Fast Approximation of Dominant Harmonics

This paper presents a fast method for estimating dominant harmonics in a sequence of data. In a stochastic sense, the proposed method finds the autoregressive scheme with a pure point spectrum that best describes the data, while from a deterministic point of view, the method is a special case of the Lanczos algorithm for finding eigenvalues of a symmetric matrix. Eigenvalue approximations come into play because every circulant matrix is diagonalized by the discrete Fourier transform matrix, and so using the Lanczos algorithm with the given data as the initial vector on a simple circulant matrix, the eigenvalues that are first approximated are the eigenvalues corresponding to eigenvectors which are dominant in the initial vector. It is shown that this method is related to “lattice methods” for linear prediction and to Prony’s method for exponential approximation.