📚 Signal-flow Graphs | 信号流图
Signal-flow graphs (SFGs) are directed graphs used to represent linear algebraic equations. Originally developed by Samuel Mason in the 1950s for feedback control systems, they provide a visual method to solve systems of linear equations without matrix inversion. In the context of IB Mathematics, SFGs connect graph theory with linear algebra, offering an elegant way to determine unknown variables by tracing paths and loops.
信号流图（SFG）是一种用于表示线性代数方程的有向图。最初由 Samuel Mason 在 20 世纪 50 年代为反馈控制系统开发，它提供了一种无需矩阵求逆即可求解线性方程组的可视化方法。在 IB 数学背景下，信号流图将图论与线性代数联系起来，通过追踪路径和环路，以一种优雅的方式确定未知变量。
1. What Are Signal-flow Graphs? | 什么是信号流图？
A signal-flow graph consists of nodes representing variables and directed branches representing linear relationships between them. Each branch has a gain (a constant multiplier) that multiplies the signal at its input node to produce a contribution at its output node. The value at each node equals the sum of all incoming signals. This additive property makes SFGs naturally suited for systems of linear equations.
信号流图由表示变量的节点和表示它们之间线性关系的有向支路组成。每条支路都有一个增益（常数乘数），它将输入节点的信号乘以该增益，产生对输出节点的贡献。每个节点的值等于所有输入信号的总和。这种加法性质使信号流图天然适合线性方程组。
2. Constructing an SFG from Equations | 从方程构建信号流图
To convert a set of linear equations into an SFG, first assign a node to each variable (including input constants such as 1). Then, for each equation, express one variable as a sum of other variables multiplied by coefficients. Draw a directed branch from each right-hand variable to the left-hand variable, labeling the branch with the corresponding coefficient. If a constant term appears, it can be represented as a branch from a source node with value 1 and gain equal to the constant.
要将一组线性方程转换为信号流图，首先为每个变量分配一个节点（包括值为1的输入常数节点）。然后，对于每个方程，将一个变量表示为其他变量乘以系数之和。从右侧的每个变量到左侧变量绘制一条有向支路，并在支路上标出相应的系数。如果出现常数项，可以用一条从值为1的源节点出发、增益等于该常数的支路来表示。
For example, the equation x = 2y + 3z leads to branches from y to x with gain 2, and from z to x with gain 3. A constant term such as x = 2y + 5 would add a branch from source 1 to x with gain 5.
例如，方程 x = 2y + 3z 会产生从 y 到 x 增益为 2 的支路，以及从 z 到 x 增益为 3 的支路。常数项如 x = 2y + 5 则会添加一条从源节点 1 到 x、增益为 5 的支路。
3. Basic Terminology | 基本术语
Node: A point representing a variable. Input nodes (sources) have only outgoing branches; output nodes (sinks) have at least one incoming branch, but any node can be considered.
节点 (Node): 表示变量的点。输入节点（源点）只有流出支路；输出节点（汇点）至少有一条流入支路，但任意节点都可被考察。
Branch: A directed edge with a gain (also called transmittance). It transmits the signal from its start node to its end node multiplied by the gain.
支路 (Branch): 带有增益（又称传递系数）的有向边。它将起始节点的信号乘以增益后传送至终止节点。
Path: A continuous sequence of branches following the arrow directions, where no node is visited more than once. The path gain is the product of all branch gains along it.
路径 (Path): 沿箭头方向由支路组成的连续序列，途中不重复经过任何节点。路径增益为其上所有支路增益的乘积。
Forward path: A path from an input node to an output node, traversing each node only once.
前向路径 (Forward path): 从输入节点到输出节点且每个节点仅经过一次的路径。
Loop: A closed path that starts and ends at the same node, without visiting any other node more than once. The loop gain is the product of gains of the branches forming the loop.
环路 (Loop): 起点和终点为同一节点且不重复经过其它节点的闭合路径。环路增益为构成环路的所有支路增益之积。
Non-touching loops: Two or more loops that share no common node. They are called non-touching because their node sets are disjoint.
不接触环路 (Non-touching loops): 彼此没有公共节点的两个或多个环路，即其节点集合互不相交。
4. Mason’s Gain Formula | 梅森增益公式
The transfer function (gain) from an input node to an output node is given by Mason’s formula:
T = ( Σ Pk Δk ) / Δ
从输入节点到输出节点的传递函数（增益）由梅森公式给出：
T = ( Σ Pk Δk ) / Δ
where Pk is the gain of the k-th forward path, Δ = 1 − ΣLi + ΣLiLj − ΣLiLjLk + … (the sum of all individual loop gains, minus the sum of products of gains of non-touching loops taken two at a time, plus the sum for three at a time, and so on), and Δk is the value of Δ for the portion of the graph that does not touch the k-th forward path (i.e., delete all nodes and branches that lie on that forward path, then compute Δ for the remaining subgraph).
其中 Pk 是第 k 条前向路径的增益，Δ = 1 − ΣLi + ΣLiLj − ΣLiLjLk + …（所有单个环路增益之和，减去所有两两不接触环路增益乘积之和，加上所有三三不接触环路增益乘积之和，依此类推，符号交替），Δk 是图中不与第 k 条前向路径接触的部分的 Δ 值（即删除该前向路径上的所有节点和支路后，剩余子图的 Δ）。
5. Calculating Δ: The Determinant of the Graph | 计算 Δ：图的特征式
The determinant Δ encapsulates the algebraic effect of all feedback loops. To compute it, list every individual loop and sum their gains. Next, identify all pairs of non-touching loops and subtract the sum of their gain products. Then add the sum of gain products for every triple of mutually non-touching loops, and continue with alternating signs. If there are no loops, Δ = 1.
特征式 Δ 概括了所有反馈环路的代数影响。计算时，先列出每个单环路并求增益之和。然后找出所有不接触的环路对，减去其增益乘积之和。接着找出所有三个不接触的环路组，加上其增益乘积之和，如此正负交替进行。如果没有环路，Δ = 1。
It is crucial to ensure loops are truly non-touching: they must not share any node. Even sharing a single node makes them touching, and their product must not appear in the sum.
确保环路真正不接触至关重要：它们不能共享任何节点。哪怕仅共享一个节点也算接触，其乘积不得出现在求和中。
6. Example: Solving Two Linear Equations | 示例：解两个线性方程
Consider the system: x = a y + b, y = c x + d. Treat b and d as inputs from a source node valued 1. Create node 1 (constant 1), node x, and node y. Draw branches: 1 → x (gain b), 1 → y (gain d), y → x (gain a), x → y (gain c).
考虑方程组：x = a y + b，y = c x + d。将 b 和 d 视为从值为 1 的源节点出发的输入。创建节点 1（常数 1）、节点 x、节点 y。画出支路：1 → x (b)，1 → y (d)，y → x (a)，x → y (c)。
Forward paths from 1 to x: P₁: 1 → x (gain b). P₂: 1 → y → x (gain d × a). Loops: L₁: x → y → x (gain c × a = a c). There are no non-touching loops, so ΣLiLj = 0. Thus Δ = 1 − a c. Both forward
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