A high impedance relativistic klystron amplifier with a cold solid cathode is designed and investigated by 2.5-D particle-in-cell simulations. The model is composed of a diode with a shielding ring, five pill-box cavities and a cone collector. In order to simplify and minimize the cathode structure, a cold cathode is designed and adopted in a high-impedance relativistic klystron amplifier with rather high conversion efficiency. A shielding ring is introduced to achieve the desired high impedance and reduces the surface electric field to avoid unexpected explosive emissions and ensures the emission uniformity. In the beam-wave interaction region, with optimization of the position of the last two cavities and the magnetic field, the performance of the klystron is validated. It reveals that microwaves with a power of 81 MW are generated at a frequency of 11.424GHz when the beam voltage and current are 525kV and 328 A respectively, under a guiding magnetic field of 0.35 T. The corresponding power conversion efficiency is as high as 47%, and the gain reaches 49 dB when the net injection power is 1 kW.