A high performance fast-Fourier-transform (FFT) spectrum analyzer, which is developed for measure spin noise spectrums, is presented in this paper. The analyzer is implemented with a field-programmable-gate-arrays (FPGA) chip for data and command management. An analog-to-digital-convertor chip is integrated for analog signal acquisition. In order to meet the various requirements of measuring different types of spin noise spectrums, multiple operating modes are designed and realized using the reprogrammable FPGA logic resources. The FFT function is fully managed by the programmable resource inside the FPGA chip. A 1 GSa/s sampling rate and a 100 percent data coverage ratio with non-dead-time are obtained. 30534 FFT spectrums can be acquired per second, and the spectrums can be on-board accumulated and averaged. Digital filters, multi-stage reconfigurable data reconstruction modules, and frequency down conversion modules are also implemented in the FPGA to provide flexible real-time data processing capacity, thus the noise floor and signals aliasing can be suppressed effectively. An efficiency comparison between the FPGA-based FFT spectrum analyzer and the software-based FFT is demonstrated, and the high performance FFT spectrum analyzer has a significant advantage in obtaining high resolution spin noise spectrums with enhanced efficiency.