Timeline for Fourier Transform of a short signal
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2021 at 12:00 | history | tweeted | twitter.com/StackPhysics/status/1404770676095991809 | ||
| Jun 14, 2021 at 1:45 | comment | added | alephzero | If you know your short-length signal has only one sine wave component and you want to estimate the frequency accurately, don't use a numerical Fourier transform. Use a method based on autocorrelation. That way you can get a good result even if your "short" signal is less than one period of the sine wave. Fourier transforms are nice mathematics, but they are not the only tool, and often not the best tool, for digital signal processing. | |
| Jun 14, 2021 at 0:31 | answer | added | nanoman | timeline score: 2 | |
| Jun 13, 2021 at 17:37 | comment | added | uhoh | @orthocresol en.wikipedia.org/wiki/F._D._C._Willard so I guess a little bit of his attitude rubbed off on me :-) | |
| Jun 13, 2021 at 17:36 | comment | added | uhoh | @orthocresol I'm just remembering the day I learned that the FT of a Gaussian is a Gaussian, of a Lorentzian is a Lorentzian, of a step function is an exponential for $t\ge0$, of a circle is an Airy function, and the reciprocals of the widths of the formers are always the widths of the laters, 2's and 𝜋's be damned. The board was covered with FT and convolution integrals, the prof would drop the $1/\sqrt{2 \pi}$ regularly and when a student pointed it out too many times he called him an "anal retentive" and got in a little trouble He also put his cat as a co-author on a Physical Review Letter | |
| Jun 13, 2021 at 17:23 | comment | added | uhoh | @orthocresol You are right of course. I'm hoping that the way I've carefully wishy-washy-worded it, it reads okay; as long as we leave "width" less-than-clearly defined and don't worry about factors of 2 or 𝜋 it gives us information rather than uncertainty. I seem to remember asking Is it “common practice in Fourier transform spectroscopy to multiply the measured interferogram by an apodizing function”? If so, why? a few years ago. I think the words windowing and apodization are the same thing mathematically at least. | |
| Jun 13, 2021 at 16:57 | comment | added | orthocresol | @uhoh: Your comment about lifetimes is related but is subtly different. OP is describing a sinusoidal function which is truncated i.e. a piecewise function $x = \cos(t)$ for $0 \leq t \leq t_0$ and $x = 0$ for $t > t_0$. You're talking about a damped sinusoid $x = \cos(t) \exp(-t/\tau)$ which gives you a linewidth (or FWHM) of $1/(\pi\tau)$. (From the perspective of a NMR spectroscopist, these are different problems: truncation is an experimental limitation and often has to be "fixed" using window functions, and relaxation is actually useful, as you say.) | |
| Jun 13, 2021 at 15:05 | answer | added | AccidentalTaylorExpansion | timeline score: 1 | |
| Jun 13, 2021 at 13:13 | comment | added | J... | A sine wave is infinitely long. A sound you hear for a few seconds is not infinitely long. A sound you hear for a few seconds is not a sine wave. | |
| Jun 13, 2021 at 11:26 | vote | accept | QuantumQuasar | ||
| Jun 13, 2021 at 1:50 | comment | added | uhoh | Once you get used to thinking that way you'll find this added information really useful and start applying or extending it elsewhere in Physics. For example, the width of an atomic spectral line will be roughly the reciprocal of the lifetime of the excited state. | |
| Jun 13, 2021 at 1:46 | comment | added | uhoh | Simple answer: after we measure the distance between two peaks, we might assume that the pattern repeats forever backward and forward in time, but it won't. You turned on the signal generator recently and you'll soon turn it off again. A single sine wave does not and can not contain that information, but a packet of closely-spaced sine waves can reproduce that, or the much shorter sample window you used. You are getting more information, not uncertainty! The width of the peak in Hz will be roughly the reciprocal of the length of the sample in seconds. | |
| Jun 13, 2021 at 0:56 | history | became hot network question | |||
| Jun 12, 2021 at 20:24 | answer | added | Hans Wurst | timeline score: 2 | |
| Jun 12, 2021 at 18:51 | comment | added | Žarko Tomičić | jackschaedler.github.io/circles-sines-signals/… | |
| Jun 12, 2021 at 18:39 | answer | added | Žarko Tomičić | timeline score: 3 | |
| Jun 12, 2021 at 18:22 | answer | added | JEB | timeline score: 8 | |
| Jun 12, 2021 at 18:17 | answer | added | Andrew | timeline score: 14 | |
| Jun 12, 2021 at 18:17 | comment | added | PM 2Ring | Your finite duration signal consists of an eternal sine wave combined with a window function. | |
| Jun 12, 2021 at 17:49 | answer | added | garyp | timeline score: 5 | |
| Jun 12, 2021 at 17:49 | comment | added | PM 2Ring | Does this help? physics.stackexchange.com/q/311663/123208 | |
| Jun 12, 2021 at 16:55 | review | First posts | |||
| Jun 12, 2021 at 17:44 | |||||
| Jun 12, 2021 at 16:55 | history | asked | QuantumQuasar | CC BY-SA 4.0 |