  The J-PARC Main Ring (MR) is a high intensity proton synchrotron, which
accelerates protons from 3 GeV to 30 GeV. The MR delivers $2.6\times10^{14}$
protons per pulse, which corresponds to the beam power of 500 kW, to the
neutrino experiment as of May 2018, and the studies to reach higher beam
intensities are in progress. During studies, we observed the longitudinal
dipole coupled-bunch instabilities in the MR for the beam power beyond 470 kW.
To mitigate them for higher beam intensities, we have developed a longitudinal
mode-by-mode feedback system. The feedback system consists of a wall current
monitor, a FPGA-based feedback processor, RF power amplifiers, and a RF cavity
as a longitudinal kicker. In the feedback processor, we utilize the single
sideband filtering technique to detect the oscillation components of the
individual coupled-bunch mode in the beam signal. The frequency response of the
filters in the feedback processor matched well with the simulation. The
oscillation amplitude of the coupled bunch oscillation measured by the system
agreed with the oscilloscope analysis.
