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Higher order modes self-excitation can seriously degrade klystron performance and cause pulse shortening in the high-gain relativistic klystron amplifier. In order to suppress higher order modes self-oscillation in the device, some special measures are taken to increase the threshold current of such modes. An output power of 1.02 GW with an efficiency of 24.3% and a gain of 53.7 dB is obtained in the simulation with an input power of 4.3 kW and a frequency of 2.88 GHz. The experiment is performed on the linear transformer driver. Under conditions of an RF power of 30 kW, the peak output power has achieved 1.2 GW, with an efficiency of 28.6% and a gain of 46 dB; the corresponding pulse length is 120 ns.