Spectral analysis of reltivistic bunched beams

Particles in a storage ring are oscillating in the longitudinal and transverse dimensions, and therefore, the frequency domain is natural for analyzing many beam generated signals. Information ranging from oscillation frequencies to beam phase space distributions can be extracted from the spectral content of these signals. The spectrum of a single particle is like a Green`s function, and it is the key to understanding the spectrum produced by a beam. Three separate cases are consider in an order of increasing complexity: (1) constant revolution frequency, (2) Frequency Modulation introduced by synchrotron oscillations, and (3) Amplitude Modulation introduced by betatron oscillations.