Polynomial Implementation of the Taylor–Fourier Transform for Harmonic Analysis

Recently, the Taylor-Fourier transform (TFT) was proposed to analyze the spectrum of signals with oscillating harmonics. The coefficients of this linear transformation were obtained through the calculation of the pseudoinverse matrix, which provides the classical solution to the normal equations of the least-squares (LS) approximation. This paper presents a filtering design technique that obtains the coefficients of the filters at each harmonic by imposing the maximally flat conditions to the polynomials defining their frequency responses. This condition can be used to solve the LS problem at each particular harmonic frequency, without the need of obtaining the whole set, as in the classical pseudoinverse solution. In addition, the filter passband central frequency can follow the fluctuations of the fundamental frequency. Besides, the method offers a reduction of the computational burden of the pseudoinverse solution. An implementation of the proposed estimator as an adaptive algorithm using its own instantaneous frequency estimate to relocate its bands is shown, and several tests are used to compare its performance with that of the ordinary TFT.