Spin noise spectroscopy (SNS) is a new kind of Faraday rotation technique, which does not need spin injection to generate polarized spin. This method uses a linearly polarized laser to detect the spontaneous spin fluctuation in a thermal equilibrium state. However, the signal of spontaneous spin fluctuation is so weak (~V) in the thermal equilibrium system that a big signal-noise ratio (SNR) is often demanded. Here, we report on the build-up and improvement of a spin noise spectrum measurement system. A home-made field-programmable gate array (FPGA) based data-acquisition card with real-time fast Fourier transform (DAC-FFT) is used to improve the SNR of the SNS measurement system. The reduction of intrinsic noise in the experimental system is discussed in detail. Both the dependence of background noise and the dependence of spin noise on the intensity of probe laser are analyzed. We find that the background noise is proportional to the intensity of the probe laser, while the spin noise signal shows square dependence on probe laser intensity. The spin noise indeed comes from the spontaneous spin fluctuation as experimentally confirmed via an acousto-optic modulator (AOM) inserted in the measurement system. The measurement performances of two FPGA based DAC-FFTs (the 8-bit FFTsDAC1 and the 12-bit FFTsDAC2, respectively) are compared. Several factors are found to affect the SNR of the system, including the measurement efficiency and the acquisition resolution. The FFTsDAC2 has longer single acquisition time and faster data transmission speed (with USB 3.0) than the FFTsDAC1, when the total measurement time is set to be the same, the effective measurement time realized in FFTsDAC2 is longer than in FFTsDAC1. With better measurement efficiency and sampling depth and longer single acquisition time, the FFTsDAC2 has a better SNR and finer frequency resolution with a much narrower full width at half maximum (FWHM) value. Moreover, the simulations of the measurement process show the effect of the single acquisition time on the FWHM of spin noise peak, further clarifying the reason why the spin noise spectrum measured by FFTsDAC2 is more accurate.