Solar imaging spectral detection is a main method for diagnosing the magnetic field and thermodynamic parameters of the solar atmosphere. The Fourier Transform Solar Spectrometer (FTSS) has the advantage of wide band and is the best choice for the current mid-infrared high-resolution solar spectrum detection. FTSS acquires spectra through the inversion of the observed interferogram from the target radiation, and the sampling interval of the optical path difference determines the wavelength range of the inversion spectrum. Therefore, from the requirement of tunable optical path difference interval for FTSS's broadband spectral observations, we designed a sampling system with fully digital frequency division and multiplication scheme based on FPGA (Field Programmable Gate Array) technology. The distributed remainder compensation method is adopted to effectively solve the problem of error accumulation in the process of frequency multiplying of the reference laser signal, and the non-uniformity of the output sampling signal is also reduced. After simulations and experimental measurements, the system frequency error δ is less than 0.04%, in the frequency range of 200Hz--50kHz, which meets the FTSS's broadband spectral observation requirements of 300nm--25μm. This study has laid a solid technical foundation for the development of FTSS in visible and infrared bands.