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I'm studying what I believe to be a system that produces a (noisy) quantal response (i.e. there should be a minimum increment of response). To do this, I've plotted a histogram of some ~6000 amplitude values from the system into small bins, and performed an FFT on the histogram, expecting that a common quantal value should appear as the lowest frequency "spike". However, my spectral response: https://www.dropbox.com/s/ivbsa660ins4lj7/WhatWouldMakeThis.jpg?dl=0 looks odd to me. What I see are what look like harmonics.

Why does this occur in the quantal release problem? Is this lowest frequency actually my quantal response, or is it a typical electrical response/artefact from the FFT of a signal with so many discontinuities? I thought that odd-numbered harmonics are the most powerful, but here they seem to be linearly declining. Why am I seeing other smaller spikes appear 1/3rd of the way through each of the "main" harmonic bands?

enter image description here

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  • $\begingroup$ Can you talk a little bit more about the application itself? If you FFT the histogram, you get back the harmonics that make up the histogram "waveform". Unless the histogram was periodic (in other words, multimodal at regular intervals), the FFT would not return anything "sensible". Furthermore, if the histogram is "fine" to the extent of having zeros in between the bars it will present a set of spikes to the FFT. Yes, what you see there looks like a square pulse. Can you post the histogram that was used as the input to this too? $\endgroup$ – A_A Jun 26 '18 at 11:04
  • $\begingroup$ Exactly, I'm looking for a common periodicity in the histogram. Ideally, quantized responses should show up in the histogram (larger numbers at 5, 10, 15, etc). However I'm expecting a bit of noise (some responses might be 7, 12, 17, etc). I can edit the original post to include the histogram. $\endgroup$ – Mark Jun 27 '18 at 0:20
  • $\begingroup$ Was this resolved? $\endgroup$ – A_A Jun 29 '18 at 15:39
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The key problem with this histogram is that it looks more like a collection of impulses rather than a "waveform" and this "shows" at the output of the Discrete Fourier Transform (DFT).

If you have to use the DFT then you need to preprocess the histogram so that it becomes smoother. You can do this with a moving average filter that will effectively integrate the impulses into smooth "humps". The downside of this is that you will have to tune the integrating period of the moving average filter to something that makes sense for your problem. And while doing this you will also notice that the waveform might change a bit from the expected depending on the time scale (or in the case of the histogram the resolution) that results from the use of a filter.

You can then pass the smoothed waveform to the FFT or, if your objective is to determine modes occuring at regular intervals, pass the smoothed histogram "waveform" to something like Kernel Density Estimation which seems more suited to your application.

Kernel Density Estimation will try to approximate the smoothed histogram output as a sum of gaussians with different widths and centers. Ideally, if these modes resemble gaussians, you might get one gaussian per mode you are trying to locate and then the regularity of the modes will show up in the centers of the fitted gaussians.

Hope this helps

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