<p>As <a href="https://electronics.stackexchange.com/a/135383/162027">this answer</a> explains, the Fourier transform of the autocorrelation of $x$ is the same as the magnitude squared of the Fourier transform of $x$. The autocorrelation of x can be expressed as $x(t)*x(-t)$, where * means <a href="https://en.wikipedia.org/wiki/Convolution" rel="nofollow noreferrer">convolution</a> and the Fourier transform of the autocorrelation is then $X(w)X^*(w)=|X(w)^2|$, which of course is the Fourier transform of the original signal $x(t)$.</p>

<p>As for references, see <a href="https://cogsci.stackexchange.com/questions/8385/reference-request-in-circuit-and-signals-for-computational-neuroscience/8395#8395">this answer</a>.</p>

CC BY-SA 3.0
2017-09-05T15:36:04.973Z
2017-09-05T15:36:04.973Z
StrongBad
<p>I want to characterize a binary time-series signal <em>x</em> (derived from neuron action potential data) in the frequency domain. </p>

<p>Should I use the FFT of the original signal <em>x</em>, or the power spectrum (FFT of the autocorrelation of <em>x</em>, which is defined as $k(n)=E[x(t)\cdot x(t-n)]$ )?</p>

<p>Also, does anyone know any literature which explicitly talks about this, or at least took sides?</p>

CC BY-SA 3.0
2014-05-28T21:43:07.957Z
What is your ultimate objective with "characterizing" the data?  Definitely read [Spikes](http://www.amazon.com/Spikes-Exploring-Neural-Computational-Neuroscience/dp/0262681080/) when you have a chance.  The short answer from there is that it depends on what you are looking to get out of the signal.  If you're looking to drive a device or something of that nature, the FFT is going to be computationally burdensome, and may not give you sufficient information.
Chuck Sherrington
CC BY-SA 3.0
2014-05-28T21:43:31.023Z
As an aside, for papers to look at, [George Gertstein](http://www.med.upenn.edu/mulab/) is the guru of the analysis of spike trains.
Chuck Sherrington
CC BY-SA 3.0
2014-05-29T19:45:18.833Z
@ChuckSherrington I just want to detect oscillatory activity in the spike train. Computational burden isnt an issue. Why would the FFT give me insufficient/different info than the autocorrelation?
DankMasterDan
CC BY-SA 3.0
2014-05-29T20:02:43.673Z
I'm using "information" as a loaded term.  I was referring to both measures that you are talking about, sorry if that wasn't clear.  Neither is going to tell you anything meaningful.  If you're locking to a specific event (sensory input or specific motor output - e.g., an EMG of a muscle) the frequency variation will tell you something, but a "free form" recording really isn't.  You'd be better off examining local field potentials (most extracellular recording systems can separate these out in the spike sorting processing) if you are trying to tease out frequency changes.
Chuck Sherrington
CC BY-SA 3.0
2014-05-29T20:03:07.023Z
Otherwise, this is really just an exercise in signal processing, to be frank.
Chuck Sherrington
CC BY-SA 3.0
2014-05-29T21:00:53.783Z
(not trying to be an overbearing curmudgeon here, just offering some constructive criticism. I'm happy to see a methods question on here!)
Chuck Sherrington
CC BY-SA 3.0
2014-05-29T21:53:36.900Z
thnx. in our work we concatonate spike train activity rather then add it up as done in perievent histograms. the reason we do this is because we do spike-train input/output modeling of these concatenated signals.
DankMasterDan
CC BY-SA 3.0
2014-05-29T22:14:38.233Z
If systems identification is what you are after ("input/output modeling"), you'll definitely need some heavier hitting analyses -- check out Bob Kearney's work http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471274569.html
Chuck Sherrington
CC BY-SA 3.0
2017-10-09T01:38:44.977Z
I suggest Chronux package if you use MATLAB: http://chronux.org/
Memming
<theoretical-neuroscience><computational-modeling>
For binary (spike train) signals, take FFT of signal or autocorrelation of signal?
DankMasterDan
# For binary (spike train) signals, take FFT of signal or autocorrelation of signal?

(Asked by: username_0 on 2014-05-28)

I want to characterize a binary time-series signal *x* (derived from neuron action potential data) in the frequency domain.

Should I use the FFT of the original signal *x*, or the power spectrum (FFT of the autocorrelation of *x*, which is defined as $k(n)=E[x(t)\cdot x(t-n)]$ )?

Also, does anyone know any literature which explicitly talks about this, or at least took sides?

**Tags:** theoretical-neuroscience, computational-modeling

> **Comments:**
> **username_1**: What is your ultimate objective with "characterizing" the data?  Definitely read [Spikes](http://www.amazon.com/Spikes-Exploring-Neural-Computational-Neuroscience/dp/0262681080/) when you have a chance.  The short answer from there is that it depends on what you are looking to get out of the signal.  If you're looking to drive a device or something of that nature, the FFT is going to be computationally burdensome, and may not give you sufficient information.  
> **username_1**: As an aside, for papers to look at, [George Gertstein](http://www.med.upenn.edu/mulab/) is the guru of the analysis of spike trains.  
> **username_0**: @ChuckSherrington I just want to detect oscillatory activity in the spike train. Computational burden isnt an issue. Why would the FFT give me insufficient/different info than the autocorrelation?  
> **username_1**: I'm using "information" as a loaded term.  I was referring to both measures that you are talking about, sorry if that wasn't clear.  Neither is going to tell you anything meaningful.  If you're locking to a specific event (sensory input or specific motor output - e.g., an EMG of a muscle) the frequency variation will tell you something, but a "free form" recording really isn't.  You'd be better off examining local field potentials (most extracellular recording systems can separate these out in the spike sorting processing) if you are trying to tease out frequency changes.  
> **username_1**: Otherwise, this is really just an exercise in signal processing, to be frank.  
> **username_1**: (not trying to be an overbearing curmudgeon here, just offering some constructive criticism. I'm happy to see a methods question on here!)  
> **username_0**: thnx. in our work we concatonate spike train activity rather then add it up as done in perievent histograms. the reason we do this is because we do spike-train input/output modeling of these concatenated signals.  
> **username_1**: If systems identification is what you are after ("input/output modeling"), you'll definitely need some heavier hitting analyses -- check out Bob Kearney's work http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471274569.html  
> **username_2**: I suggest Chronux package if you use MATLAB: http://chronux.org/  

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## 1 answers

### Answer by username_3 on 2017-09-05. Score: 1

As [this answer](https://electronics.stackexchange.com/a/135383/162027) explains, the Fourier transform of the autocorrelation of $x$ is the same as the magnitude squared of the Fourier transform of $x$. The autocorrelation of x can be expressed as $x(t)\*x(-t)$, where \* means [convolution](https://en.wikipedia.org/wiki/Convolution) and the Fourier transform of the autocorrelation is then $X(w)X^\*(w)=|X(w)^2|$, which of course is the Fourier transform of the original signal $x(t)$.

As for references, see [this answer](https://cogsci.stackexchange.com/questions/8385/reference-request-in-circuit-and-signals-for-computational-neuroscience/8395#8395).


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