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 ts 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, ts>t0, a special procedure must be used to find t0 and to compensate for the lack of information about the signal between t0 and ts. The new technique is highly accurate and radically different from that used by CERN in its muon g-2 experiment.