Betatron Tune Determination: Interpolation Formulas

In order to obtain accurate estimations of the betatron tune, interpolation formulas have been derived that give estimates that approaches the real tune as $1/N^2$ and $1/N^4$ for signals of constant amplitude. In this document interpolation formulas for signals with exponential decaying amplitudes are derived, and its errors are analyzed as a function of the input signal size and the decay constant. We obtain the same scaling law of $1/N^2$ for the case of constant amplitude, and an improvement over the previous methods in the case of decaying amplitude. Lower boundaries for the errors were observed, and methods for surpass this were analyzed.