We propose and demonstrate an ultrahigh-speed optical frequency domain reflectometry (OFDR) system based on optical frequency-to-time conversion by pulse time stretching with a linearly chirped fiber Bragg grating (LCFG). This method will be referred to as OFDR based on real-time Fourier transformation (OFDR-RTFT). In this approach, the frequency domain interference pattern, from which the desired axial depth profile is reconstructed, can be captured directly in the time-domain over the duration of a single stretched pulse, which translates into unprecedented axial line acquisition rates (as high as the input pulse repetition rate). We provide here a comprehensive, rigorous mathematical analysis of this new OFDR approach. In particular, we derive the main design equations of an OFDR-RTFT system in terms of its key performance parameters. Our analysis reveals the detrimental influence of nonlinear phase variations in the input optical pulse (including higher-order dispersion terms and group delay ripples introduced by the LCFG stretcher) on the system performance, e.g. achievable resolution. A simple and powerful method based on Hilbert transformation is successfully demonstrated to compensate for these detrimental phase distortions. We show that besides its potential to provide ultrahigh acquisition speeds (in the MHz range), LCFG-based OFDR-RTFT also offers the potential for performance advantages in terms of axial resolution, depth range and sensitivity. All these features make this approach particularly attractive for imaging applications based on optical coherence tomography (OCT). In our experiments, single-reflection depth profiles with nearly transform-limited approximately 92.8 mum (average) axial resolutions over a remarkable 18 mm depth range have been obtained from OFDR-RTFT interferograms, each one measured over a time window of approximately 50 ns at 20 MHz repetition rate. Improved sensitivities up to -61 dB have been achieved without using any balanced detection scheme.