Hello!
I'm working with a signal that is periodic but also has a distinct behaviour that identifies the passage of time in a specific direction. As an analogy, consider if I'm making a video where a car drives from left to right across a figure window and then reappears on the left side, as though the boundaries themselves were periodic. The car's location is periodic, but the car must be moving from left to right if time is progressing. If time is going backward, then the car will appear to move backward, from right to left and reappearing on the right side.
The issue I am having is that somewhere during the following process:
- Fourier transforming the signal
- Turning that result into a single-sided spectrum
- Turning that single-sided spectrum back into the double-sided spectrum
- Inverse-Fourier transforming that double-sided spectrum back into [what should be] the original signal
I end up "reversing" the passage of time; if I start with my car going from left to right, I end up with my car going from right to left.
Now, I've seen from experimentation that I can seemingly solve this problem by introducing an additional complex conjugate operation between Steps 3 and 4; however, I don't understand why this works and why I'm having the issue in the first place. As far as I can tell, I am using the standard MATLAB procedures for working with time signals and single-sided spectra, as described here and here, among other places.
Here are the functions I'm using:
function [freqs, spectra] = getSingleSidedSpectra(signal,time)
Fs = 1/mean(diff(time));
L = length(signal);
freqs = Fs*(0:(L/2))/L;
rawfft = fft(signal);
P2 = rawfft/L;
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
spectra = P1;
function [time,signal] = getSingleSidedTimeSeries(spectra,freqs)
T = 1/mean(diff(freqs));
testsize1 = size(spectra);
if testsize1(2) == 1
spectra = spectra';
end
spec_bothsides = [spectra(1) spectra(2:end-1)/2 spectra(end) fliplr(conj(spectra(2:end-1)))/2];
signal = ifft(spec_bothsides * length(spec_bothsides));
time = (0:length(signal)-1)*T;
Essentially, my process is as follows:
[freqs,spectrum] = getSingleSidedSpectra(original_signal,original_signal_times)
[~,output_signal] = getSingleSidedTimeSeries(spectrum,freqs);
I would expect the original and output signals to be the same, but no; some kind of time reversal has occured. In the analogy, the orginal car movement from left to right has somehow become backward travel from right to left.
"original_signal" and "original_signal_times" are both 1D arrays.
The "time" output from the second function doesn't actually have any utility in this context; I just kept it for completeness.
As mentioned before, if I uncomment the line in the second function where I take an additional complex conjugate, the problem seems to be fixed. However, I don't understand why that would be the case and I'm unwilling to just accept that line without understanding its physical necessity.
I will note that I've tried to take care in keeping the spectra values associated with f = 0 and f = Nyquist frequency (= Fs/2) are included in the single-sided spectra. I've seen different people keep or remove those components in their definitions. There is actually a discrepancy between the two MATLAB examples that I added links to above - in the fft page, the single-sided spectrum includes the Nyquist frequency, but the single-sided spectrum in the ifft page does not.
I will also note that I've tried removing the transpose in the second function (within the if testsize1(2) == 1 statement) and changing the bracket-based horizontal concatenation to the vertcat function. The result of this is now my final value for "output_signal" remains a complex array, as opposed to having the complexity removed from the ifft() function. I've tried specifying the dimension over which the ifft should be calculated, but the result is unchanged. Given this is the only part of the code that I can clearly identify as being the different from the specified MATLAB procedure and the fact that horizcat vs vertcat seems to have an effect on the complexity of the ifft result regardless of whether or not I specify the ifft dimensionality, this is my best guess for the source of the problem. However, even if it is I don't understand why this would be an issue.
Any help would be appreciated. Please let me know if you need any further information from me. If necessary, I can try directly linking to a movie of the data I'm attempting to plot if the time dependency I'm trying to convey isn't clear.
Edit: typo
