Restorative Up-Sampling Discussion

Our Restorative Up-Sampling algorithm (RUP) is embodied in the MATLAB file {rup.m}, which also makes use of the sub-routines {forksub.m} and {spoon.m}. This, in the works, as yet, algorithm may be called 'alchemical' in that it aims to restore the 'gold' of lost high-frequency components in a severely under-sampled signal, while, at the same time, purifying the signal of the 'dross' of aliasing noise.

Our work could also be called 'alchemical' in the sense that it aims to accomplish what had previously been regarded by science as theoretically impossible.

At least as we aim to accomplish it, up-sampling is had by 'opening up' the IDFT (inverse discrete Fourier transform) of a signal at the halfway point, to arrive at what might be called one of the 'holographic levels.' Following this, we separate out the real and imaginary components, and then use a version of our Fourier Forcasting sub-routine {forksub.m} to 'forecast' an ifft-domain transform of the lost higher-frequency components.

To reiterate, this process is done for the real and imaginary components separately, as they should be orthogonal, and then the components are combined to build a new half-IDFT, which corresponds to double the sampling frequency (after being 'closed up' again.)

For results, we use File "C" as an input file to RUP. It consists of a severely under-sampled (4x) recording of U2's Bono singing "I can't believe the news today..." from the track "Sunday Bloody Sunday," and downloaded, again, from Wikipedia. This recording is so badly under-sampled that Bono's voice and accompanying instrumentals are barely audible. We pass this input file through {rup.m}, our restorative up-sampling algorithm with four sampling frequency doublings set to get back to the original sampling frequency of 22050Hz. If the algorithm and its subroutines are successful, the output File "D" should contain a faithful and restored rendering of the original to the listening ear.