Measurement of multi-bunch transfer functions using time-domain data and Fourier analysis

Multi‐bunch transfer functions are principal ingredients in understanding both the behavior of high‐current storage rings as well as control of their instabilities. The measurement of transfer functions on a bunch‐by‐bunch basis is particularly important in the design of the active feedback systems. Traditional methods of network analysis that work well in the single bunch case become difficult to implement for many bunches. We have developed a method for obtaining empirical estimates of the multi‐bunch longitudinal transfer functions from the time‐domain measurements of the bunches’ phase oscillations. This method involves recording the response of the bunch of interest to a white‐noise excitation. The transfer function can then be computed as the ratio of the fast Fourier transforms (FFTs) of the response and excitation sequences, averaged over several excitations. The calculation is performed off‐line on bunch‐phase data and is well‐suited to the multi‐bunch case. A description of this method and an analysis of its performance is presented with results obtained using the longitudinal quick prototype feedback system developed at SLAC.