In this article, we discuss fundamentals of the spectrum analysis in beam diagnostics, where several important particle motions in a circular accelerator are considered. The properties of the Fourier transform are presented. Then the coasting and the bunched beam motion in both longitudinal and transverse are studied. The discussions are separated for the signal particle, multiple particle, and the Schottky noise cases. To demonstrate the interesting properties of the beam motion spectrum, time domain functions are generated, and then the associated spectra are calculated and plotted. In order to show the whole picture in a single plot, some data have been scaled, therefore they may not be realistic in an accelerator.