The elementary principle of a fiber Mach-Zehnder interferometer and the structure of a Fiber Fourier Transform Spectroscopy(FFTS) were introduced. Then, a novel demodulation scheme for the fiber Bragg grating sensor based on the fiber Mach-Zehnder interferometer and the FFTS was designed to achieve highly stable and accurate wavelength demodulation. In demodulation,the light emitted from a broadband light source was through the fiber coupler into a fiber Bragg grating,and the reflected light of fiber Bragg grating came back from the fiber coupler into the FFTS. The optical path difference of FFTS is up to 10 cm and the maximum spectral resolution is 0.05 cm-1 when the unilateral Fourier transform algorithm is used, which means the wavelength resolution is 0.012 nm in near infrared scope of 1 550 nm. The strain characteristics and temperature characteristics of a fiber Bragg grating were measured by the FFTS, respectively. Experimental results indicate that the strain sensitivity is 0.833 pm/με and the temperature sensitivity is 19.78 pm/℃. Obtained data show that the FFTS system has advantages of high spectral resolution and wide measuring ranges and it can satisfy the wavelength demodulation requirements of fiber Bragg grating sensors.