Frequency analysis using the discrete Fourier transform

The analysis of arbitrary time samples of signals of interest in terms of a Fourier series in effect forces the signal to be periodic with a fundamental period equal to the sample length. This causes sinusoidal components in the signal that are not harmonic in the sample interval to appear to be discontinuous at the ends of the periods; each such component leads to a complete set of the harmonic terms determined by the analysis. The determination of the inharmonic sinusoidal components can be improved by taking suitable combinations of the coefficients determined by the analysis, or by a weighting of the input data to remove the discontinuity. It is shown that improvements of the convergence are accompanied by a corresponding broadening of the principal response.