 rather than treating the summing point as an unweighted connection.
This representation makes the graph’s arithmetic explicit. Before counting paths or loops, check that every branch gain, direction, and signal node matches the original diagram.
Find the forward paths and their gains
A forward path runs from the input node to the output node without revisiting a node. Find every such path, not only the most visually obvious route. For path i, multiply the gains of its branches to obtain its path gain, Pi.
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Separate paths count separately even if they share some branches or nodes. Each will have its own contribution in Mason’s formula.
Find feedback loops and nontouching combinations
A loop is a closed route through the graph that does not repeat a node along the way, other than returning to its starting point. Its loop gain is the product of its branch gains. A pair of loops is nontouching if the loops have no signal node in common. Larger nontouching sets must likewise have no shared nodes among their member loops.
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Record each individual loop and then identify every mutually nontouching pair, triple, and larger set that exists. Nested feedback and visually busy graphs make omissions easy; inspect the node memberships rather than relying on a quick glance.
Build the full graph determinant, Δ
The determinant accounts for the loops and their nontouching combinations, with signs alternating by the number of loops in each combination:
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Δ = 1 − (sum of individual loop gains) + (sum of products of pairs of mutually nontouching loop gains) − (sum of products of triples of mutually nontouching loop gains) + …
Continue the pattern for every possible mutually nontouching set. If no loops in a particular size form a nontouching set, that term contributes nothing.
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Calculate a path-specific determinant, Δi
For each forward path, exclude every loop that touches any node on that path. Use the remaining loops to construct Δi with the same alternating-sign rule, including only nontouching combinations among those remaining loops. A loop that touches the path cannot contribute to that path’s determinant.
Assemble the transfer function
Once every path gain and its path-specific determinant are known, combine them with the full graph determinant:
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H(z) = [Σi PiΔi] / Δ
In practical terms, multiply each forward-path gain by its corresponding Δi, add those products, and divide by the full Δ. Simplify the resulting expression to obtain Y(z)/X(z).
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Use a checklist to avoid enumeration errors
- Translate: Label all nodes and directed branch gains, including z−1 delays and negative gains for subtraction.
- Enumerate paths: List every input-to-output route that does not revisit a node; calculate each Pi.
- Enumerate loops: List each closed route that does not repeat a node and calculate its gain.
- Check nontouching sets: Find all combinations of loops with no shared nodes.
- Build each Δi: For a given forward path, remove loops that touch it, then apply the alternating-sign rule to the rest.
- Build Δ: Apply that rule to all loops and all nontouching combinations in the graph.
- Substitute and verify: Apply Mason’s formula and independently check the graph and algebra where possible.
When Mason’s Rule is useful—and what it does not do
The method is especially helpful when feedback is nested or multiple signal paths interact: it keeps each path’s contribution visible while accounting systematically for feedback. That visibility comes at a cost. Counting every path, loop, and nontouching combination can be tedious, and missing one invalidates the derivation.
Direct algebraic reduction is another way to derive a transfer function. Mason’s Rule offers a graph-based accounting of the contributions; the available examples do not establish a measured speed or accuracy advantage over algebraic reduction. After deriving H(z), use it as the basis for separate frequency-response or stability analysis—the graph-counting procedure itself does not perform those analyses.
Further reading and historical tool note
Richard (Rick) Lyons’s article, “Analyzing DSP networks with Mason’s Rule”, illustrates the method with a biquad IIR filter, a DC-bias-removal network with nested loops, and a multiple-feedback network involving nontouching loops. Lyons describes Mason’s Rule as, “For my money, Mason’s Rule is the single most powerful network analysis tool at our disposal.” That is his opinion, not a measured comparison.
Lyons also mentions a MATLAB function by Rob Walton that accepted a text description of network paths. Its current availability, compatibility, and maintenance are not established here, so treat the mention as historical rather than as a recommendation of a currently supported tool. Lyons names his book Understanding Digital Signal Processing as further reading.
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