Fourier series expansion and DFT on a moving window

The recursive calculation of Fourier series spectra and the discrete Fourier transform (DFT) on a moving window is investigated. The basic problem is to calculate the spectra coefficients on a one-step-shifted-to-the-right window using the old coefficient values. Several formulas are presented giving the adaptation law of the time-varying spectra. The development in Fourier series expansion can be extended to the case of orthogonal polynomial series.<>