Computing Power Spectra with FFT

This is a tutorial paper describing the fast Fourier transform and practical aspects of the digital computation of power density spectra using FFT. Discussed first is the relation of the discrete Fourier transform (DFT) to the Fourier integral. Then we describe an FFT algorithm, showing its computational saving over the naive calculation of the DFT. We then consider the estimation of the power density spectrum, describing briefly the indirect method (used before FFT) and then the direct method of power spectrum estimation via the periodgram estimate now commonly used. We continue with a discussion of bandwidth and stability of spectral estimates and how these properties are related to the time and frequency windows used to smooth the periodgram estimates. Finally, we consider briefly the estimation of cross spectra and coherency and discuss some of their properties.