We believe we may have validated the reversibility of time highlighted by David Bohm's famous ink drop in glycerin demonstration. It is commonly understood that, in the domains of continuous time and the Fourier transform, time reversal corresponds with frequency reversal. When we are working with a real, finite length sample of discrete time data, things are a little more complicated. For the discrete Fourier transform, both the frequency profile and the time signal are periodic over N, such that reversing the frequency simply reverses this periodic discrete time signal, and doesn't unlock the actual past that occurred before the data set began at (n=0).

However, the generic relation between frequency reversal and time reversal may still provide a clue as to how to proceed to achieve this unlocking of the Implicate Order and the hidden past. For an odd length data series, the frequency components of the DFT are conjugate symmetrical about half of the sampling frequency (fs/2). The second half of the frequencies, in a limited sense are redundant and contain no new information per Se. While we didn't know where we were headed when we started, we suspected that dispensing with this mathematical redundancy by redefining our "window of concern" for our DFT to include only the dc component plus the first half of the frequency components proper might provide a starting point. This makes the new data set values complex, but we fully expected that at the end of our musings, we would reinstate the real, by reflecting the conjugate of whatever DFT we came up with about (fs/2), or half the original sampling frequency.

To be honest, our starting point wound up being in the middle of our final analysis, with forward and backward steps being added in the kind of "guess and check" procedure that I judged with some disdain in college. But, oh well, confessions are confessions, and judgement is the root of all evil. ;) The mid-point in the analysis is a frequency reversal of only the frequencies that we have selected for our near half-length DFT. Later, it was found that a preliminary time reversal was required in the analysis, an analysis which is of itself a bit holographic in reasoning. Perhaps not surprising, since it has to do with unlocking the past of the holomovement.

After these two steps, we have an ([(N-1)/2]+1)-point DFT data set, along with the eventual requirement of lengthening it with it's mirrored conjugate frequencies, to "reinstate the real." For heuristic purposes, if we view our scaling of time and frequency as being somewhat arbitrary, we can envision extending the frequency domain of our signal larger and larger, while, of necessity the time domain, contracts. As the modified top frequency component expands toward "infinity", the sampling time (ts) contracts more and more to become an infinitesimal (dt), or (0+). We can then imagine that "reinstating the real" adds an additional (N-1) even higher frequencies, and flips this infinitesimal sampling interval about the point of no extension (0), to give a negative (-dt/2) sampling interval, and, potentially a view into the hidden past. A final "flipping" of the time data was done to give these points from (N-1) to (0), oriented in a forward traveling direction.

One of my concerns is that working without a formal human partner in this is somewhat dangerous for my psyche, so if willing, I will choose the reader as my partner to check my sound files, crunch the final numbers, calculate percentage RMS errors, and overlook or correct my mistakes. I have only been able to obliquely look at results to facilitate the work in progress, but a final look into the narcissistic mirror of success or failure without another human consciousness holding my hand to assure me that none of it really matters in the eyes of God is more than I can currently stomach. God cares about us, not our egos: either their accomplishments or failings. So I haven't looked. Efforts geared toward communication, I feel, are safe, while fascination with my own reflection beyond a certain point is not.

Thank you,

...Deserdi Chapas

Given an N-point (N length odd) discrete signal indexed from (n=0) to (n=[N-1]), the following MATLAB code may generate, within a measure of uncertainty, the signal points for (n=-[N-1]) to (n=0). The validity of the reversal can be checked by running this second set of points through the MATLAB forward forecasting routine {postsr.m}, and comparing the forecast derived from the negatively indexed points with the original data set.

function y = negindex(x)

APr=(x);

NPrs=size(APr);

NPr=NPrs(2:2);

NNPr=(((NPr-1)/2)+1);

zc=complex(0.0,0.0);

APrF=flip(APr);

FPP=fft(APrF);

FPPr=[zc,(flip(FPP(2:NNPr))),(conj(FPP(2:NNPr)))];

APrn=ifft(FPPr);

y=flip(APrn);