## ABSTRACT

This chapter considers the fast Fourier transform (FFT) that obtains the spectral model of the data grid for insights on the data's derivative and integral properties with minimal assumptions and maximum computational efficiency and accuracy. The spectral model represents the data as the series summation of cosine and sine waves over a finite sequence of uniformly spaced discrete wavelengths (or equivalent frequencies called wavenumbers) that are fixed by the number and spacing of the gridded data. The FFT estimates the amplitudes of the cosine and sine waves as the cosine and sine transforms, which commonly are rendered into the power spectrum of summed squares of the cosine and sine amplitudes. The gridded data are modeled by inversely Fourier transforming the cosine and sine transforms together, or equivalently the power spectrum together with the phase spectrum of arctangents of the ratios of the negative sine to positive cosine amplitudes. The FFT dominates modern data analysis because of its computational advantages, even though numerical implementations must contend with reducing Gibbs’ oscillating approximation errors, and data gridding uncertainties that propagate as wavenumber leakage, aliasing, and resolution errors. However, window carpentry involving simplistic numerical modifications of the data and wavenumber components often minimizes these errors.