This paper introduces a method for efficiently calculating signal transforms using circulant matrices, which can replace traditional Fourier and inverse Fourier transforms. The method allows for the transformation of signals with arbitrary digital dimensions by reducing the transform to a vector-to-circulant matrix multiplication. It also establishes a connection between harmonic equations in rectangular and polar coordinate systems, enabling a robust iterative algorithm for conformal mapping calculations. A new ratio of two oscillatory signals is proposed, along with an efficient way to compute it. The traditional method of transforming a signal involves using Fourier transforms to obtain the amplitude-frequency characteristic (AFC), modifying the coefficients, and then applying the inverse Fourier transform. However, this method becomes computationally expensive as the number of sampling points increases. The proposed method uses a circulant matrix operator to perform both the AFC transformation and the transformation of analytical function values from the unit circle to a concentric circle with a different radius and starting argument. This approach is more efficient than the traditional Fourier-based method. The paper also discusses the use of circulant matrices for evaluating analytical functions based on their real part values on the unit circle. It presents a method for calculating the harmonic conjugate of a function using a circulant matrix. The paper further explores the connection between harmonic equations in rectangular and polar coordinate systems, demonstrating how the properties of harmonic functions can be used to solve conformal mapping problems. The method is applied to solve Riemann's problem of finding conformal mappings from the unit disk to a simply connected area bounded by a Jordan curve. The algorithm converges exponentially and can be implemented efficiently, with results obtained in a time proportional to N × ln(N) operations. The paper also introduces a new concept of harmonic covariation and correlation, which can be used to calculate the ratio of signals when only one component of a complex-valued variable is available. The method is demonstrated using market data to find companies with similar share price behavior.This paper introduces a method for efficiently calculating signal transforms using circulant matrices, which can replace traditional Fourier and inverse Fourier transforms. The method allows for the transformation of signals with arbitrary digital dimensions by reducing the transform to a vector-to-circulant matrix multiplication. It also establishes a connection between harmonic equations in rectangular and polar coordinate systems, enabling a robust iterative algorithm for conformal mapping calculations. A new ratio of two oscillatory signals is proposed, along with an efficient way to compute it. The traditional method of transforming a signal involves using Fourier transforms to obtain the amplitude-frequency characteristic (AFC), modifying the coefficients, and then applying the inverse Fourier transform. However, this method becomes computationally expensive as the number of sampling points increases. The proposed method uses a circulant matrix operator to perform both the AFC transformation and the transformation of analytical function values from the unit circle to a concentric circle with a different radius and starting argument. This approach is more efficient than the traditional Fourier-based method. The paper also discusses the use of circulant matrices for evaluating analytical functions based on their real part values on the unit circle. It presents a method for calculating the harmonic conjugate of a function using a circulant matrix. The paper further explores the connection between harmonic equations in rectangular and polar coordinate systems, demonstrating how the properties of harmonic functions can be used to solve conformal mapping problems. The method is applied to solve Riemann's problem of finding conformal mappings from the unit disk to a simply connected area bounded by a Jordan curve. The algorithm converges exponentially and can be implemented efficiently, with results obtained in a time proportional to N × ln(N) operations. The paper also introduces a new concept of harmonic covariation and correlation, which can be used to calculate the ratio of signals when only one component of a complex-valued variable is available. The method is demonstrated using market data to find companies with similar share price behavior.