Talk:Fourier transform

I think the table and the definition is not consistent -- there are some $2\pi$ factors that don't tally. I won't fix it for fear of making it worse .... fiddly stuff. Lupin 15:16, 12 Jun 2004 (UTC)

It's a pain to keep things consistent through convention changes in the text, but nothing is jumping out at me right now. Which rules concern you? —Steven G. Johnson 15:48, Jun 12, 2004 (UTC)

There is at least an inconsistency with the Dirac $\delta$:

"Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function 1."

Yet in the table the ${\frac {1}{\sqrt {2\pi }}}$ convention is used. Eldacan 20:57, 6 Jul 2004 (UTC)

The text is inconsistent with the FT definition used here; I'll correct it. —Steven G. Johnson 04:04, Jul 7, 2004 (UTC)

Convolution Theorem

Looks like I forgot about this page for a while :) Isn't the Fourier transform of a convolution the product of the fourier transforms, without any $2\pi$ factors? Unless you want your convolutions to have constant factors too... this appears to need fixage on the convolution page as well. Lupin 10:21, 7 Jul 2004 (UTC)

Whether you have constant factors in the convolution theorem depends upon your FT definition (and the convolution definition). With the FT and convolution definitions here, there are constant factors. Proof:

Let:

$h(t)=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )d\tau$

$f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(\omega )e^{i\omega t}d\omega$

$g(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }G(\omega )e^{i\omega t}d\omega$

Then the Fourier transform of h is:

$H(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt$

$={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }dt\int _{-\infty }^{\infty }d\tau {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega 'F(\omega ')e^{i\omega '\tau }{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega ''G(\omega '')e^{i\omega ''(t-\tau )}e^{-i\omega t}$

$={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega 'F(\omega ')\int _{-\infty }^{\infty }d\omega ''G(\omega '')\left({\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{i(\omega ''-\omega )t}dt\right)\left({\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{i(\omega '-\omega '')t}d\tau \right)$

$={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }d\omega 'F(\omega ')\int _{-\infty }^{\infty }d\omega ''G(\omega '')\left({\sqrt {2\pi }}\delta (\omega ''-\omega )\right)\left({\sqrt {2\pi }}\delta (\omega '-\omega '')\right)$

$={\sqrt {2\pi }}F(\omega )G(\omega )$

—Steven G. Johnson 17:11, Jul 7, 2004 (UTC)

Quite right, I dropped a constant in my rough calculation :-) By the way, there's a quicker proof where you don't need to use icky delta functions. The second line is obtained from the first with the change of variables $x'=x-t$.

$H(y)={\frac {1}{\sqrt {2\pi }}}\int \int f(t)g(x-t)\,dt\,e^{-ixy}\,dx$

$=\left({\frac {1}{\sqrt {2\pi }}}\int f(t)e^{-ity}\,dt\right)\left({\frac {1}{\sqrt {2\pi }}}\int g(x')\ e^{-ix'y}\,dx'\right)\cdot {\sqrt {2\pi }}$

$=F(y)G(y)\cdot {\sqrt {2\pi }}.$

Change normalization to simplify formulas?

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)

main definition

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)

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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)

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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)

P.S. Meant to include that this beautiful property of the Fourier transform -- that iterating it four times is equal to the identity transform -- is but one example of a property that holds only when the 1/sqrt(2π) constants are used. (It will not hold for any other choice of real positive constant.)

So it's not really the case that "the only important thing is that the product of the constants before the integrals be equal to 1/2π." This is true, I'm sure, for certain applications, but not true for others.Daqu 04:57, 13 December 2005 (UTC)

Generalization

I changed the wording under generalization, because the forward transform has been defined as the one resulting in F(ω). PAR 3 July 2005 02:28 (UTC)

Yep, no problem - feel free to do so anytime. I overlooked that it was previously defined earlier in the article. --HappyCamper 3 July 2005 02:53 (UTC)

Hello - I removed the properties list because they are all mentioned and defined in the "Table of important Fourier tranforms" section. All except "duality". What is "duality" in this context? PAR 3 July 2005 14:53 (UTC)

I put the list up earlier, because I think the article does not emphasize the property of linearity early enough. "Duality" refers to the application of the forward transform twice. Actually, I would like to participate in changing the organization of the 4 Fourier transform articles, where is this being done? --HappyCamper 3 July 2005 15:57 (UTC)

Right here on the talk pages is probably best. What kind of reorganization did you have in mind? I agree linearity is fundamentally important and should be more than just mentioned in the table, maybe some of the other properties as well, but other fundamental properties like orthogonality and Plancherels theorem should not be sidelined. PAR 3 July 2005 16:47 (UTC)

Reorganization

There are a few things that I have in mind, what do you think about them? (These are in no particular order)

1. Emphasize in the beginning the historical use and development of continuous Fourier transforms. Although the article Fourier transforms mentions this, it should be reiterated here. It would be worthwhile to mention the historical difficulty in placing Fourier transforms on a firm mathematical grounding. It is also worthwhile to mention its tremendous impact on engineering applications, harmonic analysis and functional analysis. The article needs to mention that in some contexts, Fourier transforms are used with the implicit assumption that most practical signals are "Fourier transformable"
2. The sufficient conditions for convergence and existence of the Fourier transform should be mentioned
3. The article should be rewritten somewhat so that it conveys a more introductory tone in the beginning, and then goes progressively deeper into the theory. For example, I don't think Lebesgue integrable functions and Hilbert spaces should be mentioned so early.
4. Mention all the basic properties of the Fourier transform earlier
   • These include linearity, time shifting, and frequency shifting
5. Move all the deeper mathematical properties and definitions to later into the article.
6. Make a table of properties which contains
   • Transform pairs, with the most common (a,b) definition in use all written out. At minimum, I feel that the definition where (a,b) = (1,1) needs to be included. Even though this material might be redundant (since all the transform pairs are proportional to each other), the article as it stands is nontheless difficult to use for someone who has grown akin to using a particular definition of the Fourier transform.
   • Table of all properties which are common to all 4 Fourier transform types. Although I understand the motivation to split the Fourier transform articles into 4 subarticles, I feel that the deep interrelations between the 4 are being obscured somewhat. We need to find a better balance for them.
7. The interpretation of Parseval's theorm in terms of energy needs to be stated. --HappyCamper 3 July 2005 18:51 (UTC)

To respond:

1. (Emphasize in the beginning the historical use and development of continuous Fourier transforms.) I agree - but only a paragraph or so. People looking for a detailed history should be looking elsewhere.
2. (The sufficient conditions for convergence and existence of the Fourier transform should be mentioned) I agree - this should go in the "completeness" section, right?
3. (The article should be rewritten somewhat so that it conveys a more introductory tone in the beginning, and then goes progressively deeper into the theory. For example, I don't think Lebesgue integrable functions and Hilbert spaces should be mentioned so early.) - I agree with that.
4. (Mention all the basic properties of the Fourier transform earlier. These include linearity, time shifting, and frequency shifting) I agree.
5. (Move all the deeper mathematical properties and definitions to later into the article) Well, we need to figure out what is "deep".
6. Make a table of properties which contains
   • (Transform pairs, with the most common (a,b) definition in use all written out) We need to keep one normalization throughout, and that has been decided (after some heated argument) to be the (0,1). This was not my favorite but it is now, since its both common and unitary. With regard to inclusion of other normalizations in the table only, I'm doubtful it wont be cluttered. I would like to see the unitary normalization for all Fourier articles, but there is resistance to that in the DFT article.
   • (Table of all properties which are common to all 4 Fourier transform types.) Agree - although I don't know the best place for it. The articles need to be kept separate, or else there will be one huge confusing article. Maybe in the Fourier transform page?
7. (The interpretation of Parseval's theorm in terms of energy needs to be stated) Strongly disagree - energy is a physics concept and the relationship needs to be drawn in the appropriate physics article, not here. Perhaps a mention, but no more.

PAR 3 July 2005 19:43 (UTC)

Thanks for your response...I'll follow suit just like the way you have it set up...

1. (I agree - but only a paragraph or so. People looking for a detailed history should be looking elsewhere.) - This was sort of what I had in mind too. An article on the History of the Fourier transforms would be nice. For these articles however, I agree that it's best to focus on their mathematical properties.
2. I agree - this should go in the "completeness" section, right? - I was actually thinking of making a little "Convergence" section, and maybe briefly talk about Dirichlet's conditions. But since the article already exists on these conditions, a mentioning in passing might be sufficient.
3. Yay, we have consensus!
4. Yay, we have consensus!
5. Well, we need to figure out what is "deep" - Let's try to answer this question like this. Who would search on Wikipedia for "continuous Fourier transforms"? I think it's reasonable that this person would likely be a university student. This person would be familiar with calculus, but not necessarily all its subtleties. Let's try to gear the article towards this person. Yes, this is sort of vague, but maybe we'll try moving some stuff around, adding new sections, and see if that works out. If not, we can always revert.
6. Make a table of properties which contains
   • (We need to keep one normalization throughout...) - Well, maybe I'll try to make a table in my own sandbox and then once I'm done with it, I'll present it here for more discussion. I'm not surprised there is a good reason why there is resistance to a parcular normalization. Sometimes a non-normalized Fourier transform pair is better suited for particular applications. Can you direct me to the discussion page where the normalization scheme was chosen? I think its reasonable to feel that choosing a particular normalization scheme is a subtle form of "POV".
   • (..although I don't know the best place for it...Maybe in the Fourier transform page?) - I was actually thinking of making a separate page for this table, since I think it would be quite big. I don't want the table to take away from the content of the main article. On the Fourier transform page, we could write maybe 2 or 3 common properties for all the transforms, and then on a separate page, list all of them.
7. (...Strongly disagree - energy is a physics concept...) - I'd be satisfied with a mentioning in passing too.

Well, it looks like to me that we basically agree on all the changes. What is next you think? --HappyCamper 3 July 2005 20:26 (UTC)

Edit away. Just this article to start. Read the whole article carefully first. Start out with rearranging the sections, then fill in one section as much as you can. Wait until tuesday or wednesday when people get back to work and see what happens (so you don't waste a lot of time editing just to get reverted, and people don't have to dig through a million edits to figure out what happened). In other words, consider the people who have contributed to this article a lot over the last few months or years, and now there's been a change they need to understand. Make it easy on them. I'm not on vacation like many, so I'll agree or complain pretty soon. Then after the dust settles, it will probably be clear what to do next.

With regard to your responses - Concerning normalization, I think the DFT talk page is the one where the discussion got pretty heated. Not here, that was wrong. With regard to the university student statement, yes, a university student should be able to get good information, and that info should be up front, but the article should DEFINITELY include the real mathematics, and not just be a book of recipes. PAR 3 July 2005 21:07 (UTC)

Of course - none of the rigorous math should be removed. Well, I'll slowly add edits in, but I'll go at my own pace. I'll make sure to put in good edit summaries :-) --HappyCamper 3 July 2005 22:10 (UTC)

Normalization and notation

An appeal to change the normalization so to be the least inconsistent with physicists and engineers.

Greetings,

I have done a few edits to this article as it is, but I do not presume to have much "ownership" of it (I know, nobody owns WP articles, but I don't want to waste a bunch of time changing things just to get it reverted). Anyway, we've been discussing this a bit at Wikipedia talk:WikiProject Electronics and we feel pretty strongly that the normalization in the Continuous Fourier transform article should, for the principal definition, toss the entire ${\frac {1}{2\pi }}\$ scaling factor into the inverse Fourier Transform (but leave the notes of other scaling conventions, including the unitary one now shown as the principal definition, and also leave that that "generalization" section). The reasons are three-fold:

1. that unitary convention with ${\frac {1}{\sqrt {2\pi }}}\$ is not common in engineering and physics texts (none that I have seen, but I certainly recognize that I do not have an exhaustive library).

2. In the cases were the angular frequency version of the Fourier transform is perferable, we should have the Fourier Transform to be as much compatible with the common double-sided Laplace transform definition as possible. The F.T. would be a degenerate case of the bilateral L.T. with little change in notation and no messy scaling:

$F_{\mathcal {F}}(\omega )=F_{\mathcal {L}}(i\omega )\$ where $F_{\mathcal {L}}(s)\$ is the bilateral L.T. of $f(t)\$

3. Even though there are some nice symmetry properties (such as duality) with the unitary convention, there remain some inelegant scaling with convolution and such operations. (The common EE convention, shown below, is both unitary and has desirable scaling properties - some of us think it is superior in elegance and compactness of expression and manipulation, but we know it is rarely used outside of the engineering discipline. That's why it will be an ancillary article.)

So the principle definition we propose for the Continuous Fourier transform would be:

$F(\omega )\equiv {\mathcal {F}}(f,\omega )=\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\,dt\$

$f(t)\equiv {\mathcal {F}}^{-1}(F,t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\ e^{+i\omega t}\,d\omega \$

We are also planning on introducing another article to link to this and to Fourier transform to present the unitary F.T. operator most often used in electrical engineering communications texts:

$X(f)=\int _{-\infty }^{\infty }x(t)\,e^{-j2\pi ft}\,dt\$

$x(t)\equiv \int _{-\infty }^{\infty }X(f)\,e^{+j2\pi ft}\,df\$

because with that symmetry of definition and no extraneous scaling factor, the use of the Duality theorem and Parseval's theorem (as well as the DC value vs. integral of transformed function theorem) are trivial. Also transforms of rectangular pulses, sinc functions, gaussian pulses, and chirp functions are trivial (and elegant). Note the different meaning of the symbol $f={\frac {\omega }{2\pi }}\$, which is non-angular frequency (likely Hz) so it is not used for the "time-domain" input to the Fourier Transform and, instead, we use $x(t)\$ and EEs use $j\equiv {\sqrt {-1}}\$ rather than $i\equiv {\sqrt {-1}}\$. But all this will be in the ancillary article we'll be making for the benefit of electrical engineers (in fields such as control theory, communications systems, and signal processing).

I know there has been some past discussion about this scaling, but the present principal definition is really a concern for us neanderthal engineers and we would like it to be a little more conventional. Can I get a little feedback on this before I put a lot of work into it? Thanks. r b-j 04:11, 12 December 2005 (UTC)

For what it's worth, I agree with r b-j. I am in favor of changing from the current convention of normalizing both the forward and inverse transforms to 1 over sqrt(2*pi) to the electrical engineering convention of normalizing only the inverse transform by a factor of 1 over 2*pi. I am also in favor of creating the ancillary article discussing the alternative form of the transform using frequency f in place of angular frequency omega such that omega = 2*pi*f. -- Metacomet 05:22, 12 December 2005 (UTC)

What you (r b-j) have said so far seems eminently reasonable to me and I can see no objections at the moment. Engineers are reasonable folk anyway (usually) and want to make things as simple as possible (but no simpler!). I dont know what reaction you may get from physicists and mathematicians tho'!--Light current 06:58, 12 December 2005 (UTC)

The $1/{\sqrt {2\pi }}$ normalization, contrary to the above statement, is probably the most common convention. (e.g. Kaplan, Weiner, etc.) Also in response to "Even though there are some nice symmetry properties (such as duality) with the unitary convention, there remain some inelegant scaling with convolution and such operations." - The duality and symmetry are the most fundamental properties of the transform, the convolution properties are not. Also, with regard to modifying the notation to conform with the Laplace transform, why do that? Nobody uses the Laplace transform, which is, by the way, a special case of the Fourier transform, not vice versa. The asymmetric normalization is for people too lazy to take a square root.

I beg to differ PAR with your statement that 'no one uses Laplace Transforms'. Electrical/electronics engineers use them all the time for solving steady state and transient network problems.--Light current 22:15, 12 December 2005 (UTC)

Ok - I'm done being aggravating. The point I am trying to make is that from my side (math-physics) you guys talk about engineers the way New Yorkers talk about New York city. There are the five boroughs, and then there is upstate (i.e. the rest of the known physical universe). There is a culture clash here, and I think that two articles may be the answer. I do think that the present article should be maintained as the math-physics angle, and not be changed to asymmetric normalization. PAR 10:30, 12 December 2005 (UTC)

Let's not argue pointlessly, let's find an effective solution

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)

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)

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)

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)

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)

Informal proposals

Proposal #1: three separate articles

how about 3 new articles:

Continuous Fourier transform (Mathematics)

$F(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\,dt\$

$f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(\omega )\ e^{i\omega t}\,d\omega \$

this would have most of the stuff from the present article.

Continuous Fourier transform (Engineering and Physical Sciences)

$F(\omega )=\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\ dt\$

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\ e^{i\omega t}\ d\omega \$

Continuous Fourier transform (Communications Engineering)

$X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-j2\pi ft}\ dt\$

$x(t)=\int _{-\infty }^{\infty }X(f)\ e^{j2\pi ft}\,df\$

all three would have their definitions clearly stated, their properties (theorems) and, finally, a table of common transform pairs. the present article would have the generalized Fourier transform and then show what pairs of a and b are used for these three different common conventions, their definitions (on the same page for comparison), and then links to specific articles. how does that sound? r b-j 02:12, 13 December 2005 (UTC)

Some good news: I believe the the second and third versions that you have presented are mathematically equivalent forms, and that the difference in the argument, meaning $\omega \,$ versus $f\,$, and the difference in scaling, meaning the extra factor of $2\pi \,$, both have a physical interpretation in terms of units of measure and power spectral density. So I believe that you could cover both of these forms in a single article that could show the equivalence of the two forms and discuss the physical meaning of each in comparison with the other. You could also use a single table with three columns, one for the time domain, the second for the $f\,$ domain, and the third for the $\omega \,$ domain. -- Metacomet 11:02, 13 December 2005 (UTC)

they're all mathematically equivalent, differing only in some small detail in scaling on the "x-axis" and/or "y-axis". but those differences mean that there are scaling differences in the table of transform pairs and the table of theorems (like duality, Parsevals, and convolution). there are real scaling differences and if you are using one definition of the F.T. and applying the table applicable to the other, scaling errors will result. i think they should be in three different articles, all dispatched from the Fourier transform or Continuous Fourier transform article (after a definition and brief explanation of the differences) and then the specific articles (whether they be in math, science, or engineering) that refer to the F.T. or C.F.T. and use some particular theorem or transform pair, will link to the specific F.T. article with the definition and result being used. r b-j 19:00, 13 December 2005 (UTC)

What exactly is the physical interpretation of the ${1 \over {\sqrt {2\pi }}}$ scaling factor? -- Metacomet 23:05, 13 December 2005 (UTC)

no "physical" interpretation, but there's a mathematical interpretation: it's whatever it has to be so that you wind up with the same x(t) that you started with after inverse transforming from X(&omega) (which came from x(t)). it come from the fact (from Fourier series)that if you integrate x(t)=1 for a normalized period (that is with the fundamental angular frequency of 1), you get 2π your Fourier series will add up to something too big by a factor of2π, if you don't divide it out. then to get the Fourier integral, you let the period go to infinity, but that 2π factor stays with you.

There is a physical interpretation in quantum mechanics. Integrating the probability density of a particle's wave function throughout space should result in 1, representing a 100% probability that the particle exists somewhere. Without the ${1 \over {\sqrt {2\pi }}}$ factor, the total probability density represented by the function in the conjugate space is no longer 1.

Yes, this is why, to a quantum physicist, the unitary representation is the only representation that has physical significance.

I think that reinforces a point that I made about a week ago, which I will repeat here:

If we decide to have a single article, then I think the article should treat all three forms of the CFT on an equal foorting. The article should state that three different forms are in common usage, and it should explain that each form differs only in terms of normalization factors and/or domain scaling. The article should present the definition of each form, one after the other, and it should identify the purpose and major applications of each form. And then it can link to three separate sub-articles, one for each form, which would contain a table of common transform pairs, and a list of key properties and theorems, both using the notation of the particular form of the CFT. This approach would be consistent with the Wikipedia policy on maintaining a neutral point of view.

-- Metacomet 04:40, 2 January 2006 (UTC)

Let me revise my previous observation. I believe that not only are the second and third forms mathematically equivalent, I believe they are equivalent period, meaning they are completely the same, identical in every respect. -- Metacomet 23:09, 13 December 2005 (UTC)

well, then, my friend, you are most certainly mistaken. "every respect" means every respect, and if you interchange some operation, like convolution in frequency domain with its counterpart with the other form, you will get a scaling error. so there is a difference (quantitative, not so much qualitative) in some respect. r b-j 01:18, 14 December 2005 (UTC)

Proposal #2: one article, three separate tables

Is it really necessary to have three different articles on the Fourier transform? It seems to me that you just need three copies of the table — one for each convention. On a side note, I dislike r b-j's proposed names. I've seen mathematicians use the $d\omega /2\pi$ convention and physicists use the unitary convention. -- Fropuff 03:40, 13 December 2005 (UTC)

well if there are three tables with quantitatively different results for some transform pair or theorem, in the same article, i might think that would ostensibly lead to confusion. it should be absolutely clear which table goes with which definition. the problem for us EE guys is that we are agitating for some overhaul in the electronics/signal_processing articles and, before undertaking that, we would like to be able to point to a contiuous Fourier Transform article that is useful to us. but it really should not be totally disconnected from the F.T. as used by mathematicians (frankly, i haven't seen the unitary F.T. in physics texts of mine, but they are 25 years old). all physics and applied mathematics texts and some engineering texts that i have show the inverse F.T. with the ¹/_{2 π} factor (possibly some say F(i ω) instead of F(ω)). other engineering texts use the X(f) convention. i know i'm not a formal mathematician (but i do mathematics for a living) and i have never seen the unitary convention in the present article anywhere except in my CRC Handbook.

i would be happy for better names for the articles (these came off the top of my head). BTW both the first and last conventions are unitary, yet not quantitatively equivalent. r b-j 04:10, 13 December 2005 (UTC)

We could put the tables in three separate articles; as in

• Table of continuous Fourier transforms 1
• Table of continuous Fourier transforms 2
• Table of continuous Fourier transforms 3

The numerical labelling avoids stereotyping a group of people.

Almost all of the physics texts that I own use the convention presented in this article. The only exceptions that I can think of is Peskin and Schroeder's Quantum Field Theory. That being said, I really do prefer the $d\omega /2\pi$ convention. Mostly because I hate the square-roots. -- Fropuff 04:23, 13 December 2005 (UTC)

Having three completely separate articles would be a huge cut-and-paste job that would be difficult to maintain in the long term. Having three tables is okay, I guess. (I wouldn't label them as "mathematics", "communications" etcetera because any claim that a particular definition is restricted to a particular field is likely to be bogus. Nor would I call them "1", "2", and "3" (which is "1"?). Label them by the the normalization: "unitary normalization", "1/2π normalization", and "non-angular frequency" or some such.) —Steven G. Johnson 06:14, 13 December 2005 (UTC)

i don't want to stereotype either. i don't see anyone else using the "non-angular frequency" convention other than people (and students) doing communication systems engineering and/or signal processing. nonetheless it is a sorta venerated convention within those groups and it is also a unitary convention (the ${\frac {1}{\sqrt {2\pi }}}$ convention is not the only unitary operator so this "non-angular frequency" convention also has a trivial expression of the Duality Theorem and moreover has the advantage of no crappy scaling factors in convolution or multiplication of functions (also, $X(0)$ is the integral of $x(t)$ and vise versa - you can't get that with any "angular frequency" convention). i would like it if some good names for these three conventions were somehow composed and have the tables of properties (or theorems) and transform pairs to go with the definition in a separate article, so there is no legitimate reason for some Joe Schmoe to mix it up. r b-j 06:44, 13 December 2005 (UTC)

Proposal #3: one main article following single convention and 3 sub-articles of transform tables

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)

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)

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)

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)

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)

"Circular frequency" or "Cyclic frequency" maybe? PAR 23:46, 14 December 2005 (UTC)

Another idea would be to use the term "ordinary frequency" when referring to frequency in hertz, and the standard "angular frequency" for frequency in radians per second. See Proposal #4 below. -- Metacomet 15:26, 26 December 2005 (UTC)

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)

If we decide to have a single article, then I think the article should treat all three forms of the CFT on an equal foorting. The article should state that three different forms are in common usage, and it should explain that each form differs only in terms of normalization factors and/or domain scaling. The article should present the definition of each form, one after the other, and it should identify the purpose and major applications of each form. And then it can link to three separate sub-articles, one for each form, which would contain a table of common transform pairs, and a list of key properties and theorems, both using the notation of the particular form of the CFT. This approach would be consistent with the Wikipedia policy on maintaining a neutral point of view. -- Metacomet 15:38, 26 December 2005 (UTC)

Proposal #4: two articles, one math oriented, one signals oriented

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), minor revision 15:06, 26 December 2005 (UTC)

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)

Thank you for your comments. My goal was to come up with naming conventions that did not identify the names themselves with a particular community or application area, but to be able to essentially move in two separate directions, one more theory/math oriented, the other more application/engineering oriented. These of course partly reflect my own biases and ideas, although I was hoping to create a platform that would allow a diverse group of functional communities to coexist peacefully and to optimize each article for particular objectives. In other words, I was hoping to find a solution that would allow all of us to have our cake and eat it too. Maybe I did not succeed, and maybe your approach would be better. I don't know, but I thought I would propose it and see what you and others think. -- Metacomet 22:18, 25 December 2005 (UTC)

BTW, the terminology "Continuous-time Fourier transform" is, in my experience, the standard terminology, and again in my experience, far more common than "Continuous Fourier transform". On the other hand, that may simply be a result of my background in signal processing and communications theory, whereas the physics and math communities may see it the other way around. -- Metacomet 22:26, 25 December 2005 (UTC)

In addition, the terms "angular frequency" and "unitary" are also quite common and considered standard. The only really new terminology that I introduced was the term "ordinary frequency" as an alternative to the term "cyclical frequency" or "circular frequency" that others have suggested. IMO, I think "ordinary frequency" is a pretty good way of distinguishing frequency from angular frequency. Again, you are free to agree or disagree, but it's an idea. -- Metacomet 22:26, 25 December 2005 (UTC)

Decide by a vote

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)

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)

Should we create a new WikiProject?

Does anyone think that there would be merit in creating a separate WikiProject devoted entirely to the topic of Fourier transforms and the Fourier series? Or is that excessive and unnecessary? -- Metacomet 23:57, 12 December 2005 (UTC)

excessive and unnecessary? Yes I think so--Light current 00:07, 13 December 2005 (UTC)

I am starting to think that maybe it is not so excessive and unnecessary – look at all of the discussion on this page alone, then go look at the discussions on DFT and DTFT and you will see that there are many many issues that need to be resolved, and each one is fairly complicated and very controversial. Setting up a project might be the only hope for achieving consensus and managing the workload. -- Metacomet 17:40, 24 December 2005 (UTC)

Formal proposals

I am creating this section as a central location where contributors can present formal proposals in a somewhat standard format, and where others can comment on each proposal separately. I have created templates for up to 4 proposals, but obviously we can add more space if necessary, or maybe there will only be one or two proposals. Whatever. If you don't like the format I have set-up, feel free to suggest improvements.

If you don't agree with someone's proposal, please express your opinion, but please remain fact-based and respectful. Also, please keep each proposal in its own separate sub-section, using the three equal signs to mark it as a Level 3 sub-header.

Important: If someone other than the author of a proposal wants to comment on the proposal, please use only the sections marked Advantages, Disadvantages, or Discussion to add your comments. Please do not change the section marked Concept, which should be reserved for the original author of the proposal only. As always, please sign your comments (using four tildes for name, date, and time).

Also, you can use three tildes after Proposed by to sign your name and five tildes after Date to add the current date and time.

Thanks.

-- Metacomet 12:50, 14 December 2005 (UTC)

sorry, Meta, but until some names of the new articles get more nailed down, i am not sure how to word a formal proposal. just hang on, it looks like something will get settled informally just above. r b-j 23:20, 15 December 2005 (UTC)

Proposal #1

Proposed by:

Date:

Concept:

Advantages:

Disadvantages:

Discussion:

Proposal #2

Proposed by:

Date:

Concept:

Advantages:

Disadvantages:

Discussion:

Error?

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)

I don't think there is an error - set a=1/2 and it works out. PAR 21:47, 12 December 2005 (UTC)

Alternative forms

angular frequency $\omega \,$ (rad/s)	unitary	$X_{1}(\omega )\equiv {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\,dt\ ={\frac {1}{\sqrt {2\pi }}}X_{2}(\omega )={\frac {1}{\sqrt {2\pi }}}X_{3}({\frac {\omega }{2\pi }})\,$ $x(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }X_{1}(\omega )\ e^{i\omega t}\,d\omega \$
	non-unitary	$X_{2}(\omega )\equiv \int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\ dt\ ={\sqrt {2\pi }}\ X_{1}(\omega )=X_{3}({\frac {\omega }{2\pi }})\,$ $x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X_{2}(\omega )\ e^{i\omega t}\ d\omega \$
ordinary frequency $f\,$ (hertz)	unitary	$X_{3}(f)\equiv \int _{-\infty }^{\infty }x(t)\ e^{-j2\pi ft}\ dt\ ={\sqrt {2\pi }}\ X_{1}(2\pi f)=X_{2}(2\pi f)\,$ $x(t)=\int _{-\infty }^{\infty }X_{3}(f)\ e^{j2\pi ft}\,df\$

-- Submitted by Metacomet 23:41, 3 January 2006 (UTC)

Modified by Bob K

Nice looking table. --Bob K 02:11, 4 January 2006 (UTC)

I am not sure that the scaling factors you have used in converting from one form to another are correct. For example, I am pretty sure that the scaling factor in converting between Form 2 and Form 3 should be 2π, not 1. -- Metacomet 14:55, 10 January 2006 (UTC)

No problem. For instance, the definition of function $X_{2}(\omega )\,$ must hold for any value of $\omega \,$, including $\omega =2\pi f\,$. It's just a simple substitution, and voilà, the result is the very definition of $X_{3}(f)\,$:

$X_{2}(2\pi f)=\left.X_{2}(\omega )\right|_{\omega =2\pi f}\equiv \int _{-\infty }^{\infty }x(t)\ e^{-j(2\pi f)t}\ dt\equiv X_{3}(f)$   Q.E.D.

Note that the variable of integation $(dt)\,$ does not change. (Were you thinking otherwise?)

--Bob K 19:03, 10 January 2006 (UTC)

The end result is not correct, and I believe the error in the math has to do with the chain rule. I am pretty sure (95 percent) that the correct result is:

$X_{2}(\omega )=2\pi X_{3}(f)\,$

where

$\omega =2\pi f\,$

I am working on a simple proof of this result, but it is not quite finished yet.

The scaling factor is also related to conservation of energy and Parseval's theorem.

-- Metacomet 19:58, 10 January 2006 (UTC)

Example

Suppose we have a signal

$x(t)=cos(\omega _{o}t)\,$

Then Form 3 of the CFT is:

$X_{3}(f)={\delta (f+f_{o})+\delta (f-f_{o}) \over 2}$

whereas Form 2 of the CFT is:

$X_{2}(\omega )=2\pi \left[{\delta (\omega +\omega _{o})+\delta (\omega -\omega _{o}) \over 2}\right]$

$=2\pi X_{3}(f)\,$

-- Metacomet 20:17, 10 January 2006 (UTC)

Conservation of energy is the very principle that you have violated by equating $\delta (\omega +\omega _{o})\,$ with $\delta (f+f_{o})\,$. But you might have to go back to the definitions of $\delta \,$ as a limit function to convince yourself.

So what is it about my iron-clad proof that you don't understand? You can avoid the limit functions by accepting my trivial proof in conjunction with your $X_{3}(f)\,$ and $X_{2}(\omega )\,$. (Because the limit functions are already embedded in the transforms.) Then it easily follows that:

$2\pi \left[{\delta (2\pi f+2\pi f_{o})+\delta (2\pi f-2\pi f_{o}) \over 2}\right]={\delta (f+f_{o})+\delta (f-f_{o}) \over 2}$

and of course:   $2\pi \cdot \delta (2\pi f+2\pi f_{o})=\delta (f+f_{o})\,$

--Bob K 21:43, 10 January 2006 (UTC)

A different table

Fourier Transforms... definition & conversion chart

	input
	$x(t)\,$	$X_{1}\,$	$X_{2}\,$	$X_{3}\,$
$X_{1}(\omega )=\,$	${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\,dt\$		${\frac {1}{\sqrt {2\pi }}}X_{2}(\omega )\,$	${\frac {1}{\sqrt {2\pi }}}X_{3}({\frac {\omega }{2\pi }})\,$
$X_{2}(\omega )=\,$	$\int _{-\infty }^{\infty }x(t)\ e^{-i\omega t}\,dt\$	${\sqrt {2\pi }}\ X_{1}(\omega )\,$		$X_{3}({\frac {\omega }{2\pi }})\,$
$X_{3}(f)=\,$	$\int _{-\infty }^{\infty }x(t)\ e^{-j2\pi ft}\ dt\$	${\sqrt {2\pi }}\ X_{1}(2\pi f)\,$	$X_{2}(2\pi f)\,$
$x(t)=\,$		${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }X_{1}(\omega )\ e^{i\omega t}\,d\omega \$	${\frac {1}{2\pi }}\int _{-\infty }^{\infty }X_{2}(\omega )\ e^{i\omega t}\,d\omega \$	$\int _{-\infty }^{\infty }X_{3}(f)\ e^{j2\pi ft}\,df\$

--Bob K 02:11, 4 January 2006 (UTC)

Proof of equivalence

Never mind, needs more work... -- Metacomet 00:02, 4 January 2006 (UTC)