Algorithms for a Precise Determination of the Betatron Tune

In circular accelerators the precise knowledge of the betatron tune is of paramount importance both for routine operation and for theoretical investigations. The tune is measured by sampling the transverse position of the beam for N turns and by performing the FFT of the stored data. One can also evaluate it by computing the Average Phase Advance (APA) over N turns. These approaches have an intrinsic error proportional to 1/N. However, there are special cases where either a better precision or a faster measurement is desired. More efficient algorithms can be used, as those suggested by E.Asseo [1] and recently by J. Laskar [2]. They provide tune estimates by far more precise than those of a plain FFT, as discussed in Ref. [3]. Another important isssue is the effect of the finite resolution of the instrumentation used to measure the beam position. This introduces a noise and the frequency response of the beam is modified [4,5} thus reducing the precision by which the tune is determined. In Section 2 we recall the methods based on the APA approach. In Section 3 we introduce the FFT based methods and the estimate of the algorithms error as a function of N. In Section 4 we discuss the effect of noise.