It is well known that Fourier-based Near-field Acoustic Holography fails to produce good source reconstructions when the aperture size of the microphone array is smaller than the source size. In this paper this problem is overcome by pre-conditioning the spatial hologram data using Linear Predictive Border Padding (LPBP) before it is Fourier-transformed to the wave-number domain. It is shown that LPBP allows for very small aperture sizes with a good reconstruction accuracy. An exhaustive analysis of LPBP is presented based on numerical experiments and measured data. The numerical experiments are performed on two different source types: modal patterns and point sources. These two types of sources represent the two limit situations that one can find in practice: modal patterns have a tonal spectrum in the spatial wave-number domain and are relatively easy to reconstruct accurately, while point sources have a broad-band wave-number spectrum which makes them very challenging to reconstruct. In order to illustrate the accuracy of the method in practice, results of measurements on a hard disk drive are presented as well. For a given distance to the source, the position and size of the hologram plane apertures is varied and the reconstructed source information is compared to the original source data. The reconstructed sources are compared both qualitatively and quantitatively. The results show that LPBP is an efficient and accurate extrapolation method, which leads to accurate reconstructions even for very small aperture sizes. r order value suffices for source identification purposes.