Transform Matrix Approach for Path Gain in Signal Flow Graph

Mason's gain formula can grow factorial because of growth in the enumeration of paths in a directed graph. Each of the (n - 2)! permutation of the intermediate vertices includes a path between input and output nodes. This paper presents a novel method for analyzing the loop gain of a signal flow graph based on the transform matrix approach. This approach only requires matrix determinant operations to determine the transfer function with complexity O(n³) in the worst case, therefore rendering it more efficient than Mason's gain formula. We derived the transfer function of the signal flow graph to the ratio of different cofactor matrices of the augmented matrix. Example of feedback networks demonstrates the intuitive approach to obtain the transfer function for both numerical and computer-aided symbolic analysis, which yields the same results as Mason's gain formula. The transfer matrix offers an excellent physical insight because it enables visualization of the signal flow.