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I am trying to understand why Fast Fourier Transform (FFT) is used in the analysis of raw EEG channel data.

My understanding (at the 30,000 ft view) is that FFT decomposes linear differential equations with non-sinusoidal source terms (which are fairly difficult to solve) and breaks them down into component equations (with sinusoidal source terms) that are easy to solve. It then combines each of these component/partial solutions to solve the original equation. In laymen's terms, it's like taking a smoothie, and breaking it down into its recipe of ingredients. Doing so would allow us to study and analyze each ingredient as it relates to the final product (the smoothie).

But how does this relate to raw (non-FFT-decomposed) EEG data? My understanding of a single channel of EEG data is that it is essentially a measurement of voltage over time. What is to be gained by using FFT to break this voltage/time signal down into its constituent frequencies (alpha/beta/gamma/delta/mu waves), etc.? What additional information can each frequency tell us about the raw data?


Update:

Here is my updated understanding of how FFT relates to raw EEG data per @Mark's description below:

enter image description here

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  • $\begingroup$ I edited the question slightly; questions on signal processing of electrophysiological data are on topic. $\endgroup$ – AliceD Apr 6 '16 at 18:57
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    $\begingroup$ Somewhat simplistic answer so I'll post it as a comment, but Electrical Engineers Fourier Transform EVERYTHING. "Huh that's an interesting signal, what happens when I Fourier Transform it?" My guess is in the case of EEGs case some one did that and it showed patterns so they kept doing it. FYI, "it is essentially a measurement of voltage over time." Can describe literally everything a computer can do. Camera? voltage over time, Force sensor? voltage over time, Current? voltage over time (through a very precice inductor or resistor), Rotation? voltage over time $\endgroup$ – Sam Apr 6 '16 at 20:54
  • $\begingroup$ Thanks @Sam I think I get what you're saying! However, when you say "My guess is in the case of EEGs case some one did that and it showed patterns so they kept doing it", can you elaborate a bit more? What patterns might "they" have found? Perhaps reading a concrete example of what patterns/information emerge when EEG data is ran through FFT might help clue me in to its underlying value proposition. Thanks again! $\endgroup$ – smeeb Apr 7 '16 at 0:06
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    $\begingroup$ You may have success approaching this differently. Rather than trying to learn the specifics about how a FFT might be used in a given scenario analyzing EEG data, you might benefit from learning what a FFT does (to any signal), what the results look like, and why we do them, and only then worry about how it lines up with EEGs. FFT is actually only one of many transforms we use, each has its own flavor. FFT happens to be an easy one that shows a lot of data that is hidden in the time-domain solution, but wavelts are another approach with different tradeoffs. $\endgroup$ – Cort Ammon Apr 7 '16 at 7:12
  • $\begingroup$ Thanks @CortAmmon (+1) - I've already (sorta) done that. But the fact that no one here (even Christiaan) has been able to yield a clear-as-day explanation as to what the benefits of FFT are (that is, specifically what new information is exposed via FFT) tells me that Sam is on to something above. It tells me that people just use FFT because it presents some interesting new ways of looking at the same data. $\endgroup$ – smeeb Apr 7 '16 at 8:40
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Fast-Fourier Transform (FFT) transforms a signal from the time domain into the frequency domain. Basically, any time-dependent signal can be broken down in a collection of sinusoids. In this way, lengthy and noisy EEG recordings can be conveniently plotted in a frequency power-spectrum. By doing so, hidden features can become apparent. By adding all the sinusoids up after FFT, the original signal can be restored, so no information is lost.

A notable application of FFT in EEG is shown in Fig. 1, which shows an EEG in an awake person (top blue trace) and an EEG in a propofol-sedated person (bottom red trace). The traces are different, but exactly how different? Scientists like to quantify stuff.

Now look in Fig. 2, which shows the same data but filtered in the delta band (low-pass filtered EEG with a cut-off frequency of 1.5 Hz, left panel). Here it already becomes more apparent what's going on, but what exactly is the difference between the two traces? That difference becomes readily apparent in the frequency domain by using FFT (Fig. 2, right panel); The frequency spectrum has a peak at 0.2 Hz in both traces, but that peak is about twice as big in the anesthetized state than in the normal state. In other words, the anesthetized brain reveals more low-frequency activity.

raw EEG
Fig. 1. Raw EEG of an awake person (blue) and propofol-anesthetized person (red). source: Wang et al (2014).

enter image description here
Fig. 2. Filtered EEGs (<1.5 Hz) of an awake person (blue) and propofol-anesthetized person (red) (left panel) and corresponding FFT spectra (right). source: Wang et al (2014).

This is reminiscent of the drowsiness encountered in slow-wave sleep; which is yet another example of why FFT is useful; various stages of sleep are markedly different in their EEG. For example, early stages of sleep are characterized by slow-wave EEG, while REM sleep is characterized by high-frequency EEG activity. By using FFT, these differences in frequency content can be captured in simple, quantifiable data.

Another widely applied FFT-based application is filtering in the frequency domain. Look at the sleep EEG in Fig. 3. By splitting the raw signal up in frequency bands, the noise can be removed (high-frequency components), but even better, the k-complex (a characteristic hallmark of normal sleep) can be beautifully isolated from a messy EEG signal (bottom trace, 12-15 Hz).

k complex
Fig. 3. EEG FFT-based filtering. source: Neurology

Reference
- Wang et al., Front Syst Neurosci (2014); 00215

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    $\begingroup$ Thanks @Christiaan (+1) - now I think we're starting to get somewhere! A few followup questions if you don't mind: (1) my understanding of the processing flow was: Capture Raw EEG Data -> Perform FFT -> Analyze Component Frequencies (Alpha, Delta, etc.). But it sounds like you are saying that the typical "flow" is: Capture Raw EEG Data -> Filter by a Particular Frequency -> Perform FFT on that Frequency -> Analyze the Frequency. Can your confirm this or clarify it for me? $\endgroup$ – smeeb Apr 6 '16 at 19:47
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    $\begingroup$ And then, (2) OK great so now we see that in the sedated patient, the Theta Frequency wave is - on average - twice as big as the awake/conscious patient. So it sounds to me like the whole point of using FFT on raw EEG data is to just give a researcher more information than what is contained in the original waveform. Can you confirm this for me as well? Thanks again for such an excellent answer! $\endgroup$ – smeeb Apr 6 '16 at 19:49
  • $\begingroup$ comment 1: It can be either way - option 2 is basically FFT-based filtering followed by frequency analysis. useful when there is noise in your data and you wish to clean it up first. There is no typical flow, whatever milks your Jersey. $\endgroup$ – AliceD Apr 6 '16 at 19:50
  • $\begingroup$ comment 2: Thanks for your comments! I agree with that statement. Often, raw data needs signal processing (tag added) to be able to deduce useful information from it. $\endgroup$ – AliceD Apr 6 '16 at 19:51
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Any periodic waveform can be produced by adding up a series of sin waves of the appropriate frequency and amplitude. The FFT looks at a complex waveform and calculates those frequencies and amplitudes. The result is a new curve which plots amplitude vs frequency. Thus, it transforms the signal from the time domain into the frequency domain.

I don't have any knowledge of EEG signals, but have worked with FFTs. If you wanted to know what frequencies make the largest contribution to the original waveform, the FFT will provide that information.

Here is a paper I found which uses FFT analysis of EEG waveforms to look at the relative contributions of different frequencies. It includes the following table which indicates that "Delta" waves are in the 0.5 to 4 Hz range, Theta waves are in the 4 - 8 Hz range, etc. The FFT analysis quickly breaks the overall waveform down into its constituent frequencies to identify which of these ranges contribute the most or least.

enter image description here

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  • $\begingroup$ Thanks for the insightful answer @Mark (+1) - however a few followup questions for you, if you don't mind: (1) once the FFT is performed on the original waveform, is each component "rhythm" (alpha, beta, gamma, etc.) now represented by it's own graph of amplitude over frequency? Meaning, the original waveform was a single graph of voltage/time: do we now have five graphs of amplitude/frequency, 1 for each rhythm? And (2) this still seems unnecessary to me! Why would I care if a beta rhythm contributed more or less to a subject thinking about birds? In other words: why do this?!? $\endgroup$ – smeeb Apr 6 '16 at 17:02
  • $\begingroup$ Obviously, that second question above is quasi-rhetorical, as I suspect that there is a very good reason for decomposing brain waves with FFT, I'm still just not seeing it! Thanks again! $\endgroup$ – smeeb Apr 6 '16 at 17:02
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    $\begingroup$ @smeeb - (1) The FFT creates a single graph of amplitude vs frequency. To actually generate the original time series from the FFT result, you would have to create a sin wave of the right amplitude for each frequency in the answer, and then add them all together. But the FFT result is just a single graph. $\endgroup$ – Mark Apr 6 '16 at 17:32
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    $\begingroup$ @smeeb - (2) Sorry, I can't help there as far as the biology goes. FFTs are used in a lot of applications where the particular frequencies are important, including mass spectrometry where the specific frequencies allow gas components to be identified; machine analysis where the individual frequencies can be used to identify defects in couplings or bearings; and audio processing where it can be used to remove noise, or analyze the harmonics of a violin. Often, there is a great deal of information within the frequency spectrum $\endgroup$ – Mark Apr 6 '16 at 17:32
  • $\begingroup$ Thanks again @Mark, please see my updated question. I have provided an illustration of my current understanding, given your excellent answers here (thanks!). First off: please let me know if something about my illustration seems wrong to you! But if my understanding/illustration seems to be correct, then my last question for you is: can we then assume (given your data table) that the x-axis (frequency) will always be the same for any given FFT-transformed EEG data - with ticks at the 4Hz, 8Hz, 13Hz, 30Hz units... $\endgroup$ – smeeb Apr 6 '16 at 18:07
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EEG is simply measuring voltage fluctuations over the scalp caused by electric dipole(s) within the brain tissue. Communication between neurons can happen at different spatial and temporal scales. At each scale, this communication results in a distinct pattern of current flow. Because EEG measures voltage fluctuations over a relatively large patch of the brain, it reflects current flow created by communication at various scales between millions of neurons (each scale presumably associated with a different frequency of oscillations). Current research is attempting to understand what neural processes in the brain tissue result in the EEG. There is still not a complete understanding of what each frequency band represents, but studying similar types of oscillations within brain tissue (called local field potentials) have provided invaluable insights (for instance, small range communications are associated with higher frequency bands). How the oscillations recorded inside the brain tissue are related to the EEG is an active area of research.

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Some of the information contained in this post requires additional references. Please edit to add citations to reliable sources that support the assertions made here. Unsourced material may be disputed or deleted.

  • $\begingroup$ provide references to sustain your claims. $\endgroup$ – have fun Apr 27 '18 at 7:46

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