Muon revolution frequency distribution from a partial-time Fourier transform of the g-2 signal in the muon g-2 experiment

A new method of precise measurement of the revolution frequency distribution, F(f), in the muon storage ring, and hence muon momentum p, energy E, and equilibrium radius R distributions, has been developed and used in analyzing data in the muon g-2 experiment at Brookhaven National Laboratory. The method is partly based on the Fourier transform of the observed electron decay signal, which is known in this experiment only after some time t_s after injection. It is shown that the standard Fourier transform would give a wrong frequency distribution even if the signal were known immediately after injection. Only the cosine Fourier transform with the properly determined initial time t₀ (different for different detectors placed along the orbit) gives the correct frequency distribution in such a case. As for a later starting time, t_s > t₀, a special procedure must be used to find t₀ and to compensate for the lack of information about the signal between t₀ and t_s. The new technique is highly accurate and radically different from that used by CERN in its muon g-2 experiment.