Analytical Expression for the Spectrum of a Bunched Beam with Reguéar Trains and Sub-Trains

We present a relatively simple analytical expression for the beam spectrum of a bunched beam formed by a sequence of bunch trains composed of regular sub-trains themselves. The location of the spectral lines and their relative importance can be deduced immediately. the entire spectrum on a high-resolution spectrum analyser can be calculated easily avoiding to sum up long series, also telling the absolute RF current in each line. Furthermore, the same expression predicts the excitation of a cavity in the beam, including the case of low Q modes covering many spectral lines or overlapping modes. Using the basic idea, this formalism can be applied with success to any beam structure with a minimum of regularity. We have explicitely applied it to the LHC beam.