Talk:Hilbert transform

message: The article should say what the Hilbert transform is used for; currently is does not, and this is a significant omission. I suggest a section be added entitled 'Applications of the Hilbert transform' which gives practical examples of what the transform can be used for.  —Preceding unsigned comment added by 81.168.113.121 (talk) 08:53, 27 February 2008 (UTC)
thread_title:  Practical Uses 

message: The notation \hat{s} should be removed.  It is not standard across mathematics and signal processing.  And conflicts with the notation more standard notation for the fourier transform.  Seeing how the Hilbert transform almost always involves a discussion of the Fourier transform I believe this to be a poor choice of notation.
: Please sign your entries with "~~~~".
: Regarding your suggestion, the argument that it is "not standard across mathematics and signal processing" is not a strong argument, since very few notations of any kind enjoy that unique status.  The hat notation is quite common in signal processing, at least.  It's no accident that it appears here.  Furthermore, I for one have seen quite a few Fourier transforms, and I don't recall any that used the hat notation.  I am not saying it doesn't exist, but it is certainly far from being the standard you claim it to be.  --Bob K 05:21, 14 July 2006 (UTC)
replies: text:  Regarding my suggestion.  Here are a list of references I might mention that use a hat to denote the Fourier transform.  M. Pinsky "Introduction to Fourier Analysis and wavelets"  B. Hubbard "The world according to wavelets"  C. Blatter "Wavelets: A primer"  W. Rudin "Real and Complex Analysis"  Conte and de Boor "Elementary Numerical Analysis An algorithmic Approach"  E. Stien and R. Shakarchi "Fourier Analysis: An introduction"  And the list continues, see also papers published by R. and C. Fefferman, T. Tao, E. Stein, etc.  Now I don't doubt that there is plenty of evidence for the Hilbert Transform. But the notation definitely conflicts, and we can agree that H(f) is a reasonable way to denote it.  We can definitely add comments about different notations in different fields.   
Thenub314 20:24, 17 July 2006 (UTC)
replies: text:  That's a good list.  (Thanks.)  It's not an encyclopedia's job to suppress any of the common notations.  On the contrary, its job is to try to describe the real world in all its ugly complexity and inconsistency.  The really hard part, of course, is trying to maximize its own internal consistency at the same time.  Maybe some day each reader will have a "profile", of his own choosing... and articles will be automatically translated through that profile to create the view he wants to see.  --Bob K 03:41, 18 July 2006 (UTC)
thread_title:  Hat notation 

message: Math world has the plus and minus reversed in {1 \over \pi} ln \left | {t+{1 \over 2} \over t-{1 \over 2}} \right |
The way it is now coincides with the MATLAB function when plotted, and also coincides with the diagram in my book (going from −∞ to +∞ during a positive side of the square wave), and the equation in my book {A \over \pi} ln \left | {t \over t-\tau} \right | for a pulse of width τ delayed by τ/2, so I'm leaving it this way. - Omegatron July 2, 2005 17:03 (UTC)
Very strange. In the mathworld we see that hilbert transform is the convolution with -1/t*pi function, not 1/t*pi. But the table with hilbert transform is nearly the same. Somebody just have written it down without thinking. 
I will correct it as soon as I create an account.
--83.25.155.136 16:41, 18 September 2005 (UTC)
thread_title:  MathWorld & Matlab 

message: I'm trying to figure out what is going on with the discrete HT.  That section now says that there is an ideal discrete HT, so and so, but this operation cannot be realized in the signal domain.  Then, it presents a filter which seems to do the job, derived from the DFT.  This seems contradictory. --KYN 21:41, 10 November 2005 (UTC)
replies: text: I think it's not contradictory, because his h[n]\, is non-realizable, as written.  But maybe what you are saying is that n does not go to ± inf, because it is just a DFT.  Therefore it can be made realizable, with sufficient delay.  And the "ideal" filter would not have that characteristic.  Is that the problem? --Bob K 09:30, 7 December 2005 (UTC)

text: Please clarify this definition, if you can.  I'm not getting it yet.  --Bob K 20:36, 7 December 2005 (UTC)

text: Hopefully it's all OBE now, since I just re-wrote that section of the article. --Bob K 01:52, 8 December 2005 (UTC)
thread_title:  Discrete HT 

message: I don't really have an experience with the DHT and therefore I must ask the following questions, some of which hopefully can work their way into the article to make it more understandable.
* Exactly how is the DHT related to the continuous HT?  I can see that there are similarities, but not that the DHT follows straightforward from the CHT.  The CHT is defined as it is simply because it make sense in a number of applications.  I guess that it can be shown that the DHT also has this property.  Examples?
:>>>Probably its main use is to convert real-valued discrete signals into the analytic representation. --Bob K 06:21, 4 January 2006 (UTC)
replies: text: >>>But the discrete version of the signal is no longer band-limited.  The DTFT is periodically extended (to infinity).  A CHT would shift all the positive freq components by -90 deg (and all the negative freq components by +90 deg).  But (as the article states) what's actually needed is to shift (0,π] by -90, (π,2π] by +90, (2π,3π] by -90, etc.  The derivation of the corresponding h[n] sequence is indicated at slide 21.  Note that the \sin^2()\, function takes on alternating 1,0,1,0,1,0,... values for odd and even values of n. --Bob K 06:21, 4 January 2006 (UTC)

text: >>>Yes, except for the fact that it has to be approximated by something with finite duration.  --Bob K 06:21, 4 January 2006 (UTC)

text: >>>Filter-order vs. frequency-response and latency is always a tradeoff.  Another issue is truncation vs. numerous methods of gradual tapering off to zero-valued coefficients.  These topics are of general enough interest that they should be treated elsewhere and only referenced here.  --Bob K 06:21, 4 January 2006 (UTC)

text: >>>Sorry, I haven't investigated that myself.  --Bob K 06:21, 4 January 2006 (UTC)
replies: text: >>>I just took a peek, and the treatment is indeed brief.  But filter-design is a well-studied and dauntingly large subject to cover in detail.  And excellent tools are widely available so that most practitioners don't really have to understand all the issues.  So I doubt that wikipedia will take on that challenge anytime soon, if ever.  All most readers will need to know is that there are tradeoffs, so they should seek the help of a decent tool if they want to do a good job.  It's like telling people that their car needs its oil changed, rather than detailing how to change the oil in every kind of car.  I think that is valid.  --Bob K 06:55, 4 January 2006 (UTC)

text: >>>Perhaps a second reading would help.  But here is the short version:  Instead of the convolution, people often do a DFT, modify the coefficients in the obvious way, and then do an inverse DFT.  That is equivalent to a circular convolution with the approximation shown in the article.  --Bob K 06:21, 4 January 2006 (UTC)

text: >>>Yes.  Due to the efficiency possible with the FFT algorithm, the DFT/IDFT approach is often faster than simple convolution, even when actual multiplications (by other than ±1 and 0) are required.  Again, that topic is of sufficiently general interest that this is not the right place for it.  And again, I have not searched for the reference.  If you have looked already, then you might try looking for the names overlap-save and overlap-add, which describe specific techniques for piecing together the outputs of block-processing, an essential part of the FFT approach with streaming data.  --Bob K 06:21, 4 January 2006 (UTC)
replies: text: I have summarized the first part of this discussion in the intro of the DHT section.  Please have a look to see if it came out the right way.  --KYN 13:48, 4 January 2006 (UTC)
replies: text: Looks fine.  --Bob K 20:06, 4 January 2006 (UTC)

text: The article does contain a link to circular convolution, which goes into more depth about fast convolution.  But normally, one uses design techniques to choose a finite filter order.  Then one transforms that design into the frequency domain.  For some reason, when it comes to the DHT, there seems to be a propensity to skip that step, and just use the idealized H(e^{j \omega})\,, which makes circular convolution unavoidable.  Seems to me that it's like using an ideal rectangle for a lowpass filter, which nobody does.  Since this seems to be endemic to the DHT, I thought it was worth a little space at the expense of some redundancy.  --Bob K 23:28, 4 January 2006 (UTC)
replies: text: Ok, I will look at the fast convolution section in the circular convolution articles which you pointed out, and try to make sense of all this.  For some reason this page does not appear when I try to search the wikipedia for "fast convolution".  Any idea why?--KYN 00:42, 5 January 2006 (UTC)
replies: text: Sorry, no.  It fails for me also.  --Bob K 01:47, 5 January 2006 (UTC)

text: I just tried it again.  It still failed, but now it knows something is wrong, and it recommends I use Google or Yahoo, with the "Wikipedia" option selected.  Those work just fine. --Bob K 02:34, 24 April 2006 (UTC)
thread_title:  Discrete HT again 

message: Did he develop and define this concept? Whaa? 21:31, 23 April 2006 (UTC)
replies: text: Excellent question.  I don't exactly know myself, I suspect that either it came up during his study of integral equations.  But none of the biographies I have read explain exactly what he was thinking about in this respect.  As far as the history of the Hilbert transform, the first real theorems I know that were proven about it go back to M. Riesz.  It is also the prototypical example of a singular integral operator, and the Hilbert transform was what motivated Zygmund to study such opertators. 75.3.32.251 12:14, 18 July 2006 (UTC)
thread_title:  Why named after [[David Hilbert]]? 

message: This has undoubtedly been discussed somewhere else, so here we go again.  If it comes to a vote, I prefer this convention:
:\mathcal{H}\{s(t)\} = h(t)*s(t)\,
to this one:
:\mathcal{H}\{s\}(t) = (h*s)(t)\,
--Bob K 14:50, 17 October 2006 (UTC)
replies: text: I agree with Bob K.  First Harmonic 19:39, 17 October 2006 (UTC)

text: Clearly, both notations are common in the literature, and on WP.  The notation that I prefer is commonly used in signal processing and communication systems engineering, wheras the notation preferred by Thenub314 is more common (I think) in mathematics.  So you could argue that either notation (or both) is acceptable.  That being said, it comes down to a question of taste and familiarity.  I prefer the one notation over the other not only because it is what I am familiar with, but also because I think it is easier to understand, clearly represents what is what, and is less ambiguous than the other.  First Harmonic 16:01, 26 October 2006 (UTC)

text: I'm with Bob K on the \mathcal{H}\{s(t)\} = h(t)*s(t)\, notation. I've seen both on Wikipedia, but I find this one more clear. -Roger (talk) 17:03, 15 May 2009 (UTC)
thread_title:  (h*s)(t) or h(t)*s(t) ? 

message: I search in vain in each of several linked articles in this article including convolution, fourier transform, signal processing, etc., etc., etc., etc., finding the one thing all of these articles have in common is that not one defines ALL of the variables used. That is perfectly fine for those who are already experts, but makes it basically useless to those who are unfamiliar with the subject. If the unfamiliar reader isn't the targeted audience then what is the point of an encyclopedia anyway? Surely not just a book mark for all those equations you regularly use? 
You don't need to break into the text, just add a section below where each variable is defined such as t =, omega =, theta=, etc.. I think that is not too much to ask. Drillerguy 14:42, 3 September 2007 (UTC)
thread_title:  PLEASE DEFINE YOUR VARIABLES!!!!!

message: The exact same text seems to have been added in the page Gianfelici Transform.  This article has been nominated for deletion because it is not clear if this transform is notable.  Maybe we should make sure this section belongs here.  There doesn't seem to be a lot that comes up about it on Google besides the reference given. Thenub314 (talk) 17:33, 12 May 2008 (UTC)
replies: text: I have once again removed reference to the Gianfelici transform.  Also the editor who added the Gianfelici reference (which is not used in the text) removed a reference which is used in the text (by Charles Fefferman, one of the most influential harmonic analysts in the history of the subject.)  I would ask that any case to be made for inclusion of this material should be done on the talk page, preferable with multiple independent references supporting the assertion that there is such a thing as a Gianfelici transform.  Google scholar doesn't turn up anything.  siℓℓy rabbit (talk) 22:32, 6 August 2008 (UTC)
thread_title:  Iterated Hilbet transfrom. 

message: Just to make sure I am correct, consider the sentance...
"However, a priori this may only be defined for u a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported distributions are dense in Lp."
Shouldn't this be functions of compact support?  Distributions of compact support do not sit inside Lp, if they are dense you'd first want to intersect with Lp. Thenub314 (talk) 13:53, 18 May 2008 (UTC)
replies: text: Yes, of course that is what I mean.  The discussion is somewhat informal, but I will change it to make it more clear. silly rabbit (talk) 14:01, 18 May 2008 (UTC)

text: I think compact support is needed in the next paragraph where I use the commutativity of convolution.  The Schwartz space is not stable under convolution with tempered distributions, whereas compactly supported distributions are.  silly rabbit (talk) 14:15, 18 May 2008 (UTC)
replies: text:  Comment:  I struck out the obviously false statement above.  silly rabbit (talk) 14:30, 18 May 2008 (UTC)

text: Correction: What I mean is that as long as all but one of the distributions in the convolution are compactly supported, then the convolution is a commutative and associative operation, so that formal manipulations can be performed with it.  silly rabbit (talk) 14:33, 18 May 2008 (UTC)
thread_title:  Convolutions. 

message: Would anyone object to me changing the current notation for the Hilbert transform of a function from H(u)(t) to  H\{u(t)\}? This is more consistent with the way Fourier transforms are defined in signal processing (in my experience), and less confusing since it's currently easy to confuse these expressions for a multiplication. The only potential downside is that the Hilbert domain variable isn't immediately obvious, but that isn't really a big deal. -Roger (talk) 16:54, 15 May 2009 (UTC)
replies: text: Well it's been a year (!) and since I haven't heard any objections I plan to change the transform notation to \mathcal{H}\{u(t)\}. It'll be inline with the Laplace transform article (though the Fourier transform uses the notation currently in this article) and the explicit Hilbert domain variable (t)  will be dropped. -Roger (talk) 03:38, 1 June 2010 (UTC)
thread_title:  Changing notation 

message: I have added a sentence on the development of Hilbert spectroscopy to the History section, as I could not see a better place. I have added the redlinked topic at Wikipedia:Requested articles/Natural sciences#Physical chemistry because I know next to nothing about it outside of reading the topic on BBC news, a couple of paper abstracts and the Terahertz article. My experience ended at narrowband spectral analysis for specific chemical bonds using infrared. If this is the wrong place, please move it to where it fits better. -84user (talk) 17:00, 20 October 2009 (UTC)
thread_title:  Hilbert spectroscopy 

message: The introduction says: "[..] if f(z) is analytic in the plane Im z > 0 and u(t) = Re f(t + 0·i ) then Im f(t + 0·i ) = H(u)(t) up to an additive constant, provided this Hilbert transform exists." That can't be right, as it implies that both 0=\Im(\cos(t)) and sin(t)=\Im(e^{it}) are Hilbert transforms of the cos(t) (since cos(z) and e^iz are both holomorphic on the plane). Basically the same claim is repeated under "Conjugate functions".
I'm not sure what would be a correct version of the statement since I don't know much about Hilbert transforms (that's why I came here..), and don't have easy access to the cited source. I guess some condition like  lim_{y\rightarrow \infty} f(x+iy) < \infty  (or perhaps =0) is missing? --132.68.56.67 (talk) 13:42, 5 January 2012 (UTC)
replies: text: The assumption in the article is that f is in L^p for p<infinity. Sławomir Biały (talk) 17:29, 5 January 2012 (UTC)
thread_title:  Relation to holomorphic functions 

message: In sect 6.6 it says H = i (2P - I), but P has not been mantioned as far as I can see, Presumably the projection on the one component of the splitting mentioned above?Billlion (talk) 12:31, 12 April 2013 (UTC)
replies: text: Yes, but that is already given in that same sentence: "...with P being the orthogonal projection from L2(R) onto (the Hardy space) H2(R)...". Mct mht (talk) 00:08, 13 April 2013 (UTC)
replies: text: Thanks, I must have been working while too tired to miss that! Billlion (talk) 08:27, 14 April 2013 (UTC)
replies: text: No! I added it after your post! Bdmy (talk) 08:35, 14 April 2013 (UTC)

text: Sorry, Billlion, should have checked first. Mct mht (talk) 17:47, 14 April 2013 (UTC)
thread_title: Invariance

message: In the history section it says the Hilbert transform may be defined for Lp with 1 ≤ p ≤ infinity.  I think this is true, but the following sentence says that the Hilbert transform is bounded on the same range of p; I think the actual statement should exclude p=1 and p=infinity.  — Preceding unsigned comment added by Mdn33 (talk • contribs) 22:04, 15 July 2014 (UTC)
thread_title:  History 

message: In the Introduction/Notation section, we have references to Brandwood 2003 and Bracewell 2000. No such reference is listed in the references section. Does anyone have any idea what these references are supposed to refer to? There is a Bracewell 1986, possibly a different edition of the same book, but if so that would need to be checked to make sure the page numbers are still correct. —David Eppstein (talk) 21:16, 5 March 2015 (UTC)
replies: text: Google books shows the Bracewell book to have a 2000 edition; the same for the 2003 Brandwood book. Unfortunately I have neither book. --Mark viking (talk) 00:36, 6 March 2015 (UTC)
thread_title:  Two broken references 

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thread_title:  External links modified 

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thread_title:  External links modified 

message: The following comment was inserted after the section heading of the Titchmarsh's theorem section of this article, by User:24.233.245.156 (talk • contribs) 08:23, 4 May 2018 (UTC)
replies: text: I agree with the above.  (I may even be the mysterious person XXX, for all I know ;-) Sławomir Biały (talk) 17:34, 10 June 2018 (UTC)

text: Pandey is not a very good book.  I think a better reference could be found, and precise conditions given.  Sławomir Biały (talk) 11:37, 15 June 2018 (UTC)
thread_title:  ''Titchmarsh's theorem'' section<span id="Titchmarsh's_theorem"></span> 

message: The Hilbert transform can be defined in various ways, one of which leads to a change in sign of  H(u)(t). The current definition with  {du(\tau) \over t-\tau}  actually leads to a Fourier multiplier of  i \text{sgn}(\omega), not  ii \text{sgn}(\omega). At least this is my understanding from Simon's "A comprehensive course in analysis." I do not know if it is easier to change the original definition or the Fourier multiplier later, since I think the original definition is actually the nonstandard one and the Fourier multiplier given here is the standard multiplier.
Jupsal (talk) 00:36, 26 December 2018 (UTC)
thread_title:  Incorrect Fourier multiplier given the current definition of the transform 

message: Maybe it's just my ignorance, but the sentence that doesn't seem very helpful is:
Specifically, the Hilbert transform of u is its harmonic conjugate , a function of the real variable t such that the complex-valued function  admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations.
Perhaps it can be re-phrased or re-located to a more mathematical section.
--Bob K (talk) 20:42, 22 January 2021 (UTC)
thread_title:  delete a sentence from lede paragraph? 

message: Maybe someone knowledgable in the Discrete Hilbert Transform could rework the section? 
- There is one reference to MATLAB and two references to the same master thesis and nothing else
- FIR filter types are used and partially defined but no reference is given as to where these definitions come from or why this type classification is used or relevant
- a FIR approximation is mentioned and shown in Figure 1 but no reference is given where this is used or who uses this
- hilb(65) in Figure 2 is not defined
- the however bulletpoints seem to be blanket statements or specific to certain types of FIR, but no further sources or discussion is provided
- half of the section is about implementation details in MATLAB? 31.10.205.218 (talk) 04:31, 5 June 2024 (UTC)
thread_title:  Discrete Hilbert Transform