In this article the fundamentals of spectrum analysis in beam diagnostics is discussed, where several representative particle motions in a circular accelerator are considered. The general properties of the Fourier transform are introduced first, then the coasting and the bunched beam motions in longitudinal and transverse planes are presented for single particle, multiple particle, and Schottky noise cases, respectively. To demonstrate the properties of the beam motion spectrum, time domain function are generated, and then the associated spectra are calculated and plotted. © 1994 American Institute of Physics