Downsampled bunch-by-bunch feedback for PEP II

The PEP II B Factory requires a feedback system to damp out longitudinal synchrotron oscillations. A time-domain bunch-by-bunch feedback system has been proposed in which each bunch is treated as an oscillator being driven by disturbances from the other bunches. The phase is detected, filtered, and the feedback correction signal is applied by the kicker. Since we are damping energy oscillations using measurements of phase, the required feedback signal must be proportional to the amplitude of the phase oscillations but phase shifted by 90 degrees. This signal must be calculated for each of the 1658 bunches, in parallel. In the original proposal, it was estimated that a farm of approximately 480 digital signal processors (DIPS) would be required to implement the feedback system. However, using the technique of downsampling, this number can be reduced to about 50 DIPS. In what follows, we will briefly explain the basic idea of downsampling and its implementation.