An 115-180 nm far ultraviolet limb imaging spectrograph prototype was developed for the detection of the ionosphere. For the particularity of the wavelength band in far ultraviolet, the signals accepted by the instrument are very weak. So the sensitivity and signal-noise-ratio (SNR) are two important performance parameters for the spectrograph. In the present paper, based on the signal statistical detection theory and the threshold detection theory, a method for calculation of the sensitivity and SNR has been proposed. Firstly, the probabilities of the accepted signals and system noises were analyzed. Secondly, the mathematical expressions of sensitivity and SNR of the instrument were deduced by introducing the detection probability factor and the false alarm rate factor. Based on the calculation of the sensitivity and the intensity of the particles radiation wavelengths, it was found that the sensitivity can meet the need of the detection. Finally, the SNR of the instrument were analyzed by using the method, and the coherent experiment was built to prove the results. The error of theoretic calculation and experimental results can be accepted. It indicates that the SNR analysis method is feasible.