Talk:Fourier transform/Archive 2

message: Hello. I wonder if there's any support for changing the normalization convention to whatever makes the formulas come out simplest. For example, so that the convolution thm becomes F(f*g) = F(f) F(g). I believe that such a convention is commonly followed. Comments? Wile E. Heresiarch 02:02, 26 Oct 2004 (UTC)
thread_title:  Change normalization to simplify formulas? 

message: I'm wondering why there is a \frac{1}{\sqrt{2\pi}} in front of the transform and inverse integrals. Most definitions that I've seen are:
X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-i\omega t}dt
and
x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega) e^{i\omega t}d\omega
Ok, thanks!
:As noted in the article, the normalization is somewhat arbitrary and varies between authors; the only important thing is that the product of the two constants is 1/2\pi.  (The one here is common among physicists and mathematicians; see e.g. Mathematical Methods for Physicists by Arfken and Weber.  It has the nice property that the transform is unitary without scaling, so that the forward and backwards transforms are conjugates.) —Steven G. Johnson 18:06, Nov 15, 2004 (UTC)
---
I don't understand that:
"The Fourier transform is close to a self-inverse mapping: if F(?) is defined as above, and f is sufficiently smooth, then..."
if F(w) is defined as above - what does it refer to? ("as above")
"f is sufficiently smooth" - what's the idea? f will be the result of the (inverse) transformation. How we can say before the transformation if f is sufficiently smooth?
Jun 10, 2005
<btw, how to make a timestamp without registration?>
Typing four tildes (i.e, ~~~~) should do. Cburnett 05:17, Jun 11, 2005 (UTC)
------------------------------------
You don't ask the writers' guild to rule on spelling; you ask the lexicographers.
It is certainly important to mention that there are various conventions for the
factors on Fourier transforms.  But it's mathematicians who invented the thing and who understand it forwards and backwards.  This is a math article.  The definition used by mathematicians is the one that should be central.  (Which is the way this article is currently organized: it should remain this way.)
(If warranted, any pure math article could have a section of it discussing relevant applications; this would be where to emphasize the version of the Fourier transform
favored by non-mathematicians.)
The reason mathematicians use 1/sqrt(2 pi) for each factor (as is mentioned in at least one Fourier transform article) is that this puts the transform and its inverse in the group of unitary transformations, ones that preserve distance in the Hilbert space L^2 of functions (or tempered distributions) on the reals. This permits all kinds of additional machinery to be brought to bear in the study of these transforms.
Also:  it should be mentioned that iterating the Fourier transform twice takes
f(t) to f(-t) (which explains the mysterious comment that the transform is almost the same as its inverse;  having similar formulas certainly does not suffice for making two
thing "almost the same").  And it should be mentioned that (therefore) iterating the
transform four times gives the original function back; this operator iterated four times is the identity operator.Daqu 19:59, 12 December 2005 (UTC)
thread_title:  main definition 

message: : The truth is that both conventions exist, and although we could debate endlessly about which one is more common or which one is more standard, the fact is that both conventions are used frequently and widely.  We will never achieve a consensus about which one is better or which one is more correct.  I prefer the electrical engineering convention for many of the reasons that r-b-j has already mentioned.  But it doesn't really matter which one I prefer, or which one anybody else prefers.  We need to find a proper way to identify and describe both conventions, explain the differences between them, why someone might use one or the other, and a bit of the rationale for each.  It is possible to do that in a single article, although it might be difficult or the article might become too long, or we could create two separate articles, one on each of the two conventions, and then a short overview article that would point readers to each of the two alternatives.
: My recomendation is that we not spend a lot of time and energy debating which definition is right or which definition is better, but rather, let's discuss how best to present both definitions without confusing the reader and in as precise and concise a fashion possible.  
:BTW, I don't know about anybody else, but I use the Laplace transform all the time.  So it is incorrect to say that nobody uses the Laplace transform!  And I do think it is worthwhile to show the connection between Fourier and Laplace.  -- Metacomet 12:28, 12 December 2005 (UTC)
replies: text: I agree with every thing you have said above. I am leaning to the idea of two articles. The remark about the Laplace transform was meant to be a semi-humorous way to make a point (please read the next paragraph after that remark). PAR 13:56, 12 December 2005 (UTC)
replies: text:  since the discussion is about different scale factor, I think the best way is to write the article only about the most general version of the transform (the "generalized" one), adding a paragraph at the end saying which are the values of the parameters most commonly used and linking to other articles, containg more particular discussions of the Fourier transforms. This way other articles about different subjects could link directly to the page about the version of the Fourier transform they are using. Such a general approach might be quite difficult: somebody with a good knowledge of math should do it. The other more particular versions of the article will be quite easy to write after the general one has been finished. Alessio Damato 19:21, 12 December 2005 (UTC)
replies: text: The resulting article will be somewhat obscure and difficult to read (even more so than now), I'm afraid.  Let me point out that this same problem has been faced by every single textbook dealing with Fourier transforms, and none of them (to my knowledge) write all of the formulas in terms of a general scale factor as you propose (in fact, you need two general scale factors: one in front of the transform, and one to convert between f and ω as desired).  Rather, they pick one convention that they happen to find convenient, and stick with it throughout — and, if necessary, they have one section which explains how to convert to other conventions.   Similarly, I think Wikipedia should stick to one convention for the most part (I don't especially care which in this case since there is no clear consensus in the literature), and then have one section clearly explaining the different conventions that are common and how to convert between them (e.g. for the table). —Steven G. Johnson 20:03, 12 December 2005 (UTC)

text: This ain't going to be easy....there are many different ways to go, I am not sure which one makes the most sense.  I do know that in any event, all of the FT-related articles need some general improvement, clarification, and clean-up (in my opinion).  But it would be good if we could reach a consesus on the general direction we want to take before investing a lot of time and effort into revising these articles. -- Metacomet 23:53, 12 December 2005 (UTC)
thread_title: Let's not argue pointlessly, let's find an effective solution

message: :I would suggest having one article that uses a single convention and derives (or at least describes) the various mathematical properties of the transform; since this article would be more mathematically oriented, I would suggest sticking with the 1/√2π convention here.  (Duplicating this three times would lead to madness.) It would also have a short section summarizing the most common conventions.  This would link to three articles for the different conventions, containing only the definitions of the transforms as well as tables of transform pairs.  —Steven G. Johnson 02:46, 14 December 2005 (UTC)
replies: text:  i could go for that.  in fact, i could go for about anything that will allow having the other two conventions (with their transform pairs) listed.  if there is some rough consensus to do what Dr. FFTW has suggested, i might just go ahead and do it.  any objections? r b-j 03:10, 14 December 2005 (UTC)
replies: text: I like Steven's suggestion. You have my vote. What names do you propose for the alternate two articles? -- Fropuff 03:30, 14 December 2005 (UTC)
replies: text: I would suggest something like Table of continuous Fourier transforms (1/2π convention) and  Table of continuous Fourier transforms (non-angular frequency).  (I wish there were a better term than "non-angular frequency", but I can't think of one.  Just "frequency" is ambiguous because ω is often called "frequency" too.)  —Steven G. Johnson 23:29, 14 December 2005 (UTC)

text:  would the existing table of continuous Fourier transforms continue to be in the present article, or would it be spun into another article like the other two?  perhaps: Table of continuous Fourier transforms (1/√2π, unitary convention), Table of continuous Fourier transforms (1/2π convention), and  Table of continuous Fourier transforms (cyclic frequency, unitary convention)?  how does that sound? r b-j 23:26, 15 December 2005 (UTC)
replies: text: "Circular frequency" or "Cyclic frequency" maybe? PAR 23:46, 14 December 2005 (UTC)

text:  Steven's suggestion is best. It seems clear that for efficiency and consistency, there should be only one theoretically oriented article, and that article should use notations and conventions geared to expressing these theoretical concepts. Separate articles can then reformulate these concepts and results in order to address the needs of a particular discipline. That has my vote. PAR 06:16, 14 December 2005 (UTC)
thread_title: Proposal #3: one main article following single convention and 3 sub-articles of transform tables 

message: I would like to suggest a different approach with two separate but linked articles.  The first article would be called  Continuous Fourier transform, and the second would be called  Continuous-time Fourier transform:
*  Continuous Fourier transform (CFT) would follow the definitions, conventions, and notation common in the mathematics literature, and would focus on the mathematical properties and origins of the Continuous Fourier transform.
::In particular, this article would define the CFT using the  unitary, angular frequency form, and would provide common transforms and properties using this form.
*  Continuous-time Fourier transform (CTFT) would follow the definitions, conventions, and notation common in the signal processing and communications theory literature, and would focus on the engineering uses, applications, and origins of the Continuous-time Fourier transform.
::This article would define two different forms of the CTFT:
::: > The non-unitary, angular frequency form  (ω in radians per second)
::: >The unitary, ordinary frequency form  (f in hertz)
-- Metacomet 19:38, 24 December 2005 (UTC)
replies: text: I prefer my earlier proposal (have one article on the transform, and multiple transform-table articles for the multiple conventions.)  First, your naming is not especially standard, and Wikipedia needs to refrain from introducing new terminologies.  Second, as a pedagogical matter, I think it is a bad idea to suggest that these are two different transforms, as opposed to different notations.  Third, the "engineering" applications of the transform have a lot of overlap with the "physics" and "mathematics" applications, and one should refrain from imposing false dichotomies.  —Steven G. Johnson 21:59, 25 December 2005 (UTC)
thread_title: Proposal #4:  two articles, one math oriented, one signals oriented

message: there are several good suggestions, we'll never agree completely. I think the only way to take a decision is to vote: let's point out which are the main choices, and anybody will say what he prefers. Alessio Damato 19:02, 13 December 2005 (UTC)
replies: text: So what would be the question to vote on?  Shall we create three articles of "equal standing" that depicts each of these definitions, and a table of theorems and of transform pairs that is correct for that specific definition, and have the main Fourier transform page "dispatch" the reader to each of these articles?  maybe someone else can formulate a better way of wording a question for voting. Perhaps a second question (i'll put in here for political compromise) is shall the one article of the three that corresponds to the existing convention in Continuous Fourier transform remain in the same page of that name? (that would obviously boost that convention to a higher standing than the other two, but is perhaps what we have to do to get a solution acceptable to most.) r b-j 01:18, 14 December 2005 (UTC)
replies: text: I believe that to this day this discussion has already ended with a good result. The article is really well presented the way it actually is now. When I studied Continious Fourier Transform, I was teached all three different forms on three different subjects, I think this is the best way to understand it and is exactly the way which the article uses: First, the mathematical unitary approach to the transform, then the non-unitary form from an engineering point of view, (including relation to the Laplace transform), and finally the frequency form. It is complete and formal. Nevertheless, I believe the article would need, as well as the initial formal mathematical approach, a less mathematical approach to the fourier transform using the "frequecy form", I would recommend using figures of simple signals in the time domain and in the frequency domain (the spectrum). I really think that an article about the continious fourier transform cannot be complete without having a single figure with a spectrum, as this is a much more intuitive way of understanding the fourier transform, much more accesible to interested people without requiring high mathematics understanding. I would recommend having a look at this page for ideas and inspiration: http://www.jhu.edu/~signals/
81.203.133.248 22:44, 9 May 2006 (UTC)
thread_title: Decide by a vote

message: This section is moot.  See history if you want.  -- Metacomet 17:28, 28 January 2006 (UTC)
thread_title: Should we create a new WikiProject?

message: This section is moot.  See history if you want.  -- Metacomet 17:27, 28 January 2006 (UTC)
thread_title:  Formal proposals 

message: apologies for not knowing the protocol for adding a comment; this information is clearly not easy to track down.
But I write to mention that there must be an error in item 16 in the table of transform pairs, if the comment that e^(-x^2 / 2) is its own transform is true.  (Probably the "a" in the denominator of the exponent of e should be replaced by a^2.) --daqu Daqu 19:40, 12 December 2005 (UTC)
replies: text: I don't think there is an error - set a=1/2 and it works out. PAR 21:47, 12 December 2005 (UTC)
replies: text:  Your proof is almost correct except that the Fourier transform of e^{iat} is  \delta(\omega - a) , not  \delta(\omega + a)  (ignoring the scaling factors). That accounts for the sign difference between your result and the one in the table. Abecedare 02:14, 1 March 2007 (UTC)
replies: text: Thanks, I knew I was being a little careless.  Thenub314 14:54, 1 March 2007 (UTC)
thread_title:  Error? 

message: This issue is closed.  See history if you are interested.  -- Metacomet 17:30, 28 January 2006 (UTC)
thread_title: Example

message: To Bob K and PAR:
You are right, and I am wrong.  After reviewing the arguments that you both have made, and after doing some reading on my own, I have concluded that the proof that I presented is flawed, and that the result is incorrect.  I am sorry that I ever brought it up, not because it is wrong (which it is), but because of the unbelievable level of hostility that some editors (not you) have shown toward me personally.  Unfortunately, the mode of operation on WP often includes insulting people's intelligence, attacking their character, and questioning their motives.  Obviously, I am exagerrating to make a point, but this particular situation is not unique.  It is all too common, and I have no interest in taking part anymore.  -- Metacomet 17:52, 27 January 2006 (UTC)
replies: text: Thanks for the closure, a very gracious jesture.  That is not sarcasm, and I never use sarcasm in writing, because once you start down that path, people often can't tell your literal remarks from your sarcastic ones.  I am sorry you are upset with Steve.  I enjoy his contributions immensely, so I am certainly willing to cut him a lot of slack for terseness.  He earns it.  He certainly means well, he is well-informed, and his advice (to read a book) was valid.  The truth sometimes hurts.  I viewed it as bluntness, not an insult.  --Bob K 18:25, 27 January 2006 (UTC)
thread_title:  The proof is incorrect 

message: This issue is closed.  It's in the history if anyone wants to see it.  Meanwhile, I have a few crow feathers to pull out of my mouth... -- Metacomet 20:40, 27 January 2006 (UTC)
thread_title:  Proof of equivalence 

message: I've done two exerxise of Riley's Physical Math.
The questions are on 
Can anyone help me?
They're ok by my teacher's help.--HydrogenSu 19:30, 29 January 2006 (UTC)
thread_title:  Two Questions about Fourier Transforms 

message: I love how this page explains the unitary/non-unitary transform pairs. It's wonderful - thank you! --HappyCamper 17:16, 23 May 2006 (UTC)
thread_title:  Wonderful! 

message: I really like how the article is, now, but I noticed that the table of Fourier transforms is growing, and it might become the main part of the article. Since a complete table of Fourier transforms can be a useful reference, I suggest to keep on improving it but moving to a different page such as Table of Contiuous Fourier transforms, keeping on the main article just a few as an example. What do you think about it?? Alessio Damato 10:40, 15 June 2006 (UTC)
replies: text: *Sounds good to me.  Also, we have created a maintenance problem regarding the row numbering.  I think it is internally consistent now, except for the missing numbers.  But there are probably some references in other articles that are out-of-date.  Is there a better way to do it?  --Bob K 12:45, 15 June 2006 (UTC)
thread_title:  table of transforms 

message: Wouldn't it be better to represent complex conjugation with a bar instead of an asterix?  Here in the US it is the usual way it is taught in high school and calculus classes.  I understand that using an asterix is more common in physics and engineering, and and has nice connections to the dual when we are working in hilbert spaces.  But I personally feel representing it as a bar makes it more readable for a wider audience.  But what do people think? Thenub314 16:25, 17 July 2006 (UTC)
replies: text: It's a bigger mess than that.  According to http://physics.nist.gov/Pubs/SP811/sec10.html, section 10.1.2, the "correct" symbol for conjugation is neither asterisk nor bar.  It is ^.  And checkbox #7 at http://physics.nist.gov/cuu/Units/checklist.html shows bar used to designate the average value of a set of numbers.  I am familiar with that usage.  I have also used it extensively (in the U.S. school system) to designate a vector.  (http://physics.nist.gov/Pubs/SP811/sec10.html, section 10.2.1 requires boldface, italic for vectors, but that is a little impractical for homework problem-solving.)  The asterisk, as you point out, is commonly used for conjugation in engineering.  But we also used it that way in the (U.S.) schools that I went to.  Maybe I am forgetting something, but I don't recall ever using the bar for conjugation.  --Bob K 03:19, 18 July 2006 (UTC)

text: Thank you sir.  Exactly right.  I should have listened to the voice in my head that was saying "Why in the world do they mention conjugation in the same breath with "p^q means p and q".  --Bob K 07:09, 20 July 2006 (UTC)

text: Me too.  But I fear that we are well below the radar here.  Your question may as well be invisible.  Good luck!  --Bob K 07:09, 20 July 2006 (UTC)
thread_title:  Notation Question 

message: A good article needs balance between its subsections. So we probably will find it necessary for good balance to outsource other subsections. — Sebastian (talk) 02:52, 1 October 2006 (UTC)
thread_title: Sections we may want to move to separate articles

message: Shouldn't the equation for the variance be \Delta^2 A\equiv\langle A^2\rangle-\langle A\rangle ^2 , rather than \Delta^2 A\equiv\langle A^2-\langle A\rangle ^2\rangle 
-- Robert L 16:31, 24 September 2006 (UTC)
replies: text: They are the same thing, since \langle x+c\rangle=\langle x\rangle+c  when c is a constant. PAR 18:13, 24 September 2006 (UTC)
replies: text: Another equivalent expression, which is arguably better because it suggests the reason for this definition of variance, is \Delta^2 A\equiv\langle (A-\langle A\rangle)^2 \rangle . —Steven G. Johnson 18:19, 24 September 2006 (UTC)
replies: text: I agree - thats the best definition, the others seem to be derivative. PAR 01:40, 26 September 2006 (UTC)
thread_title: Variance