Short-time Fourier analysis via optimal harmonic FIR filters

Abstract

The Fourier coefficients (FCs) of quasiperiodic signals are assumed to be in random walk motion in order to represent a broader class. A state model for such quasiperiodic signals is derived. The optimal short-time estimate of the Fourier coefficients is obtained via the suggested optimal harmonic FIR filter (OHFF) based on this state-space signal model. The optimal harmonic FIR filter can be considered to be a generalization of the discrete Fourier transform (DFT) in the sense that it becomes the same as the DFT when the state model is for periodic signals and the filter length is equal to the order of the state model. The optimal harmonic FIR filter derived from the model, even with nonzero state noise and measurement noise, gives an exact harmonic estimate when an incoming signal is periodic and noiseless. It is shown by examples that the ability to suppress noise and the ability to resolve changes of the Fourier coefficients can be adjusted by the filter length and the noise covariance of the state model. Finally, the suggested scheme is compared with existing short-time Fourier analysis methods in a test signal that has time-varying Fourier coefficients.