Talk:Fast Fourier transform

message: Hey, does anyone have an issue number for the citation?  The citation reads:
Cooley, James W., and John W. Tukey, 1965, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19: 297–301
But there is no issue number.
:A scan of the paper (of uncertain legality) can be found here.  I believe it was in the April issue, but strangely enough the scanned pages don't list the journal issue or volume.  In any case, the volume and page number uniquely identify the article, and should be sufficient to look it up in your library.  —Steven G. Johnson 21:59, 2 April 2007 (UTC)
replies: text: I think breaking this into more than three paragraphs is overkill.  As for expanding it, any additional information about the FFT algorithms in question should go in a separate article, like for Prime-factor FFT algorithm and Rader's FFT algorithm. —Steven G. Johnson 23:58, 29 May 2006 (UTC)

text: Done.  At least as a start (the fancier multidimensional algorithms such as vector-radix could eventually use their own pages).  —Steven G. Johnson 04:41, August 23, 2005 (UTC)
More information about the practical uses of FFT (e.g. how it is used for digital signal processing) would be very useful. If I ever find out what it does, I may write it up myself... Jevon 12:11, 23 November 2005 (UTC)

text: As is described in the first paragraph, information about applications belongs in discrete Fourier transform. —Steven G. Johnson 16:34, 23 November 2005 (UTC)
replies: text: Ramčević@gmail.com 178.237.217.117 (talk) 12:06, 6 April 2023 (UTC)
thread_title: old comments on citations

message: The whole article seems to address floating-point arithmetics as an approximation to the complex field.
However, the same algorithms may be used in any field where there is a nth prime root of the unit, where n is the length of the vector, including finite fields GF(pk). Since the multiplicative group of a finite field is cyclic, the condition for the existence of such a root is n | pk-1. If k=1, such a field is easily implementable in machine by modular arithmetics.
There is an algorithm for multiplying long integers by
* consider them written in base \beta: a=\sum a_k.\beta^k and b=\sum b_k.\beta^k, as polynomials
* transforming them by FFT modulo p
* writing the convolution product a.b=\sum c_k.\beta^k where c_k=\sum_{j=0}^k a_j.b_{k-j}
* computing this convolution product by inverse FFT of the product of the two FFTs
* doing a little carrying.
A little web search brings up: J. M. Pollard, Mathematics of Computation, Vol. 25, No. 114. (Apr., 1971), pp. 365-374.  Stable URL: http://links.jstor.org/sici?sici=0025-5718%28197104%2925%3A114%3C365%3ATFFTIA%3E2.0.CO%3B2-U 
I'm not an expert in the field, so I don't think comfortable writing about this. I'll look how GNU MP does it. David.Monniaux 22:07, 13 October 2006 (UTC)
replies: text: Only one section of the article discusses floating-point approximation, so I'm not sure why you're saying that the "whole article" is concerned with this.  However, it's true that the article is concerned with algorithms for the discrete Fourier transform over vectors of complex numbers.   It's certainly true that many of the algorithms have direct analogues for any finite field (although I can give counter-examples of some that can't, such as the Rader-Brenner algorithm); in particular, those algorithms that only use the fact that \omega_N is a primitive root of unity (e.g. Cooley-Tukey, PFA, Rader...) are generalizable in this way.  But such transforms are generally given different names, like the number theoretic transform. —Steven G. Johnson 22:49, 13 October 2006 (UTC)
replies: text: I've added a brief mention of such generalizations in the introductions.  Any further details belong in articles like number theoretic transform.  —Steven G. Johnson 23:02, 13 October 2006 (UTC)
replies: text: Ah, interesting. I actually didn't know different names were used in English (I learned mathematics in French :-) ). David.Monniaux 08:27, 14 October 2006 (UTC)
thread_title:  in finite fields 

message: Why not add create a Wikipedia article called "Fast Fourier Transform Example" and link there?
I think the goal of the project is to preserve human knowledge. It does not seem right to me to post example elsewhere. It is guaranteed to disappear / be irrelevant.
Above looks simple and easy to understand, I'm all for keeping it. Separately is fine, if that is better stylistically.  —Preceding unsigned comment added by 68.162.147.222 (talk) 19:51, 20 December 2007 (UTC)
thread_title:  Separate pages for code examples? 

message: On the Polish Wikipedia, we had a discussion regarding the proper term for the FFT algorithm. It turned out that transformata, which some believed is closer in meaning to the English word transform, was more frequently used, even though the other word, transformacja (transformation, translated literally), was more appropriate as the name for FFT. Now I'm in doubt about which of the words: transformation or transform is the proper term for the operator or operation that maps a vector to its discrete Fourier transform. Is there a distinction between the operator (which we on pl-wiki agreed was closer to transformation) and its result, or is transform used in both meanings? Note that fast Fourier transformation is also used in English publications, online  or otherwise (like the first entry in FFT#References). Safek (talk) 08:00, 1 May 2008 (UTC)
replies: text: In English, "fast Fourier transform" is far more common than "fast Fourier transformation", but the two are used more or less interchangeably as far as I can tell.  But English is a bit funny because "transform" was originally exclusively a verb, and acquired a noun usage sometime over the last 200 years according to the OED.  And, even in this sense, "transform" originally referred to the result of a transformation, but nowadays is used for the transformation itself (the "operator") too.  Of course, in English, "FFT" can also be used as an adjective, as in "FFT algorithm."  I have no idea what Polish does, but our usual practice on Wikipedia is to go with the most common usage, and accept the fact that living languages are often illogical.  Language and notation are ultimately determined by usage and convention; there is no absolute "right" and "wrong" unlike for mathematical facts.  —Steven G. Johnson (talk) 23:54, 1 May 2008 (UTC)
replies: text: Thanks for your reply. It turns out Polish is undergoing the same process as English. However, it hasn't gone that far yet, so on pl-wiki we are trying to stand by the more accurate word and explain the difference between the words. Safek (talk) 12:37, 2 May 2008 (UTC)
thread_title:  The FFT acronym 

message: Hi!  My feeling is that the LEDE should hit all the main points, and be short enough to fit on one screen without scrolling.  The longer lede version is better at this.   Adding extra divisions, in my mind, makes it worse since it pushes introductory text off the first screen, where readers are most likely to see it.   Note that many featured articles (see for example Hubble Space Telescope have ledes of this length.  LouScheffer (talk) 11:04, 11 October 2008 (UTC)
replies: text: I agree; a lede that fits comfortably without scrolling is perfectly fine, and it's not clear how the reader will be helped by breaking up the current lede.  In any case, if you want to split it up, you can't simply add headers, you need to rewrite the text.  Just adding headers without changing the text breaks up the train of thought, and makes the lede inadequate because crucial information (e.g. a clear definition of FFTs) is relegated to a later section.  (If it really needed to be shortened, my inclination would be to put the explanation of how much slower O(N^2) is than O(N log N) into a subsection.  The main purpose of the lede should be to delineate the subject of the article.  But again, this would require some rewriting, not just adding headers and cut-and-paste.) —Steven G. Johnson (talk) 23:07, 11 October 2008 (UTC)
thread_title:  Length of LEDE 

message: "Θ(N2) to Θ(N log N) remains."  Say.  Shouldn't those thetas be capital letter Os?--72.75.86.157 (talk) 07:56, 16 October 2008 (UTC)
replies: text: Technically, no.  —Steven G. Johnson (talk) 15:12, 16 October 2008 (UTC)
replies: text:  But as the user points out we make it all the way through the article using Big Oh notation until this point.  Does it really make sense to switch to a slightly different concept without comment, link, etc?  Thenub314 (talk) 06:50, 17 October 2008 (UTC)
replies: text: I've added a comment and link.  The problem is that saying FFTs reduce the complexity from O(N^2) to O(N log N) is literally false.  FFT algorithms are still O(N^2) in addition to being O(N log N), because "O" only denotes an upper bound.  —Steven G. Johnson (talk) 14:12, 17 October 2008 (UTC)
replies: text:  I am very confused by this second sentence.  Big Theta implies Big-Oh, so saying the complexity is O(N log N) is true.  I agree that every worse upper bound is still an upper bound, so the algorithms are also O(N3), etc.  But there is nothing literally false about saying FFTs reduce the complexity from O(N^2) to O(N log N). Thenub314 (talk) 07:13, 18 October 2008 (UTC)
replies: text: If you say that they reduce the complexity from O(N^2) to O(N log N), you are literally saying that they are no longer O(N^2), which is false. —Steven G. Johnson (talk) 20:02, 18 October 2008 (UTC)
thread_title:  Oh Notation 

message: I have readed the article and I don't understand why the original transformation cost is O(N^2). It is just a sum for 0..N-1 so it should be O(N). If I want to compute X_i I just need to iterate N times.  —Preceding unsigned comment added by 62.43.167.123 (talk) 16:47, 13 January 2009 (UTC)
replies: text: Each output is a sum of N terms, so is O(N).  But there are N outputs, so you need to do N sums, so it is N O(N) = O(N2).  —Steven G. Johnson (talk) 21:37, 22 May 2009 (UTC)
thread_title:  About complexity 

message: Could someone please write a new opening paragraph with a more pedestrian explanation of what FFT is? 
Reading this, I feel like an illiterate who can't count trying to read an advanced calculus book.  —Preceding unsigned comment added by 208.127.100.174 (talk) 09:22, 22 August 2009 (UTC)
replies: text: I also think that people with finished university education in mathematical or physical area do not need Wikipedia to read about FFT as they already know about it enough. For others the article seems getting unreadable Audriusa (talk) 13:53, 29 November 2010 (UTC)
thread_title:  Huh? 

message: I attempted to add a recent result on some FFT operation bounds proven by SAT solvers to the subsection "Bounds on Complexity and Operation Counts" but my changes were reverted due to conflict of interest. (I thought I was OK since the result refers to a peer-reviewed journal.) I'd appreciate review and possible inclusion by some independent person(s). You can see the additions I tried to make under the article history (changes on 6 December 2011 by Haynals).  — Preceding unsigned comment added by Haynals (talk • contribs) 22:06, 6 December 2011 (UTC)
replies: text: I agree that your paper, which is an unusual approach to this problem, seems worth mentioning in this context, and have added back a somewhat amended description.  — Steven G. Johnson (talk) 04:13, 7 December 2011 (UTC)
thread_title:  Additions to Bounds on Complexity and Operation Counts 

message: Just wondering. Why do most of the algorithms I've seen require the input array length to be a power of 2? 68.173.113.106 (talk) 00:06, 1 March 2012 (UTC)
replies: text: 2 isthe smallest prime, so the algorithms are simplest in that case...this is why many textbooks only show that case, andin particular they usually only show Cooley-Tukey for power-of-two sizes.  FFT algorithms exist for any size, though, even prime sizes. — Steven G. Johnson (talk) 00:37, 1 March 2012 (UTC)

text: While this is a superficially plausible theory, it is not really accurate as far as I can tell (nor can I recall seeing this justification in any of the early papers).  As a practical matter, one can always arrange the computation so that the differencing in indexing performance between (e.g.) 2ⁿ and 3ⁿ is negligible: for example, in a traditional textbook-style breadth-first Cooley-Tukey implementation, the multiplication or division by the radix only needs to be done once per pass over the array (with the rest of the indexing computations being additions). — Steven G. Johnson (talk) 14:51, 30 July 2013 (UTC)
replies: text:  Even in Cooley-Tukey the factor 2ⁿ pops up in many places, e.g. when the for-loops all run from 0 to 2ⁿ-1, i.e. all-zeroes to N ones. . That's the ideal loop, and there are a lot of loops. (O(NlogN)). I'm not at all claiming that it is impossible; I'm just indicating why (as a programmer not a mathematician) you'd prefer such powers of 2.  — Preceding unsigned comment added by 2001:980:396E:1:D86C:31F5:6801:F457 (talk) 09:24, 1 August 2013 (UTC)
replies: text: I've implemented numerous FFTs, and this is a negligible issue from a practical programming standpoint as well as a mathematical viewpoint.  You have to look at where you multiply or divide by the radix, and you'll see that this is typically only done at most O(log N) times, and hence is negligible.   A more substantial issue is that the twiddle factors for power-of-two radix butterfly operations often simplify (e.g. to ±1, ±i, and (±1±i)/√2), so it is easier to implement optimized radix-4, radix-8, and radix-16 Cooley–Tukey steps than it is to implement, say, radix-10 efficiently. — Steven G. Johnson (talk) 15:10, 13 June 2014 (UTC)
thread_title:  Why power of two? 

message: Are there known exploits of the FFT, by say using the power of two nature to design visual camouflage or radio jamming so that the interference is ignored? Hcobb (talk) 12:40, 1 August 2013 (UTC)
replies: text: #1 Your comment makes no sense. #2 FFTs are rarely used in modern signal processing because you need to do something to the FFT data to get something useful out of it. Wavelets provided that kind of discrimination.  — Preceding unsigned comment added by 24.12.228.160 (talk) 04:19, 7 December 2013 (UTC)
thread_title:  Known exploits? 

message: The artice references a non-existent page on fixed point FFT, but even more the section makes assumptions about the fixed point arithmetic used. If one uses enough bits, fixed point should be just as good, or likely better, than floating point. At each stage of a radix-2 algorithm the magnitude can potentially be twice as big, so should have an additional bit. Since most DSP work is done in fixed point, this is an important consideration. Gah4 (talk) 00:09, 27 July 2015 (UTC)
thread_title:  Fixed point 

message: Hi,
I invite interested experts to review and accept the changes I made on August 19, 2015 at 10:33.  The changes discuss my new contribution to lower bounding Fourier computation using matrix entropy. They were reverted by another Wikipedia user due to conflict of interest and lack of independent sources.
For a link to a view of the changes (with comparison to current version, please go here
Thanks 
Nir.ailon (talk) 07:27, 23 August 2015 (UTC)
thread_title:  Lower bounds using matrix entropy 

message: I also believe that this article is extremely difficult to read. I read this article many times trying to understand the FFT, and gave up. After looking at other sources and experimenting myself, I came back to this page and things made sense...but only in retrospect. It had no utility in me learning the material.
One thing that this article explains INCORRECTLY in the time-frequency relationships section is the meaning of the negative frequencies, and the existence of negative frequencies when taking the transform of real valued data or complex data. Redundancy and negative frequencies only come about from the transformation of real valued data, transforming complex data does not result in half of the dataset being conjugate and redundant. As a new user, I won't delete this section, but I strongly recommend that this section is remedied.
What I did as my first wikipedia edit is to add a new section, 'Meaning of 'negative frequencies' from fast Fourier transforming real valued data.' I was only able to determine this piece of information through my own study and experimentation. The meaning of the negative frequencies is an extremely sought after piece of information that is found virtually nowhere on the internet. Dangerously, I can only find people who convey that it is describing how i.e. a wheel can be viewed as spinning clockwise at +Hz or counterclockwise at -Hz, which is perhaps intuitively pleasing but not at all the mathematical meaning of the negative frequencies. 
While I leave it to the more experienced wikipedia community to decide what to do with this section, I ask that if you have a different understanding of the meaning of these negative frequencies, please contact me and we will discuss. If you feel that this section should belong on another page, great! Whatever you do, please, leave this information up...the work of science, technology, and engineering really needs this piece of information!  — Preceding unsigned comment added by Draak13 (talk • contribs) 20:03, 23 October 2015 (UTC)
thread_title:  Very hard to read. I added one intuitive 'negative frequency' section that many people search for 

message: In looking at the example set forth by the first three images on this page, and their captions, it seems to imply that the FFT algorithm is by nature less precise than the integral version of the DFT, and that this loss in accuracy is acceptable given the decreased computational complexity. Perhaps I'm understanding something wrong, but I thought that the FFT was able to compute the DFT with the same accuracy as the integral DFT, and yet still at decreased computational complexity. If this is true, this example seems to be misleading to a reader simply browsing the beginning of this page.
:The discrete Fourier transform (DFT) has no integrals.  I think you are confusing it with a Fourier transform or Fourier series. — Steven G. Johnson (talk) 15:15, 1 October 2016 (UTC)
replies: text: Well you can integrate over delta functions if you like that better.  A digital image is assumed to have been frequency band limited before sampling. (There are low-pass spatial filters on cameras with lenses good enough to need them.) An image with a finite number of sample points over a finite spatial extent can be represented with a finite number of spatial frequencies. Those frequencies are produced by the DFT/FFT.  Gah4 (talk) 17:06, 1 October 2016 (UTC)
thread_title:  Image Examples Misleading 

message: I've been an occasional vistor to this article, but I have to admit that previously I took very little notice of the diagrams on display, until I saw your comments about them.   I agree, they are  a little misleading.    Perhaps I can make a few points to help clarify the situation. 
The first point to note is that where the FFT does provide output data, this agrees with that data provided by the DFT.   What the FFT does not do, is provide information about the intervening spaces and so the red line on the third diagram has no meaning.   
As the data from the FFT meets the Nyquist criterion, then we should be able to resurrect an analogue version of the spectrum from the frequency domain samples.   Once we have this plot, we can find the output at any intervening frequency we might choose.   However, what Nyquist does not make clear is the fact that when the waveform to be generated is near the Nyquist limit, an awful lot of samples have to be included in any algorithm we choose to use.  Just linking a few local points by a smooth line just will not do! 
One way of producing a continuous output might be to feed implulses with strengths proportional to the data values into a sharp-cut linear-phase low pass filter.   Such a filter has a very long time delay (so the filter has lots of data in it on which to derive a convolved output).  Unfortunately it is a filter that is hard to build. 
Another way forward, since we are dealing here with FFTs, is to use "zero padding", to achieve extra frequency points in the output data.   If new data is generated at a sufficient number of points, these can then be safely linked by simple linear interpolation to give us a reasonably accurate analogue plot.
Say we add to our input waveform defined at n = 0 to n = N-1 a sequence of zeros, seven times as long, to make the whole sequence n = 0 to n = 8.N-1.    We find that the output of the FFT will now have data at extra, intervening, frequency points.   This should be made clear by considering the time-frequency relationships. 
Before:   there are N data points, of duration T secs, and the frequency resolution of the FFT output  is 1/T Hz. 
After:  there are 8.N data points, of duration 8.T secs, and the frequency resolution of the output is 1/(8.T) Hz, i.e. there are now seven extra data points between the original outputs. 
Of course, there is a penalty incurred from doing this. The FFT that now has to be calculated is much, much bigger than the one we had before.   Consequently, the benefits of the FFT are less clear cut  (but, at least, values approaching the true peak values are obtained). 
One final aside, the outputs of the original FFT did not show the true peak values of the spectra (unlike the DFT).   This may occur whenever the input data is not truly cyclic within the input window of the FFT and the technical term for this is "Spectral Leakage".    As this is a common problem with signal data, techniques to deal with it are a major topic of Digital Signal Processing. 
I hope that helps!
D1ofBerks (talk) 15:22, 11 December 2015 (UTC)
replies: text: I think you're confusing the DFT with the DTFT.  An FFT computes exactly the DFT, nothing more, nothing less — the DFT, by definition, only gives discretely sampled frequency outputs.  A DTFT, in contrast, is defined for a continuous (albeit periodic) set of frequencies.   With regard to the image, that could certainly be the subject of a DFT with fine enough sampling; you're right that an FFT in this case would be overkill because most of the spectrum is zero, though.  I don't think the image is a wonderful illustration of an FFT algorithm; a butterfly diagram or similar seems like it would be more appropriate for the algorithm itself.  On the other hand, a lot of readers of this article seem to want a brief introduction of what Fourier analysis (in general) is about before delving into algorithms. — Steven G. Johnson (talk) 15:14, 1 October 2016 (UTC)
thread_title:  A Response to Image Examples Misleading 

message: The recently added first figure in the lede section is good but has two issues.
First, It is not a "Complex Cosine" which would be the cosine of a complex number.  A correct description would be "A periodic signal with four harmonics."  If this was the only problem I would just fix it.  Second, The DFT depicted is incorrect unless a window has been applied and/or something else has occurred.  The duration of the sample is an exact multiple of the signal period and its harmonics.  To be specific, the period of the fundamental is 0.1 seconds and the sample duration is 0.5 seconds.  In this case, there should be no spectral leakage into adjacent harmonics.  If a window is applied to the data prior to the DFT then you can see spectral leakage.  However, the long tail of peaks following 50Hz suggests that something else was done.  I'm not sure what it was but it might be that the actual sample duration was not a multiple of 0.1 seconds.  Either the DFT is incorrect for the time sample shown, or there is other processing that needs to be describe.  I'd rather see a straight DFT of the time sample as shown without any other processing.
I'm reverting it now because of the these issues, of I encourage the editor that posted it to make corrections and repost. Constant314 (talk) 22:20, 20 November 2016 (UTC)
thread_title:  The recently added first figure in the lede section is good but has two issues. 

message: I think it would be appropriate to include a link to this paper: https://arxiv.org/abs/1201.2501 but I'm a little intimidated by the way the article is structured, and I haven't delved deeply into Fourier transforms.  Perhaps somebody else would care to take a crack at it? MatthewWilcox (talk) 19:28, 15 February 2017 (UTC)
thread_title:  Sparse Fourier Transform 

message: The article says: All of the FFT algorithms discussed above compute the DFT exactly (in exact arithmetic, i.e. neglecting floating-point errors).  As far as I know, the FFT gives better floating point results (smaller error), than the direct (non-fast) implementation of DFT.  I don't happen to have a reference, but if true it would be nice to say so here. Gah4 (talk) 04:30, 14 May 2017 (UTC)
replies: text: I think that you are right.  The FFT gets there in fewer operations than direct computation so you would expect it to be closer than a direct computation with finite precision arithmetic, but I don't think that is what is meant.  I think it means that, given perfect arithmetic, then the FFT produces exactly the same results as any other method of computing the DFT with perfect arithmetic. Constant314 (talk) 05:35, 14 May 2017 (UTC)
replies: text: Oh, yes.  It was just that it reminded me of the separate question of rounding error in the FFT. Gah4 (talk) 16:05, 14 May 2017 (UTC)

text: This exact point is made, with references, in the "Accuracy" section of the article.   (It's not just because it requires fewer operations, by the way, it's because of pairwise summation.)  — Steven G. Johnson (talk) 21:19, 19 May 2021 (UTC)
thread_title:  Floating point 

message: Regarding the removal of the table, it seems to me that WP:CALC would allow the table, but then again, it doesn't seem all that interesting or useful. People who need it, likely have a calculator nearby, or could write a program to print out the table that they need. On the other hand, a little Javascript would allow for a calculator on the page. Gah4 (talk) 08:57, 23 June 2017 (UTC)
replies: text: You can evaluate an expression, provided that the expression is from a reliable source and properly cited.  There is no telling where those numbers came from and the are probably implementation specific. Constant314 (talk) 14:36, 23 June 2017 (UTC)
thread_title:  Table 

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thread_title:  External links modified 

message: The article says: Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used.  I believe this isn't quite right.  It isn't the finiteness, but the actual precision. If sums are done with enough extra precision, then they should be as exact as fixed point.  (Though fixed point also needs extra bits.) If you do sums in the same precision as the input, then you get rounding errors.  (But as noted in the article, usually less rounding error than non-fast DFT.)  Single precision transforms could sum in double precision (on most hardware). Some have extra precision for intermediate values for sums of double precision input data.  Gah4 (talk) 16:14, 5 November 2019 (UTC)
replies: text: No, that's not correct.  The coefficients of the FFT (roots of unity, i.e. sines and cosines) are not exactly representable in *any* precision (for N > 4), so there will almost always be some roundoff errors for any finite precision.   "Exact as fixed point" is also wrong — floating-point FFTs are generally *much* more accurate than fixed-point FFTs.  See the scalings and references in the "Accuracy" section of the article. — Steven G. Johnson (talk) 21:22, 19 May 2021 (UTC)
thread_title:  ''Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used'' 

message: The first schematic shows the highest peaks for the lowest frequency at 10 Hz while the frequency domain shows it as the lowest amplitude. It would be nice if one could clarify that, not only here but also in the article itself for the readers because it's confusing.2003:E0:5F2D:2C33:E2DC:9FCC:25BE:1149 (talk) 08:53, 6 September 2020 (UTC)
replies: text: The figure is accurate.  The 50th harmonic is the strongest.  Your eyeball analyzer is focused on the envelope. Constant314 (talk) 16:07, 6 September 2020 (UTC)
replies: text: Yes! You're actually right! I was confused! I reversed the whole thing but now it makes sense of course. 2A02:908:2D17:AD00:60F2:2C8C:BE6A:D0D (talk) 16:50, 6 September 2020 (UTC)
thread_title:  How can the amplitudes of the low frequencies in the time domain be higher, while being shown as low amplitude in the frequency domain?