This paper introduces a method for efficiently calculating transforms of signals with arbitrary digital dimensions by reducing the transform to a vector-to-circulant matrix multiplication. The method overcomes the increasing computational cost associated with traditional Fourier transforms, especially when the number of sampling points is not a power of 2. The paper establishes a connection between harmonic equations in rectangular and polar coordinate systems, leading to a robust iterative algorithm for conformal mapping calculation. Additionally, it proposes a new ratio for oscillative signals and provides an efficient way to compute it. The paper also discusses the evaluation of analytical functions and the harmonic covariation and correlation of signals, demonstrating their applications in various fields such as quantum mechanics and market data analysis.This paper introduces a method for efficiently calculating transforms of signals with arbitrary digital dimensions by reducing the transform to a vector-to-circulant matrix multiplication. The method overcomes the increasing computational cost associated with traditional Fourier transforms, especially when the number of sampling points is not a power of 2. The paper establishes a connection between harmonic equations in rectangular and polar coordinate systems, leading to a robust iterative algorithm for conformal mapping calculation. Additionally, it proposes a new ratio for oscillative signals and provides an efficient way to compute it. The paper also discusses the evaluation of analytical functions and the harmonic covariation and correlation of signals, demonstrating their applications in various fields such as quantum mechanics and market data analysis.