Comparison of Fourier Signal and Error Analysis Techniques for Identifying the Self-Modulation Frequency of a Proton Bunch

The AWAKE experiment uses an ultra-high energy proton beam to create large amplitude wakefields for accelerating electrons in plasma. The proton beam is much longer than the plasma wavelength, and must be formed into small, sub- wavelength sized beamlets before it can effectively drive the wake. These beamlets are referred to as micro-bunches and are formed by the plasma self-modulation instability. An im- portant aspect of AWAKE is to measure the depth, frequency, and stability of the modulation, as this provides critical in- formation for establishing the presence of a high-amplitude wakefield driven by a self-modulation proton bunch. This paper discusses Fourier Analysis techniques for measuring the modulation frequency and compares error estimation techniques that work for both small and large datasets.