In pulse nuclear magnetic resonance (NMR) spectroscopy, data are obtained by perturbing the nucleus from its equilibrium position and acquiring the transient response. Fourier transformation is the preferred mode used in data processing of the signals due to its ability to compute the NMR spectrum rapidly. To obtain good signal-to-noise ratio, it is common practice to average many transients. To obtain good resolution, lengthier acquisition times are favored. For insensitive nuclei, where thousands of collected transients are necessary, this is a time-consuming procedure; especially if the nuclear relaxation time constant is in the order of seconds or minutes. A faster acquisition method is proposed. Using a modified NMR pulse sequence, the proposed method acquires signals more rapidly than by conventional acquisition methods; however, the signals are truncated. In processing truncated data; the shortcomings of the Fourier transform must be overcome by alternative spectral estimation methods. An alternative processing method - linear prediction (LP) - is used to reconstruct the spectrum from the incomplete time-domain magnetic resonance data. The LP method's application to truncated, fast acquisition of data is discussed in detail. This combination of methods is a novel way of acquiring and processing NMR spectroscopic data.