We report a powerful interferometric measurement technique utilizing Spectral Interferometry using Minimum-phase Based Algorithms (SIMBA) to fully characterize the spectrum (either in reflection or transmission) of any fiber Bragg grating (FBG). This complex spectrum information is crucial to recover several important parameters of an FBG, such as its impulse response or refractive index profile. The core of our approach involves sending an unknown short laser pulse, e.g., ~1-50 ps of temporal width, into the FBG of interest, and using an optical spectrum analyzer (OSA) to record the spectrum of the interference between the reflected pulse from the grating and the time-delayed replicas of the original pulse. This measured spectrum, which yields the square of the Fourier transform (FT) magnitude of the pulse sequence's electric field envelope, is then processed to uniquely recover both the phase and amplitude of the FBG spectrum. The underlying principle of our approach is that by design of the experimental set-up, the pulse sequence sent to the OSA is close to a minimum phase function (MPF). Thus, it is possible to recover its FT phase spectrum using only the knowledge of its FT magnitude spectrum. This is an important result since by merely measuring an FT magnitude, with a rather simple set-up the full complex spectrum of the grating can be recovered. This technique has significant advantages over existing techniques, including a higher resolution, a better noise performance, and the ability to use longer laser pulses. It can also conveniently be used to simultaneously characterize more than one FBG, with a single FT magnitude measurement. We demonstrate the validity of our approach with numerical simulations.