Talk:Fourier analysis

message: This talk page has been archived at Talk:Fourier transform/Archive1. I moved the page, so the edit history is preserved with the archive page. I've copied back the most recent thread. Wile E. Heresiarch 23:58, 20 Sep 2004 (UTC)
thread_title: Untitled

message: The formula for the discrete fourier transform
:x_k = \frac{1}{n} \sum_{j=0}^{n-1} f_j e^{2\pi ijk/n} \quad \quad k = 0,\dots,n-1
can be simplified because
:e^{2\pi i}=1
into
:x_k = \frac{1}{n} \sum_{j=0}^{n-1} f_j 1^{\frac {j k}{n}} \quad \quad k = 0,\dots,n-1
This convention that an n'th root of unity is written 1^{\frac{1}{n}}
rather than e^{\frac{2\pi i}{n}}
is convenient. Introducing the analytical concepts e and i and \pi into an algebraic context is confusing and pointless.
The transform can be modified a little by taking the square root of n and by taking the complex conjugate of f. 
:x_k = \frac{1}{\sqrt{n}} \sum_{j=0}^{n-1} \bar{f_j} 1^{\frac {j k}{n}} \quad \quad k = 0,\dots,n-1
Now this transformation is an involution. The procedure that transforms f into x is exactly the same as the procedure that transforms x into f. This is nice. 
PS: I inserted the note "Moreover: reality in one domain implies symmetry in the conjugate domain."
Bo Jacoby 13:18, 8 September 2005 (UTC)
thread_title:  involutory fourier transform 

message: The introductory paragraph for any section in this encyclopedia should be accessible to all readers.
What was wrong with my input? Was it too easy to understand?:
" It is a mathematical technique for expressing a waveform as a weighted sum of sines and cosines." 
Some introductory info is appropriate as:
It usually happens in mathematical analysis that certain given functions are to be operated on, and other functions or numbers are thus obtained. Often, the operations used involve integration, and here is found a very general class of operators, the so-called integral transforms. The simplest integral transform is the operation of integrating. 
---Voyajer 04:34, 30 December 2005 (UTC)
replies: text: What was wrong is that it was repetitive and redundant.  The introduction already says that the transform re-expresses a function in terms of sinusoidal basis functions, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). —Steven G. Johnson 19:41, 30 December 2005 (UTC)
replies: text: Please try to make the intro more accessible like Voyajer's version. — Omegatron 19:51, 30 December 2005 (UTC)
thread_title: Excuse me!

message: ?can anyone tell me why the first term of Fourier series a_0=0 when expanding a square wave
I have done the exercise about it..
thanks a lot--HydrogenSu 14:53, 19 January 2006 (UTC)
replies: text: a_0 is the DC (zero frequency) offset. If the average value of a square wave is zero, a_0 will be zero. Madhu 19:37, 10 August 2006 (UTC)
thread_title:  *rush to a square-wave 

message: If you consider this article: , it appears that the definition of DFT you have here on this page is actually the inverse DFT.  Whose page is wrong?
*Answer: Neither. The DFT (or FT) can be defined with either sign in the exponent as long as the inverse is defined such that applying transform and inverse in succession gives the original sequence (or function). There is also some freedom as to wether to put the 1/N, 1/L, or 1/2\pi on the transform or inverse for discrete, finite, and infinte cases respectively. I suppose I should add this to the page.--Kevmitch 02:39, 1 April 2006 (UTC)
thread_title:  DFT definition given is actually inverse DFT? 

message: Using an arrow instead of a comma makes it more readable. 
The function f, defined by y = f(x), can be written f = (y←x), pronounced: "y as a function of x"..  This nice notation allows you to distinguish the power function, (xy←x), from the exponential function, (xy←y).
The transform 
:F = (S←s) = ((S(ω)←ω)←(s(t)←t))
means that 
:S = F(s) 
:(S(ω)←ω) = F(s(t)←t) 
and 
:S(ω) = (F(s))(ω) = (F(s(t)←t))(ω) 
This notation does not identify the function f with the function value f(x), using a special letter x to identify the independent variable. Bo Jacoby 12:07, 4 August 2006 (UTC)
replies: text: Sure, this also solves the problem and if it is more established, I will go for it.  Next issue: is it relevant to present this somewhere in the article, giving an example for a notation which both is formally correct and can be used for the occations when an explicit function of a specific variable needs to be transformed?  --KYN 18:23, 5 August 2006 (UTC)
thread_title: Suggestion

message: The problem with the notation that Bo is suggesting for the Fourier transform, and for functions in general, is that it is much less common than the usual notations which are already employed in the articles.  If the notation is established but obscure, it might be worth mentioning in function (mathematics), but a wholesale adoption in other articles is not a good idea.  
Given that Bo has a history of trying to promote his own invented nonstandard notations in Wikipedia, however, I would also request that he provide a mainstream citation for his notation before inserting it into any article.
—Steven G. Johnson 17:03, 8 August 2006 (UTC)
replies: text: To StevenJ the word standard means known to me, and the word nonstandard means unknown to me; he makes no reference to standardization documents. Stevenj has a history of deleting everything that is new to him. KYN's complaint that StevenJ's notation is unsatisfactory, makes no impression on StevenJ. Still StevenJ does me too much honor by believing that I am the original inventor of the trivial shorthand notation 1x = (e2π i)x for e2π i x, or of the straightforward explicite notation f = (y←x) for the function defined by y = f(x), or of the involutary N'th order fourier transform: FN = ( (N−1/2Σj xj* 1jk/N←k) ← (xj←j) ). The article is very bad, but, according to StevenJ, it is standard, and that is all that matters. Progress is suspended until StevenJ gets majorized by fellow wikipedians. Meanwhile, study the notation for functions used in the J programming language. (I am the inventor, however, of the ordinal fraction technology, and my WP article on Ordinal fraction was deleted on StevenJ's request). Bo Jacoby 07:30, 9 August 2006 (UTC)
replies: text: Hi Bo, should I be surprised by the vehemence your invective?  (Your protests seem rather strained, as in both the 1^x and "involutary" DFT cases, in addition to ordinal fraction, you admitted that you didn't know of any publication that used your proposed notations/definitions.  Why is Wikipedia policy so hard for you to comprehend?)  By the way, I'm aware that x^2←x is sometimes used as function notation (although I maintain that it is uncommon in this context), but I don't think one often sees it mixed with "=" in the way you suggest, as in f = (x^2 ← x) instead of f(x)=x^2.  And notice that I simply asked you to provide mainstream citations, which would have supported your case far better than a diatribe.  (If the best you can do is an uncommon programming language that appears to only loosely match the notation you suggest...)  —Steven G. Johnson 04:55, 11 August 2006 (UTC)

text: Hi Steven. You are aware that x2←x is sometimes used as function notation although you maintain that it is uncommon in this context. Has the wikipedia policy requirement changed from known to common? If a is a mathematical expression, then of course you can name it: b=(a). So if x2←x is a function, of course you can name it if you want: f=(x2←x). The point is that you do not need to name is, as you do when you write f(x)=x2. You can write a function value elegantly (x2+yx+1←x)(4) = 4y+17. (Oh I forgot, you are not interested in good or bad, you are interested in standard, although you never refer to standardization documents.)  Bo Jacoby 11:07, 11 August 2006 (UTC)
thread_title: Wikipedia is not the place to promote new/obscure notations

message: Hi, the topic of this thread is twofold: (1) is there an established notation which can be used for saying, eg, the FT of sinc is rect, in a formally correct way, and (2) should the notation issue be mentioned in the article.  Bo has proposed a notation which addresses (1) which it OK with me, but since I havn't seen any evidence that it is established I don't want to write about it since Wikipedia should not be the place to introduce new notations which does not exist in the literature.  An alternative to Bo's arrow can be found in Daubechies "Ten Lectures on Wavelets" where the anonymous variable is simply denoted with a dot: \mathcal{F}\{sinc(\cdot)\}=rect(\cdot).  To my surprise the informal notation can be found in hard core math text book, for example see Debnath & Mikusinsky "Introduction to Hilbert Spaces with Applications" (2ed, page 194).  About (2), I will try to figure out something based on the text I wrote above and put it into the article shortly.  --KYN 15:48, 14 August 2006 (UTC)
replies: text: Hi KYN. A dot as a placeholder for the independent variable is worse than x or t or ω or k. The formula F(s(t+u)←t) = F(s)·(ei·ω·u←ω) becomes F(s(·+u)) = F(s)·ei···u which is spooky. Bo Jacoby 09:15, 15 August 2006 (UTC)

text: When stating that the Fourier tranform of sinc is rect, no dot is needed: F(sinc) = rect. I know of no place where the dot notation is sufficient. It is limited to functions of one variable only, and it looks like a multiplication sign. It has no future. Bo Jacoby 12:52, 15 August 2006 (UTC)
replies: text: F(x(·)) confirms for me that x is a function.  I am not so sure about F(x).  --Bob K 14:42, 15 August 2006 (UTC)

text: "the value F(x(t))" ... that ambiguity is avoided by F(x(·)).    In certain situations, F(x(·)) is better than either F(x(t)) or F(x).  --19:49, 16 August 2006 (UTC)
thread_title:  The dot notation 

message: My conclusion of this discussion is:
* there is no consensus about notation, in particular about how, e.g., "Fourier transform of rect is sinc" should be written in a concise and formally correct way.  (Yes, we have seen proposals, but no consensus).
* the informal/ill-formed \mathcal{F}\{\mathrm{rect}(t)\}=\mathrm{sinc}(\omega) appears to be frequently used, even in math text books.  Becomes meaningful only with proper context.
* there are several proposals for a formal way to rewrite this expression, e.g., Bo's arrow and Daubechies' dot.  The arrow notation, although perfectly valid, appears not be in use in the literature, and the dot notation is also not in frequent use (Daubechies' book is the only example that I have encountered).  Given that Wikipedia explicitly should have a verifiable content, which I interpret in this particular case as "describe notations which are established in literature or in common use", I suggest to leave the arrow and the dot until more evidence of their usages can be demonstrated.
--KYN 19:50, 15 August 2006 (UTC)
replies: text: Here is my favorite excerpt from that reference:  "In most written mathematics, it is enough to just say (in prose) what the bound variable is when confusion could arise. Since the entire community of mathematicians seems to have agreed that this is precise enough, new notation is apparently not necessary."  --Bob K 14:04, 17 August 2006 (UTC)

text: Yours!  (Talk:Function (mathematics)#Standard notation. )  --Bob K 13:50, 18 August 2006 (UTC)

text: I did not claim to know the community of mathematicians.  Your reference did that.  --Bob K 13:50, 18 August 2006 (UTC)

text: Is someone pretending that?  Nothing is without disadvantages.  So finding one is not sufficient justification for change.  That just leads us into circles.  --Bob K 15:44, 20 August 2006 (UTC)

text: Clearly, I did not agree to that.  I said nothing is without disadvantages, and that includes your proposals.  What I also said to you, 13:55, 11 August 2006, and subsequently echoed by your own reference (CMummert), is to use supplementary annotation (aka prose) where clarification is needed.  --Bob K 17:48, 20 August 2006 (UTC)

text: No reason at all, except to make the article comprehensible to new readers, the so-called 'little minds'. Bo Jacoby 05:59, 21 August 2006 (UTC)
thread_title:  Conclusion 

message: It would be nifty if there were a small graph of sinc function, etc. in the tables of equations.
The continuous graphs plotted with nice smooth curves to emphasize that it is continuous, while discrete graphs plotted as more of a bar-chart or set of spikes, to emphasize its discreteness.
Do we really need 4 sets of equations and graphs:
* continuous-time/continuous-frequency in Continuous_Fourier_transform#Table_of_important_Fourier_transforms
* discrete-time/discrete-frequency in Discrete Fourier transform
* discrete-time/continuous-frequency in Discrete-time Fourier transform
* discrete-frequency/continuous-time in Fourier series
?
(If there is some significant difference in the equations and graphs, it would be more visible if they were plotted all on one page, rather than scattered across all 4 articles).
:But there are also advantages to scattering, and they outweigh the disadvantages.  One advantage is divide-and-conquer.  I.e., it allows better focus on small parts of a large picture.  Scattering, linking, and search-engines are the advantages of online technology.  "web" is a good thing.  --Bob K 15:25, 10 August 2006 (UTC)
thread_title:  Graphs and pictures 

message: I understand that the normal distribution is an eigenfunction of the Fourier transform. Is this the case? Should the article mention this? —Ben FrantzDale 19:32, 9 August 2006 (UTC)
replies: text: To me, that seems perfectly appropriate for inclusion in this article. Michael Hardy 23:19, 9 August 2006 (UTC)
replies: text: More precisely: the density function of the normal distribution is an eigenfunction of the Fourier transform. Michael Hardy 23:19, 9 August 2006 (UTC)
replies: text:  Added. —Ben FrantzDale 00:53, 10 August 2006 (UTC)
replies: text:  just keep in mind that there are an infinite number of eigenfunctions in the Fourier transform.  like sinc()+rect().  any even-symmetry function added to the F.T. of that (which is also even symmetry) is an eigenfunction.  but the Gaussian function is a nice simple and intrinsic eigenfunction and i can't think of another that is. r b-j 05:27, 10 August 2006 (UTC)
replies: text: We have also polynomials times Gaussian which as a class provide eigenfunctions to the FT. Or maybe an irreducible representation?  There is also the shah or comb function.  Anyway, if this subject is to be developed to some length, I suggest to do it in a separate article "Eigenfunctions of the Fourier transform" rather than putting it into this article since it is rather long as it is.  --KYN 10:30, 11 August 2006 (UTC)
thread_title:  Normal distribution? 

message: See Talk:Continuous Fourier transform --catslash 12:19, 4 September 2006 (UTC)
thread_title:  Merge? 

message: Considering this is such a technical article - Why do I come across the word "jaggies" 82.25.174.136 19:22, 24 November 2006 (UTC)
thread_title:  Informal Language 

message: Bob, your explanation is rather a detailed mechanical one, but it's actually a better explanation of the [[periodogram] (including Lomb's method; see Least-squares spectral analysis).  That's because it doesn't describe the use of orthogonal basis functions, nor the notion of decomposition, from which derives inversion.  By focusing on how to "measure the amplitude and phase of a particular frequency component" it presupposes the idea of particular frequency component and misses the idea of a transform, that a signal can be expressed in terms of such components.  
Coming up with simple explanations is tricky; we've all tried, and I think all failed.  Maybe we can find something in a book?  Dicklyon 16:04, 28 October 2007 (UTC)
thread_title: Bob K's simple explanation

message: The reason I changed "inverse transform" to "inverse transform (synthesis)" was not because synthesis is mentioned in the other article (which it's not).  It's because inverse tranform is not previously mentioned in this article.  It just appears out of nowhere.
--Bob K 14:50, 1 December 2007 (UTC)
replies: text: That should be easy to fix in the lead. Dicklyon 16:29, 1 December 2007 (UTC)
thread_title:  "synthesis is not mentioned in that article" 

message: Why do we separate "Applications in signal processing" form "Applications"? Thenub314 (talk) 13:59, 18 September 2008 (UTC)
thread_title:  Applications. 

message: The article currently uses both t\, and x\, for the "time" domain and f\, and \nu\, for ordinary frequency (cycles per time-unit).  It also uses f\, for the time-domain function, but not at the same time it uses f\, for frequency (thank goodness).
Predictably, it is causing confusion and editorial thrashing.  But of course it does accurately reflect the real world, which is also inconsistent and confusing.
--Bob K (talk) 13:01, 28 March 2009 (UTC)
thread_title:  Inconsistencies 

message: Maybe it's just because I'm not a layman on this subject, but the How it works (basic explanation) section really seems to grate.  At best, I think that this belongs in the cross-correlation article, but either way there are some specific points in the language that need addressing:
* "its shape (but not its amplitude) is effectively squared" - what distinction is this trying to draw?
* the section begins by describing the mechanism as squaring, and then goes on to say that it's not really squaring
* the sudden introduction of the sine+cosine basis is unjustified
* the introduction of a complex basis means that no squaring occurs, as one must multiply by the complex conjugate waveform
* the introduction of the notion of vectors is misleading/unnecessarily complicated, as it's perfectly possible to interpret complex numbers as scalars.
Basically, I think that as it stands this section has more potential to confuse a newcomer than to help them.  Any thoughts?  Oli Filth(talk|contribs) 21:35, 28 May 2009 (UTC)
replies: text: I've now removed the section in question.  If anyone feels like restoring it, could they address the points I've raised above.  Even then, there's still the question of whether such an explanation should really belong in the cross-correlation article.  Oli Filth(talk|contribs) 00:02, 4 June 2009 (UTC)
thread_title:  "How it works" section 

message: Fourier transform is a representation of a signal as a collection of sinusoidals. But when adding two periodic signals the resultant would be another periodic signal. If we have fourier transform for aperiodic signals, it implies that the sum of periodic signals results an aperiodic signal. can anyone clarify this paradox?  — Preceding unsigned comment added by 103.247.48.153 (talk) 18:31, 27 August 2015 (UTC)
thread_title:  Doubt on fourier transform 

message: It is not true that all functions can be fourier transformed. Already Euler remarked on that. The requirement is that the function is square-integrabel in the sense of Lebesgue or otherwise expressed that the function has a length in the relevant Hilbert space. 
This condition is also applicable to the Laplace transform and the Heaviside operator etc..
I do not have a reference but it was proven in 1978 ( if I remember correct) and everybody know it (except the author of this article).
Burningbrand (talk) 17:09, 6 August 2019 (UTC)
thread_title:  Limits of Fourier transforms