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Given that this method can only be feasibly useful in the era of computers, I did spend a long time wondering what the hell Fourier was up to when he came up with this stuff. He was hardly making mp3s.
So I eventually learned that it's some stuff to do with steam engines and working out why they keep exploding.
This article needs a history section for others curious why some ancient mathematician would come up with an algorithm which isn't feasibly useful without a computer! —Preceding unsigned comment added by 86.154.39.2 ( talk) 20:47, 8 September 2010 (UTC)
I have added a brief reference to characteristic functions in probability theory. I think my brief reference should be expanded to include the convention for characteristic functions used in that wiki article in the table of other conventions. However, this is problematic due to differences in notation between the two articles. The notation can be made to align only in the case when there is a probability density function. JohnQPedia ( talk) 21:29, 26 June 2009 (UTC)
There is already something on the discussion page of Characteristic_function_(probability_theory) that is related to this. There you will find comments to the effect that it is more intuitive to interpret the characteristic function as being acquired by taking the inverse Fourier transform of the probability density function (in contrast to the wiki article concerned, which says this is acquired by taking a Fourier transform). However, there are two points. First, even under that interpretation, the (forward) transform is still different from any of the three listed in the other conventions sections here, because under this interpretation it is the (forward) transform that is multiplied by , rather than the inverse. And second, some authors of textbooks currently in use may in fact refer to the characteristic function in probability theory as being produced by a Fourier transform (perhaps such as those listed in the references of that page, but not adequately cited inline), rather than by an inverse Fourier transform. In this case, I think the wiki should reflect actual usage. In either case, the other conventions section is incomplete with respect to characteristic functions in probability. JohnQPedia ( talk) 01:06, 27 June 2009 (UTC)
In its article Characteristic Function, the MathWorld website confirms the interpretation of a characteristic function in probability as the Fourier transform of the probability density function, rather than the inverse Fourier transform. (Despite a compelling intuition to prefer an inverse Fourier transform, the mathematics are unaltered whether one considers a characteristic function to be acquired by a Fourier transform or an inverse Fourier transform.) Its article Fourier Transform also clears up any confusion, by explicitly saying that there are a number of conventions in widespread use. It lists five. It also gives explicit formulae for the Fourier transform and its inverse for arbitrary choices of constants. I have reproduced equations (15) and (16) from that page here
I put these here to indicate the kind of generality that could accompany the table in the section on other conventions. I believe such general formulae should be in the section for two reasons. First, it would make it clear how the specific examples in the table are derived. Second, it would provide a tool for the reader to interpret any sources that use conventions not listed in the table.
A further point is that the examples in the section on other conventions are generalized to apply to multidimensional integrals over . This means that a heavy edit is required to expand that section, while maintaining completeness of the table. (The MathWorld formulae given above are of course only one-dimensional.) I also think the multidimesional notation could be unclear to readers who are only up to speed on the most common circumstance, when the integral is one dimensional, and expressed in the form , rather than . This makes the table not generally accessible, and that undermines its utility. I suggest separate entries for the case when the integrals are one dimensional.
Thanks, all. JohnQPedia ( talk) 03:17, 27 June 2009 (UTC)
As an aid to the general public I propose to modify the first paragraph (I don't see how) in the following way:
"In Fourier transform, both domains are continuous and unbounded. Time intervals may be arbitrarily short and frequencies arbitrarily high. These infinite limits demand erudite mathematical tools not required in Fourier analysis. An example of an application more readily approached by Fourier analysis would be the inverse filter -- a band limited filter demanding an inverse of infinite bandwidth." —Preceding unsigned comment added by Jclaer ( talk • contribs) 03:37, 3 July 2008 (UTC)
Hi all. I came here looking for a transform pairs table because I'm studying communication systems and ended up wondering why some Fourier transforms differed from the ones that I have tabulated in my textbook. I didn't understand why the 2pi factor was under a square root!!
Anyway, I freaked out, googled a lot and finally read the article (it is kinda intimidating) finding out the answer my self (good job!).
What I would like to notice is that it takes a while to locate and understand the 3 different conventions thing, and I think that it could be more simple if we give the reader a big picture wise look of that in the "contents".
For instance, putting it like:
1 Definitions
1.1 Communications and signal processing (f) convention
1.2 Mathematics angular frequency (2pi) convention
1.3 Mathematics angular frequency (2pi unitary) convention
Finally, like some textbooks do, It would be really simplifying to say "in the rest of this article, we will use convention {whatever} unless stated otherwise". And stick to that in order to preserve consistency.
I would also like to say that I agree with Feraudyh (disgruntled editor) that this article is really hard to read for students like me, it seems really different from what we have on the communications textbooks. I understand that is a mathematical thing and it should be the most formal possible, but still I guess it could be more readable.-- Dhcpy 01:41, 1 December 2007 (UTC)
Why do we have a section on " # 4 Properties" and a section " # 6 Fourier transform properties". Maybe we should merge #6 into #4. Thoughts? Thenub314 00:28, 22 February 2007 (UTC)
Bo says: "By first conjugating the function, the forward transform equals the reverse transform, a so-called involution". But the forward transform produces and the reverse transform operates of In order for Bo's claim to be true, the reverse transform would have to operate on the same thing produced by the forward transform. I see no value in this involution idea. It's just smoke and mirrors.
-- Bob K 01:25, 8 March 2007 (UTC)
Bo, Ignore the above comments : both Bob and I realized too late what you meant. :-) Abecedare 02:20, 8 March 2007 (UTC)
Exactly! Sorry for all my recent reversions on the article and talk page. I have tried to rewrite the word statement to make it clear that the Fourier transform can be deined in a form that is an involution. Please check if the point is now clear. Also I reverted to using the "overline" convention for conjugation, since it is used in the rest of the page and more importantly * is used for the adjoint operator (and AFAIK, there is no other "standard" notation for the latter). Regards. Abecedare 02:28, 8 March 2007 (UTC)
No mathematical definition is standard, as no standardization organizations standardize mathematical definitions. The transformation is obviously a conjugate-linear map. That it is also an involution was just checked by editors Abecedare and Bob K. This is sufficient documentation for correctness. Steven G. Johnson's removal of the definition is a violation of Wikipedia:Resolving disputes # Avoidance. The value of the definition is that it avoids the risk of confusing forward and reverse transformation. That's why I want to share it. I'll leave it to Abecedare and Bob K to reinstall it if they like it too. Bo Jacoby 06:53, 8 March 2007 (UTC).
Thank you, Abecedare. It is not the definition, but it is a useful variant form. I agree that the nice properties are well-known and trivially derived, so there is no question about reliability, which is the reason behind the general ban against original research in wikipedia. It may be considered original research on my part, but not on yours, so I cannot reinstall it, but you can. Just refer to me. Bo Jacoby 08:05, 8 March 2007 (UTC).
I am grateful that my humble observation is honored by the name of original research. I approve Bob K's formulation: "a non-standard type of Fourier transform that is an involution, which means that the inverse transform is the same operation as the forward transform". This does not make any promises of gold-plated mainstream references but gave a straightforward and clear explanation. The corresponding FFT is foolproof because only one procedure is needed for transforming forwards and backwards. For real x(t) it coincides with one of the other definitions. Too bad that only very few people got the chance to read before removal. I can do no more about it. Any of you are welcome to undo the edit "04:56, 8 March 2007 Stevenj" when you realize that our readers deserve the best. Bo Jacoby 13:53, 8 March 2007 (UTC).
Bo, I don't know if your transform is an involution or not. I agree that it is kind of cool and interesting, but I don't see how it furthers the cause of this article, which is to help people understand the standard Fourier transform(s). In fact, I think it is counterproductive. I wish there were fewer forms, not more. If indeed yours is an involution, then why not add your transform to that article?
-- Bob K 16:00, 8 March 2007 (UTC)
See the definition in involution is simply: "a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f". Bob K checked that the variant 4 of the fourier transform satisfies this condition, just above. The forward and backward transformation is the same. There are many involutions: the identity, the sign change, the reciprocal, the complex conjugation, &c. Bo Jacoby 16:33, 8 March 2007 (UTC).
The variables f and t are not in the same domain, but the functions X and x are in the same domain. The fourier transformation transforms x into X. The transform maps the function x of time to the function that maps the frequency f to the integral
In this expression both t and f are dummy variables. Other variable names can be substituted for t and f. They may even be swapped:
Using you get
So the transformation that transforms a function of t into a function of f also transforms a function of f into a function of t. Bo Jacoby 22:38, 8 March 2007 (UTC).
According to the article Function, a transform is also a function. It is a very general concept. The domain need not be a set of numbers, but can be a set of other functions too. Bo Jacoby 00:12, 9 March 2007 (UTC).
And besides, a brand new definition does not help people understand "Fourier transform". It's bad enough that we already have three to worry about. A fourth would be counterproductive. Why not use the involution article to showcase your involution? Why here? This article is not about involutions.
-- Bob K 04:01, 9 March 2007 (UTC)
Right. Point taken. -- Bob K 15:26, 9 March 2007 (UTC)
I propose the following section to replace sections Fourier_transform#Definition and Fourier_transform#Normalization_factors_and_alternative_forms. One thing that bothers me about the current version is the abrupt appearance of the factors. I think it can be done in a more natural looking progression.
Another thing that bothers me is the paragraph:
Wouldn't it be more logical if the components being recombined had amplitudes equal to A(·)? Well this way they do:
-- Bob K 00:46, 11 April 2007 (UTC)
There are several common conventions for defining the Fourier transform of a complex-valued Lebesgue integrable function, In communications and signal processing, for instance, it is often the function:
When the independent variable represents time (with SI unit of seconds), the transform variable represents ordinary frequency (in hertz). The complex-valued function, is said to represent in the frequency domain. I.e., if is a sufficiently smooth function, then it can be reconstructed from by the inverse transform:
Other notations for are: and .
The interpretation of is aided by expressing it in polar coordinate form: , where:
Then the inverse transform can be written:
which is a recombination of all the frequency components of Each component is a complex sinusoid of the form ei 2 π f t whose amplitude is A(f) and whose initial phase angle (at t = 0) is φ(f).
In
mathematics, the Fourier transform is commonly written in terms of
angular frequency: whose units are
radians per second.
The substitution into the formulas above produces this convention:
which also happens to be a bilateral Laplace transform evaluated at s = i ω.
The 2π factor can be split evenly between the Fourier transform and the inverse, which leads to another popular convention:
A transform whose inverse has the same multiplicative factor is called a unitary transform.
Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.
I like your version and support replacing the "Definitions" and "Normalization Factors and alternative forms" by the one above. Some comments:
Aside: I just noticed that properties section was missing the factor without which the all the completeness, parseval's convolution etc results in the section were incorrect! I have corrected that now. Abecedare 01:36, 11 April 2007 (UTC)
Why is the inverse fourier transform true?
now what can I do? How can I reduce the right side of the equation so that it equals the left? —Preceding unsigned comment added by Iownatv ( talk • contribs)
Thanks Iownatv 20:20, 5 May 2007 (UTC)
Would it be appropriate to add a heuristic "derivation" of the fourier integral and inverse as a limiting process of the exponential Fourier series of an arbitrary periodic function to aperiodic by letting the period tend to infinity? This at least makes it somewhat intuitive and plausible, before one goes on to the convergence and existence conditions and all of that. DivisionByZer0 20:35, 14 June 2007 (UTC)
What about the range and domain of the Fourier transform? The article does not say anything about which functions (or generalized functions) can be transformed or what is the range of the transform. -- KYN 20:52, 3 August 2007 (UTC)
Ok, I don't expect a complete presentation on this issue, but at least the very basic could be presented in the article. Here is a start:
Something along these lines anyway. -- KYN ( talk) 07:59, 24 April 2008 (UTC)
With incompatible I simply mean that "tempered distributions" and "L^2" are sets of generalized functions which are defined in two completely different ways and even if some elements in one set can be identified with elements in the other, this is (as far as I understand) not the case for all elements of both sets. -- KYN ( talk) 20:58, 29 April 2008 (UTC)
I see, I was not aware of this correspondence between L2 and tempered distributions. Is it mentioned anywhere in Wikipedia? To me, it appears to be a sufficiently interesting fact to mention in the Lp or the tempered distr articles. -- KYN ( talk) 10:00, 1 May 2008 (UTC)
Many, particularly engineers and physicists, may be interested in only the real case and may be confused by the use of complex exponentials instead of trigonometric functions. The MoS guideline Wikipedia:Make technical articles accessible says that articles should begin with easier sections the reader can relate to. I suggest that a section using sines be added to the beginning before going on to complex exponentials. Loom91 20:15, 13 August 2007 (UTC)
hi! does somebody know where i can find a proof of
thanks! -- 217.117.224.177 09:57, 21 September 2007 (UTC)
This talk page is now so long that I believe its practical use is reduced. Therefore I propose to archive the earlier parts of this talk page, starting from 2004 until 2005, leaving Talk:Fourier transform#Alternative forms as the first posting that remains in the talk page. I propose to use the "Subpage archive method/cut and paste procedure" described here. In principle I would like to archive a larger chunk, let's say including all 2006 postings, to make the page more manageable but there appears to be some discussions which were initiated in these early postings but still have attracted answers or comments in 2007. Any optinions? -- KYN 15:21, 23 September 2007 (UTC)
All discussions started 2006 or earlier have now been moved to "Archive 2" which, together with "Archive 1", now is reachable from the Archive icon at the top of this page. If there are active threads which have been removed, please reinsert them on this page. -- KYN 09:56, 1 October 2007 (UTC)
According to my signals and systems book (Signals and Systems by Oppenheim, Willsky and Nawab) the fourier transform for the signal is . However the wikipedia article says that the fraction in front of the sum is . I changed this a few days ago, but it was changed back, so I thought I'd start a discussion about it. There is a pretty good example in my book that shows the derivation of this (p. 299) so I'm pretty sure that square root shouldn't be there, but obviously someone thinks it should be there. —Preceding unsigned comment added by Fkp1 ( talk • contribs) 21:31, 31 October 2007 (UTC)
Unless I am mistaken, is a square-integrable function when . Why then is this Fourier transform pair in the distribution section?
Butala 13:46, 1 November 2007 (UTC)
I removed this from the "Definitions" section and reproduce it here because that is where it belongs
Feraudyh 10:40, 6 November 2007 (UTC)
Dear Feraudyh. I completely agree with you. The article should be readabel to you. I regret that it is not. I have tried to mend it, but I am opposed by another WP-editor who believes that he can do better and that WP should contain 'standard' mathematics even if that is incomprehensible mathematics. See: User:Bob_K#two_articles_for_discrete_fourier_transform_.3F. I am sorry. Keep complaining. It helps. Bo Jacoby 13:49, 6 November 2007 (UTC).
I'm afraid I have to side this time with the non-mathematicians complaining about the (lack of) accessibility of this article. It is true that there is still a huge scope to improve the presentation – and not only for the benefit of non-mathematicians but mathematicians as well. Without any attempt at completeness, I would like to make the following observations and make a few suggestions:
The above may look harsh, but it is definitely not meant as critique of the editors that have worked on the article. However, in order to move the article (rather the topic) to a new level I'm afraid that something like the above should be addressed. In other words, work remains to be done. Stca74 14:24, 7 November 2007 (UTC)
I am new to Wikipedia and am not sure that this is the proper way to contribute to the discussion on this talk page. I hope so!. To start, I was drawn in by Googling the Wiki page on FTs. I was, frankly, very disappointed in the material presented. I am a math/physical scientist professional and was looking for something a little more entertaining and, for the layman or number theorist, more inspiring than a Xerox of a textbook. Been there. I agree wholeheartedly with the "disgrundtled editor" in his comments, and the others who followed, that Wiki should, in my phrasing, be a conduit to the community to the understanding of higher, interesting concepts. This "window" into a fascinating world is guided by those of us, in any field (math, history, music, etc.), who have commited their energies to gaining a deep understanding of their field, can now gladly offer their insights over the remarkable new medium of the Internet. Kind of a win-win situation.
My own thoughts on the FT presentation will be brief since this whole business seems to have such tremendous inertia. Other than reiterating my intial comment that it doesn't really do any one any good to regurgitate textbook material on Wiki ( send them to a couple of decent web sites
http://www.youtube.com/watch?v=QQbEOjGOY5I
or sic them on Bracewell's book. Try to come up with some original material. My own personal approach to teaching the idea of the FT and transforms in general is the idea of projections and orthogonality. Vectors projecting on the x-y plane leading to random functions projecting onto orthogonal periodic functions (sin and cos). Etc.
Randy Patton —Preceding unsigned comment added by Randypatton ( talk • contribs) 06:19, 31 March 2009 (UTC)
This new section is important, but somewhat confusing. The dash in 'aperiodic-discrete' is confusing. It perhaps means that the function is aperiodic and discrete. Why not write that? As to "The Fourier Transform for this (Aperiodic-Continuous) type of signal is simply called the Fourier Transform". No, it is called the fourier integral. The reader wants to know a reason for this trouble; why not a single fourier transform to cover these cases? The problem is that a 'discrete function' as the limit of continuous functions does not exist, and the solution in terms of distributions is too advanced. Bo Jacoby ( talk) 12:08, 20 January 2008 (UTC).
I've cited this article, chosen pretty much at random as many others are similar, as an example of one that I consider useless in that it is written at too high a level to be generally accessible, in an a discussion I have sought to provoke about how Wikipedia's mathematical articles are structure in the Village pump/Policy area in the Community Portal.
David Colver ( talk) 15:15, 2 February 2008 (UTC)
The text says "If x is Hölder continuous, then it can be reconstructed from by the inverse transform". The linked page on Hölder continuity tells me that anything bounded is Hölder continuous with exponent 0. But this includes bounded discontinuous functions, which cannot be reconstructed from the inverse transform. It appears to me that one of the two pages (and I don't know which) must be in error. —Preceding unsigned comment added by 80.177.154.109 ( talk) 12:33, 7 February 2008 (UTC)
Your correct there is a lot of confusion around this sentence. Not every bounded function has a Fourier transform defined by integration. More general methods for defining the Fourier transform could be applied, but they also have to be applied to define the inverse transform. I believe the sentence here is means of f is Hölder with positive exponent, then the resulting function is integrable and the integral defining the inverse transform converges almost everywhere to f. But this seems to be there for the sake of adding some conditions to the inversion formula. I feel this this it is a bit early in the article for this level of detail. Perhaps it would be better to say "Under suitable conditions on F, f may be reconstructed from F by the inverse transfrom:" Thenub314 ( talk) 14:12, 28 August 2008 (UTC)
Folks from the popular 4chan boards and other sites are targeting the LaTeX source in higher math articles for minor vandalism, I could see this hurting homework grades and exam prep for quite a few students, something to keep an eye out for.
search google for "Vandalize_Every_Equation" to see a link to a wiki on another popular site with details (it's blacklisted on wikipedia's spam filters so I can't link to it directly)
140.247.10.17 ( talk) 06:04, 16 February 2008 (UTC)anonymous
Sorry, but I'm not sure that Bessel function are square integrable (the norm L2 of its transform diverge). If is not so, please tell me why? (Sorry if I don't sign this message) --penaz
'THERE ARE ERRORS IN THE EQUATIONS FOR MEAN AND VARIANCE OF FOURIER TRANSFORM INTEGRALS!!. YOU MUST DIVIDE THE FIRST AND SECOND MOMENTS(The Equations currently given) BY THE NORM TO OBTAIN THE CORRECT RELATIONSHIPS.' 71.254.7.2 15:06, 17 June 2007 (UTC)
I think defining the heavy side function at this point breaks up the flow a little. It makes this entry different form the others in this section. I think for the sake of simplicity it is best make the point there is a convolution here. Thenub314 ( talk) 03:37, 25 April 2008 (UTC)
I would like to raise some questions about notation that I was thinking about in reading this article. Particularly in defining the Fourier transform we use x(t) as the signal and f. By the time we are done with the section we are ω. And starting in the second section we are using f as a function. And when we discuss multi-dimensional Fourier transforms x becomes the variable. I would like to move that we change the definition. At the very least let's call the function and variable in the first definition something else. I of course would like to say more about the use of "hat's" e.g. and (we mention the first, but don't say anything about the second.) I feel this notation is a standard in mathematics, and not uncommon in Engineering or Physics. Back to the main point, for the sake of first year calculus students that read this page, I would like see the first definition changed. Thenub314 ( talk) 02:52, 14 May 2008 (UTC)
Perhaps the following way of putting parentheses is nicer?
I calculated the Fourier transform of 1/x in two ways, and I didn't get the minus sign out front. Can anyone confirm this? —Preceding unsigned comment added by 59.167.144.142 ( talk) 11:50, 28 August 2008 (UTC)
Okay I just realized the table uses a different definition of the Fourier Transform, problem solved. —Preceding unsigned comment added by 59.167.144.142 ( talk) 12:22, 28 August 2008 (UTC)
and are undefined. And according to Poisson_summation_formula, is not the unitary transform.
-- Bob K ( talk) 20:15, 31 August 2008 (UTC)
Dear Frequent Editors,
I am both a user of Fourier transforms in medical imaging and signal processing, and an occasional instructor to students with various backgrounds in mathematics (medical students are usually on the less mathematical end of the spectrum). I can empathize with some of the comments from previous posters about the need for a conceptual (i.e. non formulaic) section introducing the Fourier transform.
Here are a few pages with examples:
http://www.e-mri.org/image-formation/fourier-transform.html
http://www.e-mri.org/image-formation/2d-fourier-transform.html
Another useful analogy is the comparison to musical notation (although strictly speaking this is more like a joint time-frequency transform).
And a very good reference for Fourier transform related identities:
Linear Systems, Fourier Transforms, and Optics
By Jack D. Gaskill
http://books.google.com/books?id=ZcUPAAAACAAJ
Curtcorum ( talk) 04:22, 9 September 2008 (UTC)
Here is a possible transform pair example. Heavy line is real component. Dashed is imaginary. I am not sure if it is better to show axes to scale or not.
-- Curtcorum ( talk) 03:15, 13 September 2008 (UTC)
Here is a clarification:
and is essentially table entry #205 with an additional frequency shift.
I think showing the negative time axis is a good idea. Also showing the formulas? An abbreviated table of transform pairs with graphical examples might be good? Is there a link to one online that anybody knows of? Curtcorum ( talk) 22:07, 15 September 2008 (UTC)
This article is apparently on a list of Wikipedia articles to be released on DVD. But we have a few maintenance tags to take care of. The deadline is October 20th. The two tags ask us to clarify and use inline citations. The too technical tag came after an ip user added the statement that every thing after the lead was for "advanced mathematicians". The nofootnotes tag was added back in November of 2007. I would like to make the article more accessible. Do people have suggestions as what the best changes might be? Thenub314 ( talk) 11:07, 19 September 2008 (UTC)
Here are some comments that I am collecting that might help, and I invite comment. The first sentence in the definitions section is:
Could be shortened to:
And we cite "A Friendly Guide to Wavelets" Gerald Kaiser, which has a nice discussion of the different conventions on pages 31-32.
Also the suggestion is made in our rating that:
And I think I agree with this comment.
The section on "Generalization" doesn't really generalize, but just shows how to choose different constants. It does not cite a references, and since it follows by change of variables I don't know of any references to cite for it. At best it is not very enlightening and at worst it is "original research". I am in favor of killing the section, but if we don't we should at least reference it and move deeper into the article. Thenub314 ( talk) 16:44, 21 September 2008 (UTC)
Another thought to simplify the article for the reader would be to stick with one convention throughout the article and simply have a section on the different notational conventions that one finds in the literature. Presenting them all back to back in the definitions sections seems like it may be a lot to absorb at once. Thenub314 ( talk) 18:19, 21 September 2008 (UTC)
Now that we have eradicated all semblance of typical signal processing notation, i.e. and (instead of ), do we just stop here? It seems to me that an encyclopedia should try to represent the signal-processing perspective, because that is why a lot of readers will come here. Do we now create article Fourier transform (signal processing)?
-- Bob K ( talk) 19:50, 8 October 2008 (UTC)
That might not be what other signal processing editors want. So I'm asking. It's not what I want, but if I'm the only one who cares, then heck with it.
-- Bob K ( talk) 01:16, 9 October 2008 (UTC)
The multi-dimensional form is no longer an issue (for me), because the article no longer leads with it (as it did 15 months ago). So I don't think the current objections are about that. But in a "signal processing" article, with this one as backup, I would:
-- Bob K ( talk) 21:38, 22 January 2010 (UTC)
This bit is unclear to me: If ƒ is a function for [−L/2, L/2], then for any T ≥ L we may.... There doesn't seem to be any other reference to the quantity L; is it needed? -- catslash ( talk) 12:48, 9 October 2008 (UTC)
Your correct, that sentence makes no sense. I have tried to fix it, it simply should have said that ƒ is a function supported in [−L/2, L/2]. It is only implicitly used in the equality . Thenub314 ( talk) 13:08, 9 October 2008 (UTC)
Many people are familiar with the rising and falling "waterfall" bar displays on a stereo, showing a real-time view of the music spectrum, from bass (low-frequency) on the left, to treble (high-frequency) on the right.
However, the audio signal being amplified and fed to the speakers is simply the intensity ("loudness") of the music at any given instant - older audio equipment had just a needle display ("VU meter") that represented the sound volume -- and the actual signal -- at any given time.
So if the tuba hits a note at the same instant as the piccolo, there's only one value of the signal at that moment, which is what the needle shows -- and the speaker plays! How does the display "know" to make both a left bar and a right bar - but not much in between - rise sharply?
The Fourier transform can be thought of as a "black box" which, given the input of that stream of single values representing sound volume over time, produces a stream of many values, simultaneously: the level of each frequency present at that moment in time. So - even though the original signal had the input from the tuba and piccolo added together into a single value, the Fourier transform can tease apart the low notes from the high notes, much as humans can easily zero in on a single conversation at a noisy cocktail party.
The reason it's called a transform is that it transforms the information contained in a single signal value of intensity, which varies over time (called a "time-domain representation") into a set of values representing the distribution of frequencies at each moment (hence: "frequency-domain representation").
At the risk of encouraging conflict, may I point out that the WP:MOSMATH guideline on the question of TeX versus HTML appears to be the unilateral decision of User:Oleg_Alexandrov. See Help:Displaying a formula for a more balanced assessment. -- catslash ( talk) 00:13, 17 October 2008 (UTC)
109 Fourier Transform of
but
109
If we put in all the values the comes in the numberator not denominator. Please check and let me know
Wilkn ( talk) 21:54, 21 October 2008 (UTC)Wilkn@yahoo.com
I think the entry is correct, unless I made a careless error. We can calculate the Fourier transform of by:
Let me know if this answers your question. Thenub314 ( talk) 17:55, 22 October 2008 (UTC)
Thank you for your response. I was wrong sorry about that. I am new to wikipedia so please use your best judgement to delete the page or not. My research is centered on FT, would it be possible for you to give me your contact information ? Wilkn ( talk) 00:58, 23 October 2008 (UTC)Wilkn—Preceding unsigned comment added by Wilkn ( talk • contribs) 19:21, 22 October 2008 (UTC)
(in the section Multi-dimensional version)
I have tried to clarify the statement about the fact that the indicator function of the ball is not a multiplier for Lp, by introducing fR defined by
but it is not satisfactory: the real thing is that for some f ∈ Lp, this function fR is simply not in Lp, so the question of fR converging to f is pointless. I don't exactly see what the original intention was. Does anybody know? Bdmy ( talk) 14:31, 29 November 2008 (UTC)
I didn't even realize it was an english word! (And I looked it up, and it is) In some ways the use of this word is brilliant, but I feel the word is fairly uncommon. Is there any objection to me trying to rework this sentence? Thenub314 ( talk) 20:19, 23 December 2008 (UTC)
In the table of Fourier pairs, the third and fourth column refer to conventions of the Fourier transform that both employ angular frrequency but differ by their normalization constant. I do not really understand, why different symbold (omega) and (nu) are used to denote the angular frequency. Wouldnt it be clearer to choose consistent notation here? —Preceding unsigned comment added by 91.66.240.116 ( talk) 11:36, 29 December 2008 (UTC)
I am used to epsilon denoting infinitesimally small real values in "epsilon-delta proofs" and the like, and was therefore a little confused by the formulas in section "Definition". If this isn't standard notation (and to me it seems like it isn't: neither of the standard textbooks I checked use it) I would strongly suggest we change it Thomas Tvileren ( talk) 10:35, 8 January 2009 (UTC)
The caption of sync function is wrong! It says that it is bounded and continuos but not Lebesgue integrable. Actually this is not true! If it is bounded and continuous is even Riemann integrable, and so it is Lebesgue integrable! Probably author meant that there is no analytic expression (and if I remember well analytic function is sin(x)/x whose integral should have such an expression), but this doesn't mean that the function is not integrable! —Preceding unsigned comment added by 85.134.152.225 ( talk) 23:38, 26 October 2009 (UTC)
I removed the following entry:
Function | Fourier transform unitary, ordinary frequency |
Fourier transform unitary, angular frequency |
Fourier transform non-unitary, angular frequency |
Remarks | |
---|---|---|---|---|---|
310 | Generalization of rule 309. |
The function 1/xn fails to be a distribution. The correct statement probably holds for the tempered distribution
I'm not sure if this belongs in the article, though. At any rate, it needs to be verified. Sławomir Biały ( talk) 22:25, 18 November 2009 (UTC)
For #310, I see you added the range from 0 to 1. What is the situation with 1/x^a, where a is fractional (say, 5/3 or 7/3) and greater than 1? Any clues?
Also, rather than remove it entirely, I recommend that you restore the original 310 for integer 1/x^n, somewhere else in the tables, even if that is not suitable for the distributions section. —Preceding unsigned comment added by 74.70.96.16 ( talk) 00:25, 21 November 2009 (UTC)
This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 | Archive 5 |
Given that this method can only be feasibly useful in the era of computers, I did spend a long time wondering what the hell Fourier was up to when he came up with this stuff. He was hardly making mp3s.
So I eventually learned that it's some stuff to do with steam engines and working out why they keep exploding.
This article needs a history section for others curious why some ancient mathematician would come up with an algorithm which isn't feasibly useful without a computer! —Preceding unsigned comment added by 86.154.39.2 ( talk) 20:47, 8 September 2010 (UTC)
I have added a brief reference to characteristic functions in probability theory. I think my brief reference should be expanded to include the convention for characteristic functions used in that wiki article in the table of other conventions. However, this is problematic due to differences in notation between the two articles. The notation can be made to align only in the case when there is a probability density function. JohnQPedia ( talk) 21:29, 26 June 2009 (UTC)
There is already something on the discussion page of Characteristic_function_(probability_theory) that is related to this. There you will find comments to the effect that it is more intuitive to interpret the characteristic function as being acquired by taking the inverse Fourier transform of the probability density function (in contrast to the wiki article concerned, which says this is acquired by taking a Fourier transform). However, there are two points. First, even under that interpretation, the (forward) transform is still different from any of the three listed in the other conventions sections here, because under this interpretation it is the (forward) transform that is multiplied by , rather than the inverse. And second, some authors of textbooks currently in use may in fact refer to the characteristic function in probability theory as being produced by a Fourier transform (perhaps such as those listed in the references of that page, but not adequately cited inline), rather than by an inverse Fourier transform. In this case, I think the wiki should reflect actual usage. In either case, the other conventions section is incomplete with respect to characteristic functions in probability. JohnQPedia ( talk) 01:06, 27 June 2009 (UTC)
In its article Characteristic Function, the MathWorld website confirms the interpretation of a characteristic function in probability as the Fourier transform of the probability density function, rather than the inverse Fourier transform. (Despite a compelling intuition to prefer an inverse Fourier transform, the mathematics are unaltered whether one considers a characteristic function to be acquired by a Fourier transform or an inverse Fourier transform.) Its article Fourier Transform also clears up any confusion, by explicitly saying that there are a number of conventions in widespread use. It lists five. It also gives explicit formulae for the Fourier transform and its inverse for arbitrary choices of constants. I have reproduced equations (15) and (16) from that page here
I put these here to indicate the kind of generality that could accompany the table in the section on other conventions. I believe such general formulae should be in the section for two reasons. First, it would make it clear how the specific examples in the table are derived. Second, it would provide a tool for the reader to interpret any sources that use conventions not listed in the table.
A further point is that the examples in the section on other conventions are generalized to apply to multidimensional integrals over . This means that a heavy edit is required to expand that section, while maintaining completeness of the table. (The MathWorld formulae given above are of course only one-dimensional.) I also think the multidimesional notation could be unclear to readers who are only up to speed on the most common circumstance, when the integral is one dimensional, and expressed in the form , rather than . This makes the table not generally accessible, and that undermines its utility. I suggest separate entries for the case when the integrals are one dimensional.
Thanks, all. JohnQPedia ( talk) 03:17, 27 June 2009 (UTC)
As an aid to the general public I propose to modify the first paragraph (I don't see how) in the following way:
"In Fourier transform, both domains are continuous and unbounded. Time intervals may be arbitrarily short and frequencies arbitrarily high. These infinite limits demand erudite mathematical tools not required in Fourier analysis. An example of an application more readily approached by Fourier analysis would be the inverse filter -- a band limited filter demanding an inverse of infinite bandwidth." —Preceding unsigned comment added by Jclaer ( talk • contribs) 03:37, 3 July 2008 (UTC)
Hi all. I came here looking for a transform pairs table because I'm studying communication systems and ended up wondering why some Fourier transforms differed from the ones that I have tabulated in my textbook. I didn't understand why the 2pi factor was under a square root!!
Anyway, I freaked out, googled a lot and finally read the article (it is kinda intimidating) finding out the answer my self (good job!).
What I would like to notice is that it takes a while to locate and understand the 3 different conventions thing, and I think that it could be more simple if we give the reader a big picture wise look of that in the "contents".
For instance, putting it like:
1 Definitions
1.1 Communications and signal processing (f) convention
1.2 Mathematics angular frequency (2pi) convention
1.3 Mathematics angular frequency (2pi unitary) convention
Finally, like some textbooks do, It would be really simplifying to say "in the rest of this article, we will use convention {whatever} unless stated otherwise". And stick to that in order to preserve consistency.
I would also like to say that I agree with Feraudyh (disgruntled editor) that this article is really hard to read for students like me, it seems really different from what we have on the communications textbooks. I understand that is a mathematical thing and it should be the most formal possible, but still I guess it could be more readable.-- Dhcpy 01:41, 1 December 2007 (UTC)
Why do we have a section on " # 4 Properties" and a section " # 6 Fourier transform properties". Maybe we should merge #6 into #4. Thoughts? Thenub314 00:28, 22 February 2007 (UTC)
Bo says: "By first conjugating the function, the forward transform equals the reverse transform, a so-called involution". But the forward transform produces and the reverse transform operates of In order for Bo's claim to be true, the reverse transform would have to operate on the same thing produced by the forward transform. I see no value in this involution idea. It's just smoke and mirrors.
-- Bob K 01:25, 8 March 2007 (UTC)
Bo, Ignore the above comments : both Bob and I realized too late what you meant. :-) Abecedare 02:20, 8 March 2007 (UTC)
Exactly! Sorry for all my recent reversions on the article and talk page. I have tried to rewrite the word statement to make it clear that the Fourier transform can be deined in a form that is an involution. Please check if the point is now clear. Also I reverted to using the "overline" convention for conjugation, since it is used in the rest of the page and more importantly * is used for the adjoint operator (and AFAIK, there is no other "standard" notation for the latter). Regards. Abecedare 02:28, 8 March 2007 (UTC)
No mathematical definition is standard, as no standardization organizations standardize mathematical definitions. The transformation is obviously a conjugate-linear map. That it is also an involution was just checked by editors Abecedare and Bob K. This is sufficient documentation for correctness. Steven G. Johnson's removal of the definition is a violation of Wikipedia:Resolving disputes # Avoidance. The value of the definition is that it avoids the risk of confusing forward and reverse transformation. That's why I want to share it. I'll leave it to Abecedare and Bob K to reinstall it if they like it too. Bo Jacoby 06:53, 8 March 2007 (UTC).
Thank you, Abecedare. It is not the definition, but it is a useful variant form. I agree that the nice properties are well-known and trivially derived, so there is no question about reliability, which is the reason behind the general ban against original research in wikipedia. It may be considered original research on my part, but not on yours, so I cannot reinstall it, but you can. Just refer to me. Bo Jacoby 08:05, 8 March 2007 (UTC).
I am grateful that my humble observation is honored by the name of original research. I approve Bob K's formulation: "a non-standard type of Fourier transform that is an involution, which means that the inverse transform is the same operation as the forward transform". This does not make any promises of gold-plated mainstream references but gave a straightforward and clear explanation. The corresponding FFT is foolproof because only one procedure is needed for transforming forwards and backwards. For real x(t) it coincides with one of the other definitions. Too bad that only very few people got the chance to read before removal. I can do no more about it. Any of you are welcome to undo the edit "04:56, 8 March 2007 Stevenj" when you realize that our readers deserve the best. Bo Jacoby 13:53, 8 March 2007 (UTC).
Bo, I don't know if your transform is an involution or not. I agree that it is kind of cool and interesting, but I don't see how it furthers the cause of this article, which is to help people understand the standard Fourier transform(s). In fact, I think it is counterproductive. I wish there were fewer forms, not more. If indeed yours is an involution, then why not add your transform to that article?
-- Bob K 16:00, 8 March 2007 (UTC)
See the definition in involution is simply: "a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f". Bob K checked that the variant 4 of the fourier transform satisfies this condition, just above. The forward and backward transformation is the same. There are many involutions: the identity, the sign change, the reciprocal, the complex conjugation, &c. Bo Jacoby 16:33, 8 March 2007 (UTC).
The variables f and t are not in the same domain, but the functions X and x are in the same domain. The fourier transformation transforms x into X. The transform maps the function x of time to the function that maps the frequency f to the integral
In this expression both t and f are dummy variables. Other variable names can be substituted for t and f. They may even be swapped:
Using you get
So the transformation that transforms a function of t into a function of f also transforms a function of f into a function of t. Bo Jacoby 22:38, 8 March 2007 (UTC).
According to the article Function, a transform is also a function. It is a very general concept. The domain need not be a set of numbers, but can be a set of other functions too. Bo Jacoby 00:12, 9 March 2007 (UTC).
And besides, a brand new definition does not help people understand "Fourier transform". It's bad enough that we already have three to worry about. A fourth would be counterproductive. Why not use the involution article to showcase your involution? Why here? This article is not about involutions.
-- Bob K 04:01, 9 March 2007 (UTC)
Right. Point taken. -- Bob K 15:26, 9 March 2007 (UTC)
I propose the following section to replace sections Fourier_transform#Definition and Fourier_transform#Normalization_factors_and_alternative_forms. One thing that bothers me about the current version is the abrupt appearance of the factors. I think it can be done in a more natural looking progression.
Another thing that bothers me is the paragraph:
Wouldn't it be more logical if the components being recombined had amplitudes equal to A(·)? Well this way they do:
-- Bob K 00:46, 11 April 2007 (UTC)
There are several common conventions for defining the Fourier transform of a complex-valued Lebesgue integrable function, In communications and signal processing, for instance, it is often the function:
When the independent variable represents time (with SI unit of seconds), the transform variable represents ordinary frequency (in hertz). The complex-valued function, is said to represent in the frequency domain. I.e., if is a sufficiently smooth function, then it can be reconstructed from by the inverse transform:
Other notations for are: and .
The interpretation of is aided by expressing it in polar coordinate form: , where:
Then the inverse transform can be written:
which is a recombination of all the frequency components of Each component is a complex sinusoid of the form ei 2 π f t whose amplitude is A(f) and whose initial phase angle (at t = 0) is φ(f).
In
mathematics, the Fourier transform is commonly written in terms of
angular frequency: whose units are
radians per second.
The substitution into the formulas above produces this convention:
which also happens to be a bilateral Laplace transform evaluated at s = i ω.
The 2π factor can be split evenly between the Fourier transform and the inverse, which leads to another popular convention:
A transform whose inverse has the same multiplicative factor is called a unitary transform.
Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.
I like your version and support replacing the "Definitions" and "Normalization Factors and alternative forms" by the one above. Some comments:
Aside: I just noticed that properties section was missing the factor without which the all the completeness, parseval's convolution etc results in the section were incorrect! I have corrected that now. Abecedare 01:36, 11 April 2007 (UTC)
Why is the inverse fourier transform true?
now what can I do? How can I reduce the right side of the equation so that it equals the left? —Preceding unsigned comment added by Iownatv ( talk • contribs)
Thanks Iownatv 20:20, 5 May 2007 (UTC)
Would it be appropriate to add a heuristic "derivation" of the fourier integral and inverse as a limiting process of the exponential Fourier series of an arbitrary periodic function to aperiodic by letting the period tend to infinity? This at least makes it somewhat intuitive and plausible, before one goes on to the convergence and existence conditions and all of that. DivisionByZer0 20:35, 14 June 2007 (UTC)
What about the range and domain of the Fourier transform? The article does not say anything about which functions (or generalized functions) can be transformed or what is the range of the transform. -- KYN 20:52, 3 August 2007 (UTC)
Ok, I don't expect a complete presentation on this issue, but at least the very basic could be presented in the article. Here is a start:
Something along these lines anyway. -- KYN ( talk) 07:59, 24 April 2008 (UTC)
With incompatible I simply mean that "tempered distributions" and "L^2" are sets of generalized functions which are defined in two completely different ways and even if some elements in one set can be identified with elements in the other, this is (as far as I understand) not the case for all elements of both sets. -- KYN ( talk) 20:58, 29 April 2008 (UTC)
I see, I was not aware of this correspondence between L2 and tempered distributions. Is it mentioned anywhere in Wikipedia? To me, it appears to be a sufficiently interesting fact to mention in the Lp or the tempered distr articles. -- KYN ( talk) 10:00, 1 May 2008 (UTC)
Many, particularly engineers and physicists, may be interested in only the real case and may be confused by the use of complex exponentials instead of trigonometric functions. The MoS guideline Wikipedia:Make technical articles accessible says that articles should begin with easier sections the reader can relate to. I suggest that a section using sines be added to the beginning before going on to complex exponentials. Loom91 20:15, 13 August 2007 (UTC)
hi! does somebody know where i can find a proof of
thanks! -- 217.117.224.177 09:57, 21 September 2007 (UTC)
This talk page is now so long that I believe its practical use is reduced. Therefore I propose to archive the earlier parts of this talk page, starting from 2004 until 2005, leaving Talk:Fourier transform#Alternative forms as the first posting that remains in the talk page. I propose to use the "Subpage archive method/cut and paste procedure" described here. In principle I would like to archive a larger chunk, let's say including all 2006 postings, to make the page more manageable but there appears to be some discussions which were initiated in these early postings but still have attracted answers or comments in 2007. Any optinions? -- KYN 15:21, 23 September 2007 (UTC)
All discussions started 2006 or earlier have now been moved to "Archive 2" which, together with "Archive 1", now is reachable from the Archive icon at the top of this page. If there are active threads which have been removed, please reinsert them on this page. -- KYN 09:56, 1 October 2007 (UTC)
According to my signals and systems book (Signals and Systems by Oppenheim, Willsky and Nawab) the fourier transform for the signal is . However the wikipedia article says that the fraction in front of the sum is . I changed this a few days ago, but it was changed back, so I thought I'd start a discussion about it. There is a pretty good example in my book that shows the derivation of this (p. 299) so I'm pretty sure that square root shouldn't be there, but obviously someone thinks it should be there. —Preceding unsigned comment added by Fkp1 ( talk • contribs) 21:31, 31 October 2007 (UTC)
Unless I am mistaken, is a square-integrable function when . Why then is this Fourier transform pair in the distribution section?
Butala 13:46, 1 November 2007 (UTC)
I removed this from the "Definitions" section and reproduce it here because that is where it belongs
Feraudyh 10:40, 6 November 2007 (UTC)
Dear Feraudyh. I completely agree with you. The article should be readabel to you. I regret that it is not. I have tried to mend it, but I am opposed by another WP-editor who believes that he can do better and that WP should contain 'standard' mathematics even if that is incomprehensible mathematics. See: User:Bob_K#two_articles_for_discrete_fourier_transform_.3F. I am sorry. Keep complaining. It helps. Bo Jacoby 13:49, 6 November 2007 (UTC).
I'm afraid I have to side this time with the non-mathematicians complaining about the (lack of) accessibility of this article. It is true that there is still a huge scope to improve the presentation – and not only for the benefit of non-mathematicians but mathematicians as well. Without any attempt at completeness, I would like to make the following observations and make a few suggestions:
The above may look harsh, but it is definitely not meant as critique of the editors that have worked on the article. However, in order to move the article (rather the topic) to a new level I'm afraid that something like the above should be addressed. In other words, work remains to be done. Stca74 14:24, 7 November 2007 (UTC)
I am new to Wikipedia and am not sure that this is the proper way to contribute to the discussion on this talk page. I hope so!. To start, I was drawn in by Googling the Wiki page on FTs. I was, frankly, very disappointed in the material presented. I am a math/physical scientist professional and was looking for something a little more entertaining and, for the layman or number theorist, more inspiring than a Xerox of a textbook. Been there. I agree wholeheartedly with the "disgrundtled editor" in his comments, and the others who followed, that Wiki should, in my phrasing, be a conduit to the community to the understanding of higher, interesting concepts. This "window" into a fascinating world is guided by those of us, in any field (math, history, music, etc.), who have commited their energies to gaining a deep understanding of their field, can now gladly offer their insights over the remarkable new medium of the Internet. Kind of a win-win situation.
My own thoughts on the FT presentation will be brief since this whole business seems to have such tremendous inertia. Other than reiterating my intial comment that it doesn't really do any one any good to regurgitate textbook material on Wiki ( send them to a couple of decent web sites
http://www.youtube.com/watch?v=QQbEOjGOY5I
or sic them on Bracewell's book. Try to come up with some original material. My own personal approach to teaching the idea of the FT and transforms in general is the idea of projections and orthogonality. Vectors projecting on the x-y plane leading to random functions projecting onto orthogonal periodic functions (sin and cos). Etc.
Randy Patton —Preceding unsigned comment added by Randypatton ( talk • contribs) 06:19, 31 March 2009 (UTC)
This new section is important, but somewhat confusing. The dash in 'aperiodic-discrete' is confusing. It perhaps means that the function is aperiodic and discrete. Why not write that? As to "The Fourier Transform for this (Aperiodic-Continuous) type of signal is simply called the Fourier Transform". No, it is called the fourier integral. The reader wants to know a reason for this trouble; why not a single fourier transform to cover these cases? The problem is that a 'discrete function' as the limit of continuous functions does not exist, and the solution in terms of distributions is too advanced. Bo Jacoby ( talk) 12:08, 20 January 2008 (UTC).
I've cited this article, chosen pretty much at random as many others are similar, as an example of one that I consider useless in that it is written at too high a level to be generally accessible, in an a discussion I have sought to provoke about how Wikipedia's mathematical articles are structure in the Village pump/Policy area in the Community Portal.
David Colver ( talk) 15:15, 2 February 2008 (UTC)
The text says "If x is Hölder continuous, then it can be reconstructed from by the inverse transform". The linked page on Hölder continuity tells me that anything bounded is Hölder continuous with exponent 0. But this includes bounded discontinuous functions, which cannot be reconstructed from the inverse transform. It appears to me that one of the two pages (and I don't know which) must be in error. —Preceding unsigned comment added by 80.177.154.109 ( talk) 12:33, 7 February 2008 (UTC)
Your correct there is a lot of confusion around this sentence. Not every bounded function has a Fourier transform defined by integration. More general methods for defining the Fourier transform could be applied, but they also have to be applied to define the inverse transform. I believe the sentence here is means of f is Hölder with positive exponent, then the resulting function is integrable and the integral defining the inverse transform converges almost everywhere to f. But this seems to be there for the sake of adding some conditions to the inversion formula. I feel this this it is a bit early in the article for this level of detail. Perhaps it would be better to say "Under suitable conditions on F, f may be reconstructed from F by the inverse transfrom:" Thenub314 ( talk) 14:12, 28 August 2008 (UTC)
Folks from the popular 4chan boards and other sites are targeting the LaTeX source in higher math articles for minor vandalism, I could see this hurting homework grades and exam prep for quite a few students, something to keep an eye out for.
search google for "Vandalize_Every_Equation" to see a link to a wiki on another popular site with details (it's blacklisted on wikipedia's spam filters so I can't link to it directly)
140.247.10.17 ( talk) 06:04, 16 February 2008 (UTC)anonymous
Sorry, but I'm not sure that Bessel function are square integrable (the norm L2 of its transform diverge). If is not so, please tell me why? (Sorry if I don't sign this message) --penaz
'THERE ARE ERRORS IN THE EQUATIONS FOR MEAN AND VARIANCE OF FOURIER TRANSFORM INTEGRALS!!. YOU MUST DIVIDE THE FIRST AND SECOND MOMENTS(The Equations currently given) BY THE NORM TO OBTAIN THE CORRECT RELATIONSHIPS.' 71.254.7.2 15:06, 17 June 2007 (UTC)
I think defining the heavy side function at this point breaks up the flow a little. It makes this entry different form the others in this section. I think for the sake of simplicity it is best make the point there is a convolution here. Thenub314 ( talk) 03:37, 25 April 2008 (UTC)
I would like to raise some questions about notation that I was thinking about in reading this article. Particularly in defining the Fourier transform we use x(t) as the signal and f. By the time we are done with the section we are ω. And starting in the second section we are using f as a function. And when we discuss multi-dimensional Fourier transforms x becomes the variable. I would like to move that we change the definition. At the very least let's call the function and variable in the first definition something else. I of course would like to say more about the use of "hat's" e.g. and (we mention the first, but don't say anything about the second.) I feel this notation is a standard in mathematics, and not uncommon in Engineering or Physics. Back to the main point, for the sake of first year calculus students that read this page, I would like see the first definition changed. Thenub314 ( talk) 02:52, 14 May 2008 (UTC)
Perhaps the following way of putting parentheses is nicer?
I calculated the Fourier transform of 1/x in two ways, and I didn't get the minus sign out front. Can anyone confirm this? —Preceding unsigned comment added by 59.167.144.142 ( talk) 11:50, 28 August 2008 (UTC)
Okay I just realized the table uses a different definition of the Fourier Transform, problem solved. —Preceding unsigned comment added by 59.167.144.142 ( talk) 12:22, 28 August 2008 (UTC)
and are undefined. And according to Poisson_summation_formula, is not the unitary transform.
-- Bob K ( talk) 20:15, 31 August 2008 (UTC)
Dear Frequent Editors,
I am both a user of Fourier transforms in medical imaging and signal processing, and an occasional instructor to students with various backgrounds in mathematics (medical students are usually on the less mathematical end of the spectrum). I can empathize with some of the comments from previous posters about the need for a conceptual (i.e. non formulaic) section introducing the Fourier transform.
Here are a few pages with examples:
http://www.e-mri.org/image-formation/fourier-transform.html
http://www.e-mri.org/image-formation/2d-fourier-transform.html
Another useful analogy is the comparison to musical notation (although strictly speaking this is more like a joint time-frequency transform).
And a very good reference for Fourier transform related identities:
Linear Systems, Fourier Transforms, and Optics
By Jack D. Gaskill
http://books.google.com/books?id=ZcUPAAAACAAJ
Curtcorum ( talk) 04:22, 9 September 2008 (UTC)
Here is a possible transform pair example. Heavy line is real component. Dashed is imaginary. I am not sure if it is better to show axes to scale or not.
-- Curtcorum ( talk) 03:15, 13 September 2008 (UTC)
Here is a clarification:
and is essentially table entry #205 with an additional frequency shift.
I think showing the negative time axis is a good idea. Also showing the formulas? An abbreviated table of transform pairs with graphical examples might be good? Is there a link to one online that anybody knows of? Curtcorum ( talk) 22:07, 15 September 2008 (UTC)
This article is apparently on a list of Wikipedia articles to be released on DVD. But we have a few maintenance tags to take care of. The deadline is October 20th. The two tags ask us to clarify and use inline citations. The too technical tag came after an ip user added the statement that every thing after the lead was for "advanced mathematicians". The nofootnotes tag was added back in November of 2007. I would like to make the article more accessible. Do people have suggestions as what the best changes might be? Thenub314 ( talk) 11:07, 19 September 2008 (UTC)
Here are some comments that I am collecting that might help, and I invite comment. The first sentence in the definitions section is:
Could be shortened to:
And we cite "A Friendly Guide to Wavelets" Gerald Kaiser, which has a nice discussion of the different conventions on pages 31-32.
Also the suggestion is made in our rating that:
And I think I agree with this comment.
The section on "Generalization" doesn't really generalize, but just shows how to choose different constants. It does not cite a references, and since it follows by change of variables I don't know of any references to cite for it. At best it is not very enlightening and at worst it is "original research". I am in favor of killing the section, but if we don't we should at least reference it and move deeper into the article. Thenub314 ( talk) 16:44, 21 September 2008 (UTC)
Another thought to simplify the article for the reader would be to stick with one convention throughout the article and simply have a section on the different notational conventions that one finds in the literature. Presenting them all back to back in the definitions sections seems like it may be a lot to absorb at once. Thenub314 ( talk) 18:19, 21 September 2008 (UTC)
Now that we have eradicated all semblance of typical signal processing notation, i.e. and (instead of ), do we just stop here? It seems to me that an encyclopedia should try to represent the signal-processing perspective, because that is why a lot of readers will come here. Do we now create article Fourier transform (signal processing)?
-- Bob K ( talk) 19:50, 8 October 2008 (UTC)
That might not be what other signal processing editors want. So I'm asking. It's not what I want, but if I'm the only one who cares, then heck with it.
-- Bob K ( talk) 01:16, 9 October 2008 (UTC)
The multi-dimensional form is no longer an issue (for me), because the article no longer leads with it (as it did 15 months ago). So I don't think the current objections are about that. But in a "signal processing" article, with this one as backup, I would:
-- Bob K ( talk) 21:38, 22 January 2010 (UTC)
This bit is unclear to me: If ƒ is a function for [−L/2, L/2], then for any T ≥ L we may.... There doesn't seem to be any other reference to the quantity L; is it needed? -- catslash ( talk) 12:48, 9 October 2008 (UTC)
Your correct, that sentence makes no sense. I have tried to fix it, it simply should have said that ƒ is a function supported in [−L/2, L/2]. It is only implicitly used in the equality . Thenub314 ( talk) 13:08, 9 October 2008 (UTC)
Many people are familiar with the rising and falling "waterfall" bar displays on a stereo, showing a real-time view of the music spectrum, from bass (low-frequency) on the left, to treble (high-frequency) on the right.
However, the audio signal being amplified and fed to the speakers is simply the intensity ("loudness") of the music at any given instant - older audio equipment had just a needle display ("VU meter") that represented the sound volume -- and the actual signal -- at any given time.
So if the tuba hits a note at the same instant as the piccolo, there's only one value of the signal at that moment, which is what the needle shows -- and the speaker plays! How does the display "know" to make both a left bar and a right bar - but not much in between - rise sharply?
The Fourier transform can be thought of as a "black box" which, given the input of that stream of single values representing sound volume over time, produces a stream of many values, simultaneously: the level of each frequency present at that moment in time. So - even though the original signal had the input from the tuba and piccolo added together into a single value, the Fourier transform can tease apart the low notes from the high notes, much as humans can easily zero in on a single conversation at a noisy cocktail party.
The reason it's called a transform is that it transforms the information contained in a single signal value of intensity, which varies over time (called a "time-domain representation") into a set of values representing the distribution of frequencies at each moment (hence: "frequency-domain representation").
At the risk of encouraging conflict, may I point out that the WP:MOSMATH guideline on the question of TeX versus HTML appears to be the unilateral decision of User:Oleg_Alexandrov. See Help:Displaying a formula for a more balanced assessment. -- catslash ( talk) 00:13, 17 October 2008 (UTC)
109 Fourier Transform of
but
109
If we put in all the values the comes in the numberator not denominator. Please check and let me know
Wilkn ( talk) 21:54, 21 October 2008 (UTC)Wilkn@yahoo.com
I think the entry is correct, unless I made a careless error. We can calculate the Fourier transform of by:
Let me know if this answers your question. Thenub314 ( talk) 17:55, 22 October 2008 (UTC)
Thank you for your response. I was wrong sorry about that. I am new to wikipedia so please use your best judgement to delete the page or not. My research is centered on FT, would it be possible for you to give me your contact information ? Wilkn ( talk) 00:58, 23 October 2008 (UTC)Wilkn—Preceding unsigned comment added by Wilkn ( talk • contribs) 19:21, 22 October 2008 (UTC)
(in the section Multi-dimensional version)
I have tried to clarify the statement about the fact that the indicator function of the ball is not a multiplier for Lp, by introducing fR defined by
but it is not satisfactory: the real thing is that for some f ∈ Lp, this function fR is simply not in Lp, so the question of fR converging to f is pointless. I don't exactly see what the original intention was. Does anybody know? Bdmy ( talk) 14:31, 29 November 2008 (UTC)
I didn't even realize it was an english word! (And I looked it up, and it is) In some ways the use of this word is brilliant, but I feel the word is fairly uncommon. Is there any objection to me trying to rework this sentence? Thenub314 ( talk) 20:19, 23 December 2008 (UTC)
In the table of Fourier pairs, the third and fourth column refer to conventions of the Fourier transform that both employ angular frrequency but differ by their normalization constant. I do not really understand, why different symbold (omega) and (nu) are used to denote the angular frequency. Wouldnt it be clearer to choose consistent notation here? —Preceding unsigned comment added by 91.66.240.116 ( talk) 11:36, 29 December 2008 (UTC)
I am used to epsilon denoting infinitesimally small real values in "epsilon-delta proofs" and the like, and was therefore a little confused by the formulas in section "Definition". If this isn't standard notation (and to me it seems like it isn't: neither of the standard textbooks I checked use it) I would strongly suggest we change it Thomas Tvileren ( talk) 10:35, 8 January 2009 (UTC)
The caption of sync function is wrong! It says that it is bounded and continuos but not Lebesgue integrable. Actually this is not true! If it is bounded and continuous is even Riemann integrable, and so it is Lebesgue integrable! Probably author meant that there is no analytic expression (and if I remember well analytic function is sin(x)/x whose integral should have such an expression), but this doesn't mean that the function is not integrable! —Preceding unsigned comment added by 85.134.152.225 ( talk) 23:38, 26 October 2009 (UTC)
I removed the following entry:
Function | Fourier transform unitary, ordinary frequency |
Fourier transform unitary, angular frequency |
Fourier transform non-unitary, angular frequency |
Remarks | |
---|---|---|---|---|---|
310 | Generalization of rule 309. |
The function 1/xn fails to be a distribution. The correct statement probably holds for the tempered distribution
I'm not sure if this belongs in the article, though. At any rate, it needs to be verified. Sławomir Biały ( talk) 22:25, 18 November 2009 (UTC)
For #310, I see you added the range from 0 to 1. What is the situation with 1/x^a, where a is fractional (say, 5/3 or 7/3) and greater than 1? Any clues?
Also, rather than remove it entirely, I recommend that you restore the original 310 for integer 1/x^n, somewhere else in the tables, even if that is not suitable for the distributions section. —Preceding unsigned comment added by 74.70.96.16 ( talk) 00:25, 21 November 2009 (UTC)
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