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Hi, I think there is a small mistake in section 15 "Tables of important Fourier transforms" -> "Functional relationships, one-dimensional", property 102, time shifting of fourier transform. There should be a minus in the power of e: e^(-2*pi*i*...). That minus is missing in the entire row. I think I verified it on paper, but also with other sources, including the wikipedia fourier transform article itself (section 5.1.2 Translation / time shifting). I have no idea how to fix this. This is my first post on wikipedia ever. I hope I'm correct though and not wasting anyones time.
-- 83.130.77.27 ( talk) 12:23, 13 January 2021 (UTC)
Function | Fourier transform unitary, ordinary frequency |
Fourier transform unitary, angular frequency |
Fourier transform non-unitary, angular frequency |
Remarks | |
---|---|---|---|---|---|
Definition | |||||
102 | Shift in time domain |
Hello. There is a problem with Chrome and browsers based off of Chrome such as Edge. The wizards are aware of it. As a temporary fix you can increase the zoom factor. Also, you can get the Math Anywhere extension for both Chrome and Edge that seems to take care of the problem. Or you can wait until Chrome fixes the problem. Constant314 ( talk) 16:12, 13 January 2021 (UTC)
I have to agree with the IP editor that this is poorly written and after checking the source, I see that the source does not say what is written here. In fact, the source doesn’t give a reason, it just says that engineers prefer a certain sign convention. The source isn't even about the Fourier transform.
The reason given is nonsense. In fact, both negative and positive signs are used by different communities and there is no problem with convergence of the integral.
Since the source doesn’t give a reason, I will remove the reason. But the reason is this: it is arbitrary. It is just a choice of where you want your negative signs to appear. As an engineer, I have my preference mainly because that is the way I was taught and that is the way it appears in most of my textbooks. You may find justifications for one choice or the other, but you will not find a definitive reason the sign must be negative or must be positive. Constant314 ( talk) 12:26, 8 October 2021 (UTC)
I use the same convention that you are advocating. I have no problem with using that convention. If you want to state a reason for that convention in the article you need a reliable secondary source that states that reason. No amount of WP:OR will change that. However, I do not mind dabbling in OR here on the talk page.
Let
I hope it is obvious that
This has no physical effect because physical effects are caused by energy or power. The power of a Fourier transform is computed by multiplying the transform by its conjugate.
Again, I hope it is obvious that
So, lets look at a couple of examples. I will suppress multiplicative constants that clutter up the results.
First, consider the Fourier transform of .
Next, consider the Fourier transform of .
Now let me go way off into OR la-la land to speculate why engineers prefer the usual convention. Consider the Fourier transform of cos(ωt) + sin(ωt). It is . The component at the positive frequency of is . Notice in particular that the sign of the imaginary part is negative. Engineers prefer this because lags by 45°. When an engineer plots this in Cartesian space, it is [1,-1]. The principal argument is negative. Engineers prefer that because the phase of cos(ωt) + sin(ωt) relative to cos(ωt) is negative. Mathematicians consider cos(ωt) and sin(ωt) as basis vectors and they plot cos(ωt) + sin(ωt) as [1,1]. That is all there is to it. Engineers prefer that the Fourier component of sin(ωt) should be negative at positive frequency. Constant314 ( talk) 22:03, 13 November 2021 (UTC)
For real-valued the convention hardly matters, because every frequency has a positive equivalent (e.g. see Aliasing#Sampling_sinusoidal_functions). The concept of negative frequency is unnecessary... two-sided Fourier transforms are redundant.
Cutting to the chase, the convention determines whether is considered a positive or a negative frequency. The customary definition of
instantaneous frequency is the derivative of instantaneous phase, which is whose derivative is
Therefore is a positive frequency. And its measurement is (which means is ).
I conclude that those who claim otherwise have a different definition of instantaneous phase or instantaneous frequency, which puts them at odds with Wikipedia's sourced articles. The burden is on them to provide contradictory sources.
--
Bob K (
talk) 04:38, 15 November 2021 (UTC)
I get your point. But I disagree with your statement "But the reason is this: it is arbitrary. It is just a choice of where you want your negative signs to appear." No. It comes down to your definition of the instantaneous phase and frequency of function When you "arbitrarily" choose you are also arbitrarily rejecting the customary definitions of instantaneous phase and frequency. Therefore you need to provide sourced reasons for that whim.
--
Bob K (
talk) 05:10, 15 November 2021 (UTC)
I went through several of my text books. Here is what I found.
Constant314 ( talk) 17:28, 17 November 2021 (UTC)
Thank you. I can expand the upper list, if needed, but it seems to be coming down to signals and communication vs electromagnetics. Amazingly, I still have my undergrad copy of
It is a dense 554-page book, with not a single Fourier transform formula or even Euler's formula. My take-away is that the EM applications of Fourier transform theory don't go deep enough to matter which convention they use. To quote myself (above) "For real-valued the convention hardly matters". The concept of negative frequency is not useful. So your statement:
might be a misleading generalization based on certain limited applications of transform theory.
Anyhow, we're getting a little off track. The point is that the convention chosen for the article (which we both agree with) was not an arbitrary coin toss. It might not have any consequences for EM theory, but it does have consequences for signal theory. So I added a footnote that does not need an external citation. All it relies on is a Wikilink to our instantaneous frequency article.
--
Bob K (
talk) 12:30, 18 November 2021 (UTC)
That sounds about right, based on my distant memories (circa 1968) of one EM theory course.
--
Bob K (
talk) 11:47, 16 March 2022 (UTC)
I think this article should be improved in some manner for the general public that is scientifically minded but not taking a full calculus class in college. I think the possibility of adding it to the Simple English Wikipedia with easier to understand language is a good idea, in addition to the process of adding explanations and writing that is not mathematically centered. Maybe including something of this sort, "The Fourier Transform helps to transform functions such as cosine and sine into different output functions that behave differently than normal trigonometric functions." ScientistBuilder ( talk) 01:38, 14 October 2021 (UTC)ScientistBuilder ScientistBuilder ( talk) 01:38, 14 October 2021 (UTC)
A quick impression is that this section could be simplified, perhaps making use of the Analytic signal concept instead of Fourier series. I'll try to give that some thought.
Furthermore, the statement "every real sinusoid consists of an equal contribution of positive and negative frequency components, which is true of all real signals" is misleading. It is a cancellation, not a contribution, analogous to something like "10 apples consists of 5 apples + 5 bananas and 5 apples - 5 bananas". (See Negative_frequency#Sinusoids)
-- Bob K ( talk) 12:34, 16 March 2022 (UTC)
I think that's on the right track. I was definitely struggling with that section... I kept coming back to the question "Does it even need to be here?" IMO, the ccrma.stanford.edu viewpoint is the easy explanation, more of an engineering convenience than a true insight. I'm all in favor of conveniences, but I'm also in favor of distinguishing them from the underlying realities.
--
Bob K (
talk) 11:45, 17 March 2022 (UTC)
I was hoping to see an IPA pronunciation in the first sentence of the article (but there is not one). Wiktionary has an English pronunciation for Fourier (as a surname), which might apply to Fourier transform. - excarnateSojourner ( talk| contrib) 21:30, 11 July 2022 (UTC)
The section idiotically titled Introduction contains this passage:
"Although Fourier series can represent periodic waveforms as the sum of harmonically-related sinusoids, Fourier series can't represent non-periodic waveforms. However, the Fourier transform is able to represent non-periodic waveforms as well. It achieves this by applying a limiting process to lengthen the period of any waveform to infinity and then treating that as a periodic waveform. [1]"
The last sentence is complete mathematical nonsense, and is not taken from the cited reference. 2601:200:C000:1A0:BC00:5039:DB55:E9EC ( talk) 19:13, 29 July 2022 (UTC)
References
As I tried to write a few clarifying sentences in the section Fourier transform for functions that are zero outside an interval, in order to specify the periodic function whose Fourier series was referred to ... I realized that the article refers to the same function when discussing the Fourier transform. But: It's not the same function. One is periodic and the other is zero outside the interval [-T/2, T/2].
I hope someone knowledgeable about this subject can fix this apparent problem. 2601:200:C000:1A0:BC00:5039:DB55:E9EC ( talk) 20:08, 29 July 2022 (UTC)
I find the introduction difficult to understand. It would be far better to avoid use of the word "transform" at first in the explanation, and instead say that the Fourier transform is a mapping taking an integrable function on Rn to an integrable function.
And that according to a version of the Fourier transform F favored by mathematicians, F4 = I.
Surely this is worthy of prominent mention. 2601:200:C000:1A0:FDA6:8A26:FCCA:2C1B ( talk) 04:39, 22 September 2022 (UTC)
Well done, but it repeats a lot of information available thru WikiLinks, and now also added to Fourier transform#The analysis formula. IMO this section can be downsized or eliminated. Bob K ( talk) 16:47, 12 December 2022 (UTC)
It has been a decade since the
ξ_vs_ν_? discussion between just 2 editors. And as noted then, ν was the original preference, replaced without a consensus. Now I am asking if there is any consensus for reverting back to ν, because ξ is unnecessarily intimidating-looking (IMO).
--
Bob K (
talk) 00:59, 19 December 2022 (UTC)
The reliance on:
is no better than relying on the transform pair:
Consequently, you've made a mountain out of a molehill.
--
Bob K (
talk) 01:59, 25 December 2022 (UTC)
The result of this section seems to be the statement:
Under appropriate conditions, the Fourier series of f will equal the function f. In other words, f can be written:
At the very least it needs to clarified that the first equality only applies in the interval T.
But also we already know that can be recovered from So what you are showing is that when its domain is bounded, it can also be recovered from discrete samples of which, by the way, is the dual of the time domain
sampling theorem. This is mildly interesting, but it strikes me as a proof looking for a home, and the whole article is already pretty cluttered with stuff.
--
Bob K (
talk) 09:00, 27 December 2022 (UTC)
The function is listed as a square integrable function (Line 207). I think this is only true if . However, this condition is not mentioned, and if this condition is included, (Line 207) becomes just a restatement of (Line 206) with the replacement . 18.29.20.123 ( talk) 16:40, 10 February 2023 (UTC)
I would like to discuss the recent revert of my edit. The edit, in part, addresssed the comment from the IP editor about that our tables were incorrect. While looking at the article I noticed it was inconsistent in the placement of vs so I fixed it. It seems the objections are mostly about the later. For which I would like to point out, the prevailing culture is that if a is used, it is before the i to see this I would like to point to:
Aside from the last one, these are large professional orgainzations trying to make something for the masses. About the last one, I include the last one because I would like to return to the status quo for the article. I also feel it would benefit readers because they are (in my opinion) more likely to encouter in books/papers/references. Thenub314 ( talk) 15:47, 15 February 2023 (UTC)
I removed a reference from here because the cited reference wasn't discussing the Fourier transform in the section indicated. I will try to find something more specific to this topic. Thenub314 ( talk) 20:47, 16 February 2023 (UTC)
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:
Participate in the deletion discussion at the nomination page. — Community Tech bot ( talk) 14:53, 17 February 2023 (UTC)
In the inversion section the integrals are over the variable σ but the variable ξ appears in the integrands. Where does the ξ come from? — Preceding unsigned comment added by 2.27.171.228 ( talk) 21:25, 21 August 2023 (UTC)
I reverted this edit, because Wikipedia articles do not usually include code samples (see MOS:CODE), unless those code samples illustrate some fundamental aspect of an algorithm. In this case, the algorithm (the fast Fourier transform, for which there is already a separate article) is not actually shown. Instead, it uses a builtin function of the numpy library. So this code is very python-specific, and is not a good illustration of the Fourier transform. Tito Omburo ( talk) 10:42, 23 May 2024 (UTC)
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Hi, I think there is a small mistake in section 15 "Tables of important Fourier transforms" -> "Functional relationships, one-dimensional", property 102, time shifting of fourier transform. There should be a minus in the power of e: e^(-2*pi*i*...). That minus is missing in the entire row. I think I verified it on paper, but also with other sources, including the wikipedia fourier transform article itself (section 5.1.2 Translation / time shifting). I have no idea how to fix this. This is my first post on wikipedia ever. I hope I'm correct though and not wasting anyones time.
-- 83.130.77.27 ( talk) 12:23, 13 January 2021 (UTC)
Function | Fourier transform unitary, ordinary frequency |
Fourier transform unitary, angular frequency |
Fourier transform non-unitary, angular frequency |
Remarks | |
---|---|---|---|---|---|
Definition | |||||
102 | Shift in time domain |
Hello. There is a problem with Chrome and browsers based off of Chrome such as Edge. The wizards are aware of it. As a temporary fix you can increase the zoom factor. Also, you can get the Math Anywhere extension for both Chrome and Edge that seems to take care of the problem. Or you can wait until Chrome fixes the problem. Constant314 ( talk) 16:12, 13 January 2021 (UTC)
I have to agree with the IP editor that this is poorly written and after checking the source, I see that the source does not say what is written here. In fact, the source doesn’t give a reason, it just says that engineers prefer a certain sign convention. The source isn't even about the Fourier transform.
The reason given is nonsense. In fact, both negative and positive signs are used by different communities and there is no problem with convergence of the integral.
Since the source doesn’t give a reason, I will remove the reason. But the reason is this: it is arbitrary. It is just a choice of where you want your negative signs to appear. As an engineer, I have my preference mainly because that is the way I was taught and that is the way it appears in most of my textbooks. You may find justifications for one choice or the other, but you will not find a definitive reason the sign must be negative or must be positive. Constant314 ( talk) 12:26, 8 October 2021 (UTC)
I use the same convention that you are advocating. I have no problem with using that convention. If you want to state a reason for that convention in the article you need a reliable secondary source that states that reason. No amount of WP:OR will change that. However, I do not mind dabbling in OR here on the talk page.
Let
I hope it is obvious that
This has no physical effect because physical effects are caused by energy or power. The power of a Fourier transform is computed by multiplying the transform by its conjugate.
Again, I hope it is obvious that
So, lets look at a couple of examples. I will suppress multiplicative constants that clutter up the results.
First, consider the Fourier transform of .
Next, consider the Fourier transform of .
Now let me go way off into OR la-la land to speculate why engineers prefer the usual convention. Consider the Fourier transform of cos(ωt) + sin(ωt). It is . The component at the positive frequency of is . Notice in particular that the sign of the imaginary part is negative. Engineers prefer this because lags by 45°. When an engineer plots this in Cartesian space, it is [1,-1]. The principal argument is negative. Engineers prefer that because the phase of cos(ωt) + sin(ωt) relative to cos(ωt) is negative. Mathematicians consider cos(ωt) and sin(ωt) as basis vectors and they plot cos(ωt) + sin(ωt) as [1,1]. That is all there is to it. Engineers prefer that the Fourier component of sin(ωt) should be negative at positive frequency. Constant314 ( talk) 22:03, 13 November 2021 (UTC)
For real-valued the convention hardly matters, because every frequency has a positive equivalent (e.g. see Aliasing#Sampling_sinusoidal_functions). The concept of negative frequency is unnecessary... two-sided Fourier transforms are redundant.
Cutting to the chase, the convention determines whether is considered a positive or a negative frequency. The customary definition of
instantaneous frequency is the derivative of instantaneous phase, which is whose derivative is
Therefore is a positive frequency. And its measurement is (which means is ).
I conclude that those who claim otherwise have a different definition of instantaneous phase or instantaneous frequency, which puts them at odds with Wikipedia's sourced articles. The burden is on them to provide contradictory sources.
--
Bob K (
talk) 04:38, 15 November 2021 (UTC)
I get your point. But I disagree with your statement "But the reason is this: it is arbitrary. It is just a choice of where you want your negative signs to appear." No. It comes down to your definition of the instantaneous phase and frequency of function When you "arbitrarily" choose you are also arbitrarily rejecting the customary definitions of instantaneous phase and frequency. Therefore you need to provide sourced reasons for that whim.
--
Bob K (
talk) 05:10, 15 November 2021 (UTC)
I went through several of my text books. Here is what I found.
Constant314 ( talk) 17:28, 17 November 2021 (UTC)
Thank you. I can expand the upper list, if needed, but it seems to be coming down to signals and communication vs electromagnetics. Amazingly, I still have my undergrad copy of
It is a dense 554-page book, with not a single Fourier transform formula or even Euler's formula. My take-away is that the EM applications of Fourier transform theory don't go deep enough to matter which convention they use. To quote myself (above) "For real-valued the convention hardly matters". The concept of negative frequency is not useful. So your statement:
might be a misleading generalization based on certain limited applications of transform theory.
Anyhow, we're getting a little off track. The point is that the convention chosen for the article (which we both agree with) was not an arbitrary coin toss. It might not have any consequences for EM theory, but it does have consequences for signal theory. So I added a footnote that does not need an external citation. All it relies on is a Wikilink to our instantaneous frequency article.
--
Bob K (
talk) 12:30, 18 November 2021 (UTC)
That sounds about right, based on my distant memories (circa 1968) of one EM theory course.
--
Bob K (
talk) 11:47, 16 March 2022 (UTC)
I think this article should be improved in some manner for the general public that is scientifically minded but not taking a full calculus class in college. I think the possibility of adding it to the Simple English Wikipedia with easier to understand language is a good idea, in addition to the process of adding explanations and writing that is not mathematically centered. Maybe including something of this sort, "The Fourier Transform helps to transform functions such as cosine and sine into different output functions that behave differently than normal trigonometric functions." ScientistBuilder ( talk) 01:38, 14 October 2021 (UTC)ScientistBuilder ScientistBuilder ( talk) 01:38, 14 October 2021 (UTC)
A quick impression is that this section could be simplified, perhaps making use of the Analytic signal concept instead of Fourier series. I'll try to give that some thought.
Furthermore, the statement "every real sinusoid consists of an equal contribution of positive and negative frequency components, which is true of all real signals" is misleading. It is a cancellation, not a contribution, analogous to something like "10 apples consists of 5 apples + 5 bananas and 5 apples - 5 bananas". (See Negative_frequency#Sinusoids)
-- Bob K ( talk) 12:34, 16 March 2022 (UTC)
I think that's on the right track. I was definitely struggling with that section... I kept coming back to the question "Does it even need to be here?" IMO, the ccrma.stanford.edu viewpoint is the easy explanation, more of an engineering convenience than a true insight. I'm all in favor of conveniences, but I'm also in favor of distinguishing them from the underlying realities.
--
Bob K (
talk) 11:45, 17 March 2022 (UTC)
I was hoping to see an IPA pronunciation in the first sentence of the article (but there is not one). Wiktionary has an English pronunciation for Fourier (as a surname), which might apply to Fourier transform. - excarnateSojourner ( talk| contrib) 21:30, 11 July 2022 (UTC)
The section idiotically titled Introduction contains this passage:
"Although Fourier series can represent periodic waveforms as the sum of harmonically-related sinusoids, Fourier series can't represent non-periodic waveforms. However, the Fourier transform is able to represent non-periodic waveforms as well. It achieves this by applying a limiting process to lengthen the period of any waveform to infinity and then treating that as a periodic waveform. [1]"
The last sentence is complete mathematical nonsense, and is not taken from the cited reference. 2601:200:C000:1A0:BC00:5039:DB55:E9EC ( talk) 19:13, 29 July 2022 (UTC)
References
As I tried to write a few clarifying sentences in the section Fourier transform for functions that are zero outside an interval, in order to specify the periodic function whose Fourier series was referred to ... I realized that the article refers to the same function when discussing the Fourier transform. But: It's not the same function. One is periodic and the other is zero outside the interval [-T/2, T/2].
I hope someone knowledgeable about this subject can fix this apparent problem. 2601:200:C000:1A0:BC00:5039:DB55:E9EC ( talk) 20:08, 29 July 2022 (UTC)
I find the introduction difficult to understand. It would be far better to avoid use of the word "transform" at first in the explanation, and instead say that the Fourier transform is a mapping taking an integrable function on Rn to an integrable function.
And that according to a version of the Fourier transform F favored by mathematicians, F4 = I.
Surely this is worthy of prominent mention. 2601:200:C000:1A0:FDA6:8A26:FCCA:2C1B ( talk) 04:39, 22 September 2022 (UTC)
Well done, but it repeats a lot of information available thru WikiLinks, and now also added to Fourier transform#The analysis formula. IMO this section can be downsized or eliminated. Bob K ( talk) 16:47, 12 December 2022 (UTC)
It has been a decade since the
ξ_vs_ν_? discussion between just 2 editors. And as noted then, ν was the original preference, replaced without a consensus. Now I am asking if there is any consensus for reverting back to ν, because ξ is unnecessarily intimidating-looking (IMO).
--
Bob K (
talk) 00:59, 19 December 2022 (UTC)
The reliance on:
is no better than relying on the transform pair:
Consequently, you've made a mountain out of a molehill.
--
Bob K (
talk) 01:59, 25 December 2022 (UTC)
The result of this section seems to be the statement:
Under appropriate conditions, the Fourier series of f will equal the function f. In other words, f can be written:
At the very least it needs to clarified that the first equality only applies in the interval T.
But also we already know that can be recovered from So what you are showing is that when its domain is bounded, it can also be recovered from discrete samples of which, by the way, is the dual of the time domain
sampling theorem. This is mildly interesting, but it strikes me as a proof looking for a home, and the whole article is already pretty cluttered with stuff.
--
Bob K (
talk) 09:00, 27 December 2022 (UTC)
The function is listed as a square integrable function (Line 207). I think this is only true if . However, this condition is not mentioned, and if this condition is included, (Line 207) becomes just a restatement of (Line 206) with the replacement . 18.29.20.123 ( talk) 16:40, 10 February 2023 (UTC)
I would like to discuss the recent revert of my edit. The edit, in part, addresssed the comment from the IP editor about that our tables were incorrect. While looking at the article I noticed it was inconsistent in the placement of vs so I fixed it. It seems the objections are mostly about the later. For which I would like to point out, the prevailing culture is that if a is used, it is before the i to see this I would like to point to:
Aside from the last one, these are large professional orgainzations trying to make something for the masses. About the last one, I include the last one because I would like to return to the status quo for the article. I also feel it would benefit readers because they are (in my opinion) more likely to encouter in books/papers/references. Thenub314 ( talk) 15:47, 15 February 2023 (UTC)
I removed a reference from here because the cited reference wasn't discussing the Fourier transform in the section indicated. I will try to find something more specific to this topic. Thenub314 ( talk) 20:47, 16 February 2023 (UTC)
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:
Participate in the deletion discussion at the nomination page. — Community Tech bot ( talk) 14:53, 17 February 2023 (UTC)
In the inversion section the integrals are over the variable σ but the variable ξ appears in the integrands. Where does the ξ come from? — Preceding unsigned comment added by 2.27.171.228 ( talk) 21:25, 21 August 2023 (UTC)
I reverted this edit, because Wikipedia articles do not usually include code samples (see MOS:CODE), unless those code samples illustrate some fundamental aspect of an algorithm. In this case, the algorithm (the fast Fourier transform, for which there is already a separate article) is not actually shown. Instead, it uses a builtin function of the numpy library. So this code is very python-specific, and is not a good illustration of the Fourier transform. Tito Omburo ( talk) 10:42, 23 May 2024 (UTC)