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 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. It lays out the (N-point) time-domain data with an (N-1) mirror inverse of the same time data, as if the time-data were embodied in a simple frequency-domain rendering.

The algorithm then takes an inverse DFT of this [(2N-1)-point] representation and uses {forksub.m} to forecast it out to a [(2*{2N-1}-1)-point] rendering before undoing the inverse DFT operation with a FFT of the forecasted signal. Then, we take only the first [(2N-1)-points] to yield our doubled sampling frequency signal with high-frequency components restored and aliasing undone

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. The up-sampling routine assumes a time-delimited signal, with every value past the last point in the signal 'proper' being equated to zero. If this is not the case, variants of up-sampling as well as non-bandlimited forecasting may be had by recursive use of Fourier forecasting {fork.m} combined with restorative up-sampling {rup.m} used with signal length trimming operations.

For bare-bones results, we use File "C" (below) as an input file to RUP. It consists of a severely under-sampled (16x) 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 three 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" (again, below) should contain a faithful and restored rendering of the original to the listening ear.