Analysis of DSP-based longitudinal feedback system: trials at SPEAR and ALS

Recently a single-channel prototype of the proposed PEP-II longitudinal feedback system was successfully demonstrated at SPEAR and ALS on single-bunch beams. The phase oscillations are detected via a wide-band pick up. The feedback signal is then computed using a digital signal processor (DSP) and applied to the beam by phase modulating the RF. We analyze results in the frequency- and the time-domain and show how the closed-loop transfer functions can be obtained rigorously by proper modeling of the various components of this hybrid continuous/digital system. The technique of downsampling was used in the experiments to reduce the number of computations and allowed the use of the same digital hardware on both machines