Talk:Sinc function

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1 Definition
2 More on sinc definition
3 Removed this section:
4 incorrect use of de l'Hospital rule
5 Relationship to delta function
6 Revisit normalization and definition issue
7 The "disambiguation" of the sinc function serves no useful purpose and should be reversed.
8 i am re-asserting the issue of the base or primary definition.
9 move request to admins
10 Two different functions?
11 i have AfD the other sinc function pages ...
12 yea! PAR! that looks really nice!
13 Question
14 2D Sinc?
15 Integral proof
16 Table

[edit] Definition

I have seen definitions of sinc where the removable singularity at x=0 is removed explicity, ie.,

$\mathrm{sinc}\ x = \left\{\begin{matrix} {\sin{x}\over x} &\mathrm{if} & x\ne 0 \\ 1 & \mathrm{otherwise} & \end{matrix}\right.$

However, I've also seen definitions where it is simply left as (sin x)/x. Which is best? Shall we mention both? Dysprosia 07:39, 8 Jan 2005 (UTC)

vote for explicit removal plus comment that that makes the function analytic at 0. --Zero 12:07, 1 Feb 2005 (UTC)

I agree, I inserted it Paul Reiser 15:03, 1 Feb 2005 (UTC)

[edit] More on sinc definition

first of all, there appears to be something missing in the first paragraph. "It is defined by:" what?!

certainly in some math and physics texts, i have seen the definition of sinc() as stated in this

$\mathrm{sinc}\ x = \left\{\begin{matrix} {\sin{x}\over x} &\mathrm{if} & x\ne 0 \\ 1 & \mathrm{otherwise} & \end{matrix}\right.$

but in electrical engineering signal processing texts (this is where the sinc() function is used most often, as far as i can tell) the definition is adjusted a little:

$\mathrm{sinc}\ x = \left\{\begin{matrix} {\sin{\pi x}\over {\pi x}} &\mathrm{if} & x\ne 0 \\ 1 & \mathrm{otherwise} & \end{matrix}\right.$

it is mentioned as the "normalized sinc" function in this article, but i have never seen that term used anywhere else.

this is typical of the difference in notation between math and physics texts and electrical engineering texts. is it $i \$ or $j = \sqrt {-1} \$ ? is it $\omega \$ or $j 2 \pi f \$ in the Fourier Transform? is it $\sin (x)/x \$ or $\sin (\pi x) / (\pi x) \$ ?

i am bothered by the lack of self-consistent notation inside of wikipedia (the i vs. j thing doesn't bother me) but am not sure what can be done about it. r b-j 17:07, 11 Mar 2005 (UTC)

A google search on "sinc function" is helpful. It seems both are used. How do you suggest we fix the article? Regarding what to do, I think we just have to fix any inconsistencies whenever we run across them. Paul Reiser 17:38, 11 Mar 2005 (UTC)

believe me, i have googled it. and it comes out both ways. Wolfram likes it without the pi but there are many web pages on both sides of the definition. what i think has happened is that some electrical engineering profs have long ago decided that the $\sin (\pi x) / (\pi x) \$ definition was better, espcially in signal processing (since the sinc() function hits zero at all other sample times). Many have also redefined the Fourier Transform from the definition here to one with $j 2 \pi f \$ and no $1/( 2 \pi ) \$ in the inverse transform. This simplifies the duality property, power expressions, and Parseval's theorem. It would be nice if it were completely consistent in Wikipedia and the simplest definitions, but i am afraid that historical and practical concerns are not identical. r b-j 05:47, 12 Mar 2005 (UTC)

The "normalized" means that x would have a range of -1 to 1 and cover the unit circle. Cburnett 17:29, 14 Mar 2005 (UTC)

[edit] Removed this section:

It has the following limiting property:

$\lim_{a\rightarrow \infty}\frac{1}{\pi x}\sin\left(\frac{x}{a}\right) =\lim_{a\rightarrow \infty}\frac{1}{\pi a}\textrm{sinc}(x/a)=\delta(a)$

where δ is the Dirac delta function. This is an interesting equation since it contradicts the view of the delta function δ(x) as being zero every where except at x=0. In this representation, the delta function oscillates infinitely rapidly inside an envelope of y=±1/x. If a function is continuous in some interval that does not contain zero, then the integral times the delta function will be zero. In this case, this is due to the infinitely fast variation of the delta function which averages to zero.

The displayed equation does not make sense. The variable a is a dummy variable so the answer cannot be a function of it. The first limit in fact does not exist according to the definition of limit and so the second doesn't either (as it's the same function). The comment about "oscillating infinite often" has no mathematical meaning. --Zero 12:07, 1 Feb 2005 (UTC)

Sorry about that, its a typo and should read δ(x) on the right hand side.

With regard to the question of whether the limit exists, a quote from the Dirac delta function article says:

The delta function can be viewed as the limit of a sequence of functions

$\delta (x) = \lim_{a\to 0} \delta_a(x)$

where δ_a(x) is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (on groups more general than the real numbers, e.g. the unit circle).

With regard to "oscillating infinitely often", I agree that strictly speaking it has no meaning, but neither does this quote from the delta function article:

The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one.

The point I was trying to make is that the above informal idea of the delta function is not applicable to the above limit of the sinc function. We can write about the delta function in strictly correct mathematical terms, which will be opaque to anyone without mathematical training, or we can relax and give a newcomer an intuitive idea of the behavior of the delta function. I believe the latter is best. Maybe the "informally thought of" caveat can be added to the "oscillating infinitely often" statement? Anyway, looking at what I've just written, I think that this discussion of the delta function should go in the delta function article, not here in the sinc function article, so I will put it in there soon. Paul Reiser 15:03, 1 Feb 2005 (UTC)

I think the limit

$\lim_{a\rightarrow \infty}\frac{1}{\pi x}\sin\left(\frac{x}{a}\right) =\lim_{a\rightarrow \infty}\frac{1}{\pi a}\textrm{sinc}(x/a)=\delta(x)$

is valid, if interpreted in the distributional sense, i.e.

$\lim_{a\rightarrow \infty}\int_{-\infty}^\infty \frac{1}{\pi x}\sin\left(\frac{x}{a}\right)\varphi(x)\,dx =\lim_{a\rightarrow \infty}\int_{-\infty}^\infty \frac{1}{\pi a}\textrm{sinc}(x/a)\varphi(x)\,dx =\int_{-\infty}^\infty\delta(x)\varphi(x)\,dx = \varphi(0),$

for all compactly supported functions φ. The limit is of course invalid in the usual calculus sense, because the δ "function" is not a function but a distribution. However, I'm not absolutely sure about it, so I didn't put it back in. Anyway, if the limit is indeed correct, then this could well be mentioned in this article (either with the intuitive explanation, or with a mathematically precise explanation, or both). Personally, I would leave out the sentence "This is an interesting equation since it contradicts the view of the delta function δ(x) as being zero every where except at x=0". -- Jitse Niesen 14:43, 2 Feb 2005 (UTC)

I think it should be $a\rightarrow 0$ , not $a\rightarrow \infty$ , in which case I agree with your interpretation. --Zero 17:49, 2 Feb 2005 (UTC)

I agree with that too. Again, it might be worth mentioning here, but its mostly a discussion of the nature of the delta function and would better go in the delta function article. I will do that soon, and I will be sure to drink a cup of coffee and wake up before I do it. Paul Reiser 18:53, 2 Feb 2005 (UTC)

Very intersting. Belongs in the delta function article linas 17:30, 26 Mar 2005 (UTC)

I put a mention of it in the introduction, with an explanation in the sinc function article. The delta function article has a number of cases where it is assumed that the delta function is zero everywhere except at x=0, including the idea that the delta function is a probability distribution, and that its support is x=0. I'm not an expert in distributions, so I'm trying to get a discussion going rather than just go in and change things, but it doesn't seem to be working. If you know about this subject, or know anyone who does, I think some input would be helpful. PAR 20:48, 26 Mar 2005 (UTC)

The reason people get into a mess when talking about the delta function is that they think it is a function. It isn't! It's a distribution. Further confusion comes from the word "distribution" having at least two quite different meanings. The delta function is not a "probability distribution", since they are functions which are monotonically non-decreasing (etc). If we take the step function F(x) = 0 (x<0), 1 (x≥ 0), that is a probability distribution. Its density does not exist in the usual sense (as a function) but can be described as a distribution (other sense), in which case we get the delta function. Probabilists prefer to use the language of measures and Stieltjes integrals, so instead of writing $\int \delta(x)g(x)\,dx$ they write $\int g(x)\,dF(x)$ , both of which equal g(0). The talk about being infinite or oscillating infinitely often etc is just an outcome of the mistake of thinking of δ(x) as a function. The correct interpretation is given a few paragraphs above. And, yes, the delta function article needs fixing. I'll try to look at it before long. --Zero 02:04, 27 Mar 2005 (UTC)

I reverted to the previous version and if you revert to your previous version, I won't fight it until we've talked about this for a while. The reason I reverted is because I carefully revised the Feb.1 statement about the limit of the sinc function to produce this new one. Its not a simple reversion to the Feb.1 statement, yet you use phrases from the old statement as the reasons you reverted. For example, I never used the phrase "infinitely often" in the latest statement. I've never used the idea that the delta function is a function without the "informal" modifier. Please read the latest statment before discarding it.

Also, I am trying to make a point here, and its the same one that you are trying to make. Thinking of the delta function as a function can get you into trouble, and this is an excellent example of that. Can you see why it bothers me that every time I try to make this point you revert and leave nothing in its place? Will you please help me come up with something that shakes people out of the "zero everywhere except infinity at zero" mindset, and yet is not so esoteric as to make a newcomers eyes glaze over? PAR 03:18, 27 Mar 2005 (UTC)

PAR, could you please provide a reference for the limit. I have no problems with the current formulation, except that I'd like to add a reference and a caveat that it is not a normal limit. Zero, could you please say exactly what you disagree with? It might be nice to connect the example to the Riemann-Lebesgue lemma, which says that "oscillating infinitely often = zero". -- Jitse Niesen 12:44, 27 Mar 2005 (UTC)

Its taken from some notes I have, and I don't have the original reference. Googling "dirac delta sinc limit" yields a number of possibilities, the best of which seems to be [1] which in turn references Bracewell (1986) on the subject a number of times. I will track down more if this is not sufficient. Thanks for helping us with this. PAR 16:54, 27 Mar 2005 (UTC)

I object to it because it is wrong. The limit on the left side does not exist for any x except x=0. That means the limit claim cannot be allowed as is. We can consider presenting it as a limit in the space of generalised functions but that page is not too user-friendly and I can't see a definition of limit there which is clear enough to refer to. That leaves spelling out the definition, like Jitse did above just under the phrase "in the distributional sense". Notice how the limit is taken outside the integral; this makes all the difference. I would have no objection to seeing that on the page. However, the metaphysical comments about δ(x) we can do without. I find them also wrong (to the extent they are meaningful at all). Even if these were ordinary functions (which δ(x) is not) there is no reason that a limit should have any particular property of the functions in the sequence. It is perfectly ordinary for a limit to satisfy bounds that none of the functions in the sequence satisfy. An example which is quite similar is the Gibbs phenomenon -- we don't say that this contradicts our understanding of the square-wave function having square corners.
In summary: my preference is to put in something like this:

In the language of distributions, the sinc function is related to the delta function δ(x) by

$\lim_{a\rightarrow \infty}\frac{1}{\pi a}\textrm{sinc}(x/a)=\delta(x).$

This is not an ordinary limit, since the left side does not converge except at x=0. Rather, it means that

$\lim_{a\rightarrow \infty}\int_{-\infty}^\infty \frac{1}{\pi a}\textrm{sinc}(x/a)\varphi(x)\,dx =\int_{-\infty}^\infty\delta(x)\varphi(x)\,dx = \varphi(0),$

for any smooth function $\varphi(x)$ with compact support.

Btw, this might look nicer using sinc_N. --Zero 00:35, 28 Mar 2005 (UTC)

Ok, I get the idea. Note, the above needs to have the limit a->0 so that will change things a little, the limit doesn't exist for any x at all, right? I still think we should not pretend the "zero everywhere except at zero" idea of the delta function doesn't exist. The delta function article shows that mindset a number of times, and references with that mindset are all over the place. I think we should use this example to help dispose of that idea. That means we have to make metaphysical comments if only to point out that they are not useful. We shouldn't just "do it right" and ignore the problem. Do you have any ideas on how to do this? PAR 01:27, 28 Mar 2005 (UTC)

Yes, a->0, sorry. As for the delta function, the article on it is the place to correct errors and misconceptions, not this article which is only slightly related. --Zero 11:48, 28 Mar 2005 (UTC)

I revised the statement to conform to your statment above. I kept some "metaphysical comments but I agree, they would better go in the dirac delta function article. Right now, there is just a passing reference in the introduction to the delta function article to this sinc function topic. I couldn't figure out how transfer the metaphysical stuff, but I will eventually unless you edit the delta function article first, which, by the way, I think is a better idea. PAR 13:58, 28 Mar 2005 (UTC)

[edit] incorrect use of de l'Hospital rule

One cannot state that, by de l'Hospital rule, it is

$\lim_{x\to0}\frac{\sin x}{x}=1$

because the de l'Hospital rule uses the derivatives, and in order to derivate the $sin x$ function, one has to use that limit. In fact

$D(\sin x)=\lim_{h\to0}\frac{\sin(x+h)-\sin x}{h}$

$=\lim_{h\to0}\frac{\sin x\cos h+\cos x\sin h -\sin x}{h}$

$=\lim_{h\to0}\frac{\sin x(\cos h -1)-cos x\sin h}{h}$

$=\lim_{h\to0}\left(\frac{\sin x(\cos h -1)}{h}+\frac{\sin h}{h}\right)$

= 1

You dont NEED l'Hospital's rule to calculate the derivative of sin(x). You have just shown that it COULD be used. You could take the derivative of the Taylor expansion of sin(x) to get the derivative of sin(x) without using l'Hospital's rule. Then l'Hospital's rule could be used to prove sin(x)/x -> 1. PAR 15:07, 28 October 2005 (UTC)

I don't understand your reply. I do not use l'Hospital rule to calculate the derivative of sin(x). I have shown that in order to calculate the limit of sin(x)/x with the de l'Hospital rule you must know in advance that sin(x)/x->1, so you can't calculate the limit of sin(x)/x using the de l'Hospital rule. Roberto.zanasi 21:13, 28 October 2005 (UTC)

in that case, you would say

$\lim_{x\to0}\frac{\sin(x)}{x} = \frac{ \lim_{x\to0} \frac{d}{dx} \sin(x)}{ \lim_{x\to0} \frac{d}{dx} x} = \frac{ \lim_{x\to0} \cos(x)}{\lim_{x\to0} 1} = \frac{1}{1} = 1$

r b-j 20:00, 28 October 2005 (UTC)

Roberto, you do have a point. If one wants to use l'Hôpital's rule to calculate the limit, then one has to know the derivative of the sine at x = 0. However, the limit follows immediately from that derivative being one, by the very definition of derivative. Nevertheless, I think that many people of the intended audience will prefer using l'Hôpital's rule and the fact that the derivative of the sine is the cosine (which they accept without much thought). -- Jitse Niesen (talk) 20:15, 28 October 2005 (UTC) (via edit conflict)

Roberto, you say "I have shown that in order to calculate the limit of sin(x)/x with the de l'Hospital rule you must know in advance that sin(x)/x->1". I don't need to know that in advance, unless I use your method of calculating the derivative D(sin(x)). Suppose we calculate the derivative using

$\sin(x)=\sum_{n=0}^\infty\frac{x^{2n+1}}{(2n+1)!}(-1)^n$

Then:

$D(\sin(x))=\lim_{h\rightarrow 0}\frac{\sin(x+h)-\sin(x)}{h}$

$=\lim_{h\rightarrow 0}\sum_{n=0}^\infty\frac{(x+h)^{2n+1}-x^{2n+1}}{h(2n+1)!} (-1)^n$

$=\lim_{h\rightarrow0} \sum_{n=0}^\infty\frac{[x^{2n+1}+h(2n+1)x^{2n}+... O(h^2)]-x^{2n+1}}{h(2n+1)!} (-1)^n$

$=\sum_{n=0}^\infty\frac{x^{2n}}{(2n)!} (-1)^n$

$=\cos(x)\,$

The limit of sin(x)/x was never needed.PAR 20:22, 28 October 2005 (UTC)

i dunno what your point is meant to be PAR, and i dunno if Roberto was the "OP", but you still have a circular logic problem if you use McLauren or Taylor series to make your proof that the " $D(\sin(x))=\cos(x) \$ ". how do you get your McLaurin or Taylor series in the first place without knowing the derivatives of $\sin(x) \$ and $\cos(x) \$ at $x=0 \$ ? BTW, i hope you are not using this "Euler's notation": " $D(\sin(x)) \$ " in articles for the derivative. it's peculiar. why not the normal Newton or Liebniz? r b-j 21:05, 28 October 2005 (UTC)

I see what you are saying. But I could say "how do you get the derivative at zero without first knowing the series expansion?".

the way we did it in my first calculus class was Roberto's way with a geometric argument for why sin(x)/x -> 1 (the chord becomes the same length as the arc as x -> 0).

I guess the final answer depends on the definition of sin(x). I think sin(x) is mathematically (not geometrically!) defined in terms of its series, but I may be wrong. What is the definition of sin(x)? Once we have that, we will have the answer.

Also - I'm only using D(sin(x)) because I was responding to Roberto, who introduced the notation. PAR 21:45, 28 October 2005 (UTC)

i see that's the case. it's an okay notation when i'm doing normed metric spaces, but i don't like it for normal calculus. it's just a preference, there's little else wrong with it. r b-j 02:05, 29 October 2005 (UTC)

I have changed the wording a bit so as not to imply that l'Hospital's rule can necessrily be used as a proof of the statement in question. PAR 03:07, 29 October 2005 (UTC)

Ok, the very question is: which is the definition of sin(x)? If we define sin(x) geometrically, then we cannot use the de l'Hospital rule to calculate the limit of sin(x)/x, as I have shown. This is the normal way (pre-graduate and university courses, here in Italy). If we define sin(x) like that:

$\sin(x)=\sum_{n=0}^\infty\frac{x^{2n+1}}{(2n+1)!}(-1)^n$

then it is possible to use de l'Hospital. I think that this is an "innatural" definition (but more rigorous than the classical one), but it is my very humble opinion... Roberto.zanasi (how can I sign my posts automatically? I'm quite a newby :-))

Hi Roberto - I am quite sure that whatever the definition is, it should not be geometrical, because that involves a lot of unnessecary baggage, needing to build up a two-dimensional euclidean metric space before you can define the function. It should be very simple and not make reference to triangles, etc. I consulted "Shilov - Elementary Real and Complex Analysis" and he gives two definitions:

1. sin(x) and cos(x) are functions of complex variables x such that:

$\sin^2(x)+\cos^2(x)=1\,$
$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)\,$
$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)\,$
$0<\sin(x)<x<\sin(x)/\cos(x)\,$ for sufficiently small x>0

2.the power series

Shilov says either definition yields the other as a result, but the series definition is preferred.

Also - sign your name with four tildes: ~ ~ ~ ~ PAR 15:10, 29 October 2005 (UTC)

Very well. In that case, the use of the de L'Hospital rule is legitimate. Thanks. Roberto.zanasi 16:50, 30 October 2005 (UTC)

[edit] Relationship to delta function

Should this:

$\lim_{a\rightarrow 0}\frac{1}{\pi a}\textrm{sinc}(x/a)=\delta(x).$

actually be this?:

$\lim_{a\rightarrow 0}\frac{1}{a}\textrm{sinc}(x/a)=\delta(x).$

--Bob K 19:26, 28 March 2006 (UTC)

No, I don't think so, at least not under the definition used in the article. It follows from

$\int_{-\infty}^\infty \frac{\sin x}{x} \,dx = \pi.$

that at least the constant is correct. However, if you use the alternative definition which in the article is denoted by sinc_π(x), then the second formula is indeed correct. -- Jitse Niesen (talk) 07:50, 29 March 2006 (UTC)

[edit] Revisit normalization and definition issue

Since the article itself says: " Applications of the sinc function are found in digital signal processing, communication theory, control theory, and optics. " I can pretty much guarantee that, as the sinc() function is presently defined herein, that every appearance of sinc will have a π in it. Since nearly every communications/signal_processing text defines the sinc() function as

$\mathrm{sinc}(x) = \left\{\begin{matrix} {\frac{\sin(\pi x)}{\pi x}} &\mathrm{if} & x\ne 0 \\ 1 & \mathrm{otherwise} & \end{matrix}\right.$

why don't we change the definition to the normalized form, make mention that the historical definition and some mathematics texts will have it without the π and i am will to find all citations of this article and fix the reference to sinc() so that WP is consistent. Rbj 21:46, 30 April 2006 (UTC)

Every appearance of sinc will have a π in it In your world. In my world (math and physics), I rarely see it defined with a π. We went through the same thing with the Fourier series stuff and the resolution was that if you want to understand the mathematics and prove theorems, learn the mathematicians notation. If you want to use it in a practical sense, learn the notation that suits that field. The article should attempt to address both users: Someone coming from a communications article should not have too much difficulty finding what they want, nor should someone coming from a mathematics or physics article. Its not easy. I'm sorry if I sound contentious, but it irks me when communications/signal_processing people start talking like they are the only real people in the known universe. Seriously, I realise the sinc-sub-pi notation for the communications people is a problem that needs to be fixed, but this kind of language isn't the right beginning. PAR 00:46, 1 May 2006 (UTC)

It is a generous offer, and I happen to also inhabit Rbj's world, but unfortunately I think PAR is correct. It's not a matter of contemporary vs. historical. It's just two different conventions, overloading of the "word" sinc. A textbook has the advantage of being able to choose one and ignore all others, and that is what we are accustomed to. But Wikipedia does not have that luxury. And anything we "resolve" today can be revisited ad infinitum (as we are doing right now). That is why Wikipedia will never be high quality. We could write the perfect article today, and it would erode over time. We could make some aspect of Wikipedia totally consistent over all articles, and the inconsistencies and errors would creep back it, like weeds. So all I am saying is that there is a limit to how much we should stress over this. But each person must decide that for himself. --Bob K 14:26, 2 May 2006 (UTC)

it's okay, i give up. it's just that i clicked What links here and looked at some of the results. Only Bessel function has an expression of $\frac{\sin(x)}{x} \$ (without explicitly changing it to $\mathrm{sinc}(x) \$ ). even the Gamma function (the other ostensibly pure math article) uses the normalized $\mathrm{sinc}_N(x) \$ and all of the other uses were communications/signal_processing or sampling/interpolation related and i know for sure that the normalized sinc is what is used in those context. so i do not see evidence here at Wikipedia to support PAR's point. Rbj 16:24, 2 May 2006 (UTC)

Nyquist–Shannon sampling theorem, for instance, has an explicit π in the argument of its sinc functions, so I believe that is the unnormalized variety. But I think the only reason it was chosen is to conform with this article. I.e., the normalized sinc would have a cleaner look. Perhaps that is what you mean(?). --Bob K 17:06, 2 May 2006 (UTC)

yes. that is what i mean. until the definition is changed to include π in it, we have to put the π in everywhere else. i, for the most part, refuse to use the present normalized sinc notation $\mathrm{sinc}_N(x) \$ or $\mathrm{sinc}_\pi(x) \$ or whatever. i'll just put the π in the argument. but it seems to me silly to do so because we want to leave it out of the $\mathrm{sinc}(x) = \frac{\sin(x)}{x} \$ definition. it makes much more sense, at least from the electrical engineering and DSP POV to do what the textbooks do and put the π in the definition. i know this sounds EE and DSP centric, but i would like to see how often the sinc is used in applications in other contexts, and how many of those times there is no π in the argument (given the present definition). Rbj 19:15, 2 May 2006 (UTC)

Me too. --Bob K 01:13, 3 May 2006 (UTC)

Come to think of it, the Wikipedia standard for dealing with different concepts or definitions associated with the same [overloaded] title is to disambiguate. Therefore Sinc function should actually be disambiguation page, pointing to Sinc function, unnormalized and to Sinc function, normalized. Each article that references a sinc function would link directly to the appropriate definition. --Bob K 17:06, 2 May 2006 (UTC)

i don't like that idea. i'd rather keep it as it is and continue to put π in the argument of the sinc everywhere we use it. i want to reduce multiple definitions of the same thing. i want more unification of meaning of terms. Rbj 19:15, 2 May 2006 (UTC)

That is exactly what I have been doing, in the interest of global peace and harmony. But the world is actually a messy place. I am inclined to distinguish Wikipedia's role from that of a textbook. Wikipedia's job is to organize a messy room (so people can find what they want), but not throw anything of value away. A textbook, OTOH, achieves apparent unity by limiting its scope to just one corner of the organized room. So I think your goal for Wikipedia is too ambitious. To paraphrase PAR, unity within each discipline is more important than unity across the disciplines. By reaching for the latter, we ensure that Wikipedia's EE-oriented articles will have different-looking mathematics than all the EE textbooks. I was resigned to that fate until I realized that disambiguation is the organizational tool we need to avoid throwing away something of obvious value. --Bob K 01:13, 3 May 2006 (UTC)

[edit] The "disambiguation" of the sinc function serves no useful purpose and should be reversed.

User:Omegatron put the merge notices on the articles and, if no one else does, i'll write the first dissent.

BobK, this was a bad idea from the beginning, you made only one suggestion of doing it in advance, you got absolutely no support for the idea, and you went ahead and split it into two anyway. it makes me regret my original suggestion that the default definition be changed from

$\mathrm{sinc}(x) \equiv \frac{\sin(x)}{x}$

$\mathrm{sinc}(x) \equiv \frac{\sin(\pi x)}{\pi x}$ .

there is not enough difference in the definitions to warrant splitting it into two. i agree that the notation of $\mathrm{sinc}_N(x) \$ or $\mathrm{sinc}_{\pi}(x) \$ is useless because we can just use $\mathrm{sinc}(\pi x) \$ instead. now we have the worst of both worlds with two quantitatively different definitions of $\mathrm{sinc}(x) \$ floating around Wikipedia and now we have to explain which definition we're using each time we make a reference to sinc function instead of just including the π factor. this disambiguation was unecessary, it clutters WP, and it causes confusion. it should be reversed. r b-j 04:27, 26 May 2006 (UTC)

I'm sorry you now regret your suggestion. I thought it was a good start but just as impractical as your current suggestion, because unlike most textbooks, Wikipedia is a multi-discipline product, and its policy is to maintain a neutral POV. My bottom line argument is that "sinc(x)" is an ambiguous symbol in the world outside Wikipedia. For the people who come here to learn about it, that is the first thing they should learn. And disambiguation is the quickest, clearest, and most neutral way to do that. But if there is a merge, the two definitions should be given equal standing. Before I did the disambig, you had to read the fine print to discover the normalized definition. It was easy to miss, which is too bad for all those readers puzzling over the subsequent discrepancies with their EE textbooks. --Bob K 05:47, 26 May 2006 (UTC)

I think the disambiguation was a bad choice. We now have two pages which should have the same material except for the odd factor of π floating around. There are quite a lot of mathematical terms that are ambiguous (does the set of natural numbers contain 0 or not?); see for instance Wikipedia:WikiProject Mathematics/Conventions for one attempt to tackle these issues.

Thank you... I was unaware of Wikipedia:WikiProject Mathematics/Conventions. However, debating whether to represent concept A with notation B or notation C is not ambiguity. $\sin(x)/x\,$ and $\sin(\pi x)/(\pi x)\,$ are functions A and B, being represented by notation C. That is ambiguity. Wikipedia has a novel and very effective procedure for dealing with it. And IMO, it does give A and B equal standing. --Bob K 22:02, 30 May 2006 (UTC)

If your problem is that one of the definitions was not given enough prominence in the old article, then why not change that? But in the end, it will not be possible to give both definitions exactly equal standing; one of them has to be mentioned first.

Whatever approach we choose, I think that the definition of sinc should be given each time it is used, precisely because it is ambiguous. We cannot expect the reader to click on a link to discover what definition is being used. -- Jitse Niesen (talk) 12:45, 26 May 2006 (UTC)

Yes, that is the established standard that has evolved, rather than someone having the sense to use different symbols. It is now too late for $\mathrm{sinc}_{\pi}\,$ to catch on. I have no objection to being explicit. Repeating the definition is an acceptable way to be explicit. And as of this moment, an easier way is to mention sinc function (unnormalized) or sinc function (normalized). --Bob K 22:02, 30 May 2006 (UTC)

My bottom line argument is that "sinc(x)" is an ambiguous symbol in the world outside Wikipedia. For the people who come here to learn about it, that is the first thing they should learn.

Exactly. One of the greatest things about Wikipedia is that it presents the same idea from all different perspectives. An engineer like me reads an article about, say, the continuous Fourier transform and learns that there are many different weightings and conventions used in different fields; the ones I'm used to are specific to my field. — Omegatron 13:26, 26 May 2006 (UTC)

(Please sign your comments.) Yes, it's very cool. And I did a lot of that very sort of work on continuous [time] Fourier transform, so thank you for your appreciation! I don't have as much trouble with $\mathcal{F}$ as I do with $\mathrm{sinc}()\,$ , because nobody writes $\mathcal{F}\{h\}(f)$ (or $H(f)\,$ ) for a transform to radian frequency or $\mathcal{F}\{h\}(\omega)$ (or $H(\omega)\,$ ) for a transform to ordinary frequency. The problem has already been addressed to a large extent. The amplitude scaling differences between unitary and non-unitary transforms tend to be far less troublesome than the frequency scale, in my own experience. --Bob K 22:38, 30 May 2006 (UTC)

Cramming all the definitions into one article is not the only way to bring them to peoples' attention. For instance, in the "See Also" section, one could add a link to the disambiguation page. It's a powerful tool. There is nothing wrong with it. Let's use it. --Bob K 22:38, 30 May 2006 (UTC)

And disambiguation is the quickest, clearest, and most neutral way to do that.

No. Since they're the same thing, putting them in the same article and explaining the difference between each form is the best, most neutral way to do that. — Omegatron 13:26, 26 May 2006 (UTC)

They are not the same thing. That is the problem. --Bob K 22:02, 30 May 2006 (UTC)

How aren't they the same thing? — Omegatron 23:25, 30 May 2006 (UTC)

$\sin(x)/x\,$ is an instance of $\sin(ax)/(ax)\,$ . But it is never an instance of $\sin(\pi x)/(\pi x)\,$ . Rather they are both instances of $\sin(ax)/(ax)\,$ . If that is the "sinc" function you want to define, i.e. a generalized sinc(), then I would happily agree to merge them all into one place. You are entitled to a different opinion, so there is no need to belabor the point. --Bob K 04:11, 31 May 2006 (UTC)

in one important sense (a purely quantitative sense), they are not the same thing. and that is the problem, Bob. i don't want two quantitatively different definitions of $\mathrm{sinc}(x) \$ floating around in the same encylcopedia. especially when there is no need for it. there is some need for different scaled definitions of the continuous Fourier transform - we need that nomalized unitary definition common in communications textbookds so we can apply duality and Parseval's theorem or $\int_{-\infty}^{+\infty}{x(t) dt} = X(0) \$ so easily without having to worry about scaling factors. from a DSP and communications systems POV, it would be nice if the primary definition of sinc() was the normalized sinc. i see only one reference of it (in Bessel function) without the π. but i don't want to have a war with the pure mathematicians about it and it isn't so hard to always include the π in the argument so that our definitions are the same, both qualitatively (that's where they are no different) or quantitatively. r b-j 16:12, 31 May 2006 (UTC)

Yes, $\sin(ax)/(ax)\,$ would be the generalized sinc that sinc function is about. I doubt many people use it as such, though, so it would either be a very small note or implied by context.

For the record, this is how Mathworld does it.[2]

The two functions are pretty similar, no?

— Omegatron 05:34, 31 May 2006 (UTC)

Thanks. Shall I plot them on one set of axes for you? How do you folks feel about the $\mathrm{sinc}_{\pi}\,$ notation? I like it, but it is non-standard, and I don't think it is going to become standard in my lifetime ... besides the fact that it is not Wikipedia's charter to replace [bad] old standards with [better] new ones. --Bob K 12:26, 31 May 2006 (UTC)

Should we split sin(x) and sin(πx) into separate articles, too? Or sine and cosine?

Under what ambiguous name do sin(x) and sin(πx) both go? Ditto for sine and cosine? --Bob K 22:56, 31 May 2006 (UTC)

the $\mathrm{sinc}(\pi x) \$ notation is fine for the normalized sinc. there is no reason to bring in another definition so that someone reading $\mathrm{sinc}_{\pi}(x) \$ has to wonder what the hell that new subscripted function is about. i s'pose there is a minute danger of a reader seeing " $\mathrm{sinc}(\pi x) \$ " and thinking it's a normalized sinc without the π factor so what is displayed would be thought to be $\mathrm{sinc}(\pi x) = sin(\pi^2 x)/(\pi^2 x) \$ . but i think that danger is small. r b-j 16:12, 31 May 2006 (UTC)

I don't see why we would need any special notation (maybe in other articles?), but there are several instances where we've used conventions and notations that are not widespread in order to prevent confusion (where confusion wouldn't exist anywhere except a non-field-specific encyclopedia). I see no problem with this, as long as it's clearly explained. — Omegatron 14:51, 31 May 2006 (UTC)

In response to "an easier way is to mention sinc function (unnormalized) or sinc function (normalized)": well, I don't think that's enough. It is not clear without following those links what normalization you're talking about, and a Wikipedia article has to stand on itself as much as possible. So, I think that you have to mention the definition anyway. -- Jitse Niesen (talk) 01:58, 31 May 2006 (UTC)

and a Wikipedia article has to stand on itself as much as possible

That would lead to a lot of redundancy/clutter, and I'm not just talking about sinc(). Is that a personal opinion or [bad] policy? It is a major strength of Wikipedia that distractions from the main point of an article can be quietly relegated to links, rather than embedded in the main flow. --Bob K 04:11, 31 May 2006 (UTC)

Of course, there is a balance between explaining everything (and having a very long off-topic article) and explaining nothing (and having a short unintellible article). But explaining definitions that the reader is not supposed to know is standard practice in technical writing. In this case, I wonder how much clutter would it give. Instead of "where sinc denotes the unnormalized sinc function", we write "where sinc is defined by sinc(x) = sin(x) / x". Indeed, all articles that use the sinc function except triangular function do this.

standard practice in technical writing

This seems to be the heart of the matter. It's a fine standard for the constraints/technology under which it evolved. But evolution does not end. Change will come, but apparently not here, not now. --Bob K 12:52, 31 May 2006 (UTC)

I assume almost everybody would agree with this, which makes it policy. I don't know whether it's explicitly written down somewhere. Related issues are touched upon by Wikipedia:Guide to writing better articles, which says for instance "establish context so that a reader unfamiliar with the subject can get an idea about the article's meaning without having to check several links".

As I said before, N can denote either the set {0,1,2,3,…} or {1,2,3,4,…}, but we have only one article natural number; we don't have two articles natural number (with zero) and natural number (without zero). That seems to be the same situation.

I have no opinion on whether it is the same or not, because I do not use it. It has not been a problem I have had to cope with. But if it is problematic for its users, then I recommend they consider disambiguation as a solution. --Bob K 12:52, 31 May 2006 (UTC)

I asked on Wikipedia talk:WikiProject Mathematics for some more opinions. -- Jitse Niesen (talk) 05:22, 31 May 2006 (UTC)

I vote to merge the articles, giving both definitions in one place. They describe the same function, just with a different scaling of the x axis. One article even directs the reader to the other article for more details. Disambig is unnecessary and confusing. Gandalf61 08:14, 31 May 2006 (UTC)

Yes, do merge these into one article Sinc. What about this idea. First say there are two definitions in use, the "pure mathematician's" and the "engineer's", and give both explicitly. Then observe that they can be unified by introducing an implicit parameter z, as in sinc(x) = sin(zx)/zx, from which the originals can be retrieved by setting z to 1 or π in the Sinc entry of the Math > Parameter Options section of the Preferences menu on your mathotron. Then state (inasmuch as reasonable) the properties of sinc using z in their expression, like that the zero crossings are at the multiples of z, except at 0. Maybe it's too confusing, but my expectation is that the reader who cares about the properties of sinc has enough maths background to understand this. --Lambiam ^Talk 13:48, 31 May 2006 (UTC)

My vote is already clear. r b-j 16:12, 31 May 2006 (UTC)

I agree the dab page should go. The two versions of sinc should be merged and discussed in one article. Lambiam's suggestion seems a workable approach that would also not appear to mirror Mathworld. Gimmetrow 16:23, 31 May 2006 (UTC)

Merge as explained above. — Omegatron 17:53, 31 May 2006 (UTC)

Merge Of course, mention both definitions and where each is typically used. --Macrakis 20:50, 31 May 2006 (UTC)

No. But I will contribute a plot of both functions in different colors on one set of axes. --Bob K 22:56, 31 May 2006 (UTC)
- For what purpose? I can make one easily enough, if there's a good reason for it. — Omegatron 19:39, 1 June 2006 (UTC)

Thanks, but don't bother (see http://en.wikipedia.org/wiki/Image:Sinc_functions.png). The purpose, of course, is to graphically illustrate the differences, e.g. http://en.wikipedia.org/wiki/Image:Kaiser-window.png. It should have been done a long time ago. Instead, we have separate plots, which actually suggests separate articles [and disambiguation]. --Bob K 20:02, 2 June 2006 (UTC)

That's because I created them while the articles were split. — Omegatron 00:02, 3 June 2006 (UTC)

And now they are apparently being merged. That is the answer to your question. --Bob K 05:43, 3 June 2006 (UTC)

Merge There should be one article describing each form, with some notes on where each is used. Any other article which uses the sinc function should explicitly state which definition is being used, according to the common usage for that particular field. This way anyone interested in the sinc function per se will be aware of the differences, but anyone reading about it in a particular context won't have to be continually translating to the commonly used definition for that particular field if that definition doesn't happen to be the Wikipedia "base or primary" definition. Requiring uniformity in this situation is counterproductive. PAR 01:11, 1 June 2006 (UTC)

Merge. Are you kidding? Are we also to have 3 different articles on Fourier transform and error function because of misplaced constants? Very bad precedent. -lethe ^{talk +} 18:57, 1 June 2006 (UTC)

Merge. Sorry folks, but, to be brutally honest t to point of offensiveness, splitting these articles apart is one of the goofiest ideas I've encountered at WP. linas 21:15, 1 June 2006 (UTC)

[edit] i am re-asserting the issue of the base or primary definition.

this should get the attention of User:PAR and other non-engineering mathematics scholars and pracitioners. when we reverse the disambiguation (i think the consensus is pretty clear on that), we should emphasize, as the primary definition of the sinc function, the so-called "normalized" sinc() function:

$\mathrm{sinc}(x) = \left\{\begin{matrix} {\frac{\sin(\pi x)}{\pi x}} &\mathrm{if} & x\ne 0 \\ 1 & \mathrm{if} & x = 0 & \end{matrix}\right.$

we can mention that the historical definition of the sinc function (literally "sine cardinal") leaves the π scaling out of the definition and that this so-called "unnormalized" sinc function, without the π in its definition, is the one that is equal to the zero^th order spherical Bessel function of the first kind. other than in that reference, every other reference of application of the sinc function in the literature or on the internet (the first few pages that Google returns), either uses the normalized sinc() definition (what i advocate changing this to, as the primary definition) or uses the unnormalized definition but sticks a π factor in the argument. that strongly suggests that the natural definition of the sinc() function is the normalized version. it is the normalized sinc() function that is the natural Fourier transform of the rectangular function when the normalized and unitary definition of the F.T. is used.

now, User:PAR complained about that change of primary definition, but now i challenge those that oppose this change to show that there would be any quantity of articles that would have to put a 1/π factor in the argument of sinc() if the definition was changed to include the π. i assert that the facts show the other way around - that, other than the Bessel function article, all other articles will have to include π (or 1/2 for angular frequency) in the argument of sinc() if it is used in a real application.

so, i am formally proposing that the primary definition of the sinc function is changed to

$\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \$

and that the article mentions as an alternative and historical definition of the sinc function is

$\mathrm{sinc}(x) = \frac{\sin(x)}{x} \$

and that this might be more natural to use with the zero^th order spherical Bessel function of the first kind. but, i believe, if we do make this change, that in the Bessel function article, we be explicit and simply refer to the sinc function article but explicitly use $\frac{\sin(x)}{x} \$ without calling it " $\mathrm{sinc}(x) \,$ " so that there is no possible confusion of scaling. r b-j 21:15, 31 May 2006 (UTC)

Yes. --Bob K 22:46, 31 May 2006 (UTC)
No See the explanation after my vote above. PAR 01:11, 1 June 2006 (UTC)

okay, but as it is, there are multiple places in WP where " $\mathrm{sinc}(\pi x) \,$ " is explicitly referred to in equations and in all cases there is a π factor in it and there is no place, outside of the sinc function article, where $\mathrm{sinc}(x) \,$ is used without the π factor. it's okay with me that neither definition in the sinc function article is "primary" or "alternate" and both are co-equal, but since it appears that only Bessel function has " $\frac{\sin(x)}{x} \,$ " without the π, my proposal is that while either use can refer to the article: Sinc function, the only use of the notation " $\mathrm{sinc}(x) \,$ " in equations outside of the article Sinc function, should only refer to one common and quantitatively consistent definition and that should be the so-called "normalized" sinc function. is that acceptable to you, PAR? to be explicit, outside of the Sinc function article, whenever you see " $\mathrm{sinc}(x) \,$ " in an equation, there should be no doubt that the meaning is $\frac{\sin(\pi x)}{\pi x} \,$ . if one is creating an equation that has " $\frac{\sin(x)}{x} \,$ ", that editor should leave it in that form but should identify it as the sinc function with link to the article. is this a compromise you can live with PAR?

I am not convinced, but it is a workable solution and in this case, I can live with the consensus. PAR 15:19, 1 June 2006 (UTC)

okay, to be clear this is what i will do after waiting a little longer (for more possible objections/comments).

i'm gonna try to legitimately rename Sinc function (unnormalized) to Sinc function
merge the stuff from Sinc function (normalized)
go to all linked references of Sinc function (normalized) and Sinc function (unnormalized) and change the link to Sinc function.
- A couple of redirects could be helpful. --Bob K 23:33, 1 June 2006 (UTC)
in all cases in those other articles where there is in math equations, i will change it to .
- I just took a shot at it... --Bob K 11:51, 6 June 2006 (UTC)
There are, outside of the Sinc function articles, no present use of just .
- You mean in the unnormalized sense (right?). --Bob K 18:48, 7 June 2006 (UTC)
- Exceptions to that "rule":

triangle function (subsequently updated by me)

continuous Fourier transform (subsequently updated by me)

Diffraction of light

in all cases where $\frac{\sin(x)}{x} \,$ appear (presently it's just the Bessel function article), we will leave it as $\frac{\sin(x)}{x} \,$ without equating it to $\mathrm{sinc}(x) \,$ (which is the inconsistancy in usage i want to avoid at all costs) or to $\mathrm{sinc}(x / \pi) \,$ (which is consistent with other uses of the sinc() but maybe a little ugly) but, in the text, we will link to Sinc function.

if this is not acceptable, now is the time to complain. r b-j 16:07, 1 June 2006 (UTC)

OK, so who actually defines the sinc with a pi in it?

[edit] Without pi

mathworld
planet math
Cambridge University (page 3)
Hecht, "Optics" [3]
Modern Nmr Techniques and Their Application in Chemistry, by K Hallenga, Millicent Popov, A I Popov, 1990.
Linear Networks and Systems: Alogrithms and Computer-Aided Implementations, by Wai-Fah Chen, 1990.
A Handbook of Fourier Theorems, by D. C. Champeney, 1989.
Signal Processing Handbook, by Chen Chen, 1988

[edit] With pi

Internet citings:

MATLAB
[4]
Julius O. Smith (he's a pretty big cheese, in case you're wondering)
Rodenburg (i dunno who they are.)
The Scientist and Engineer's Guide to Digital Signal Processing
SCILAB
Connexions
sumbody at UCSD
SoundForge
Math Forum
Physics‑Astronomy Dept
Computer Science Dept
Image Synthesis (Comp Sci)
Apple Computer (page 9)

Textbook and published literature:

[edit] i believe i have found all the places with "sinc" in equations...

... and made sure they consistently meant "normalized" sinc(). this was the basic inconsistency i wanted fixed from the beginning. i might suggest that where sinc() is in the nascent Delta functions, we change all of the Fourier integrals to the unitary normalized Fourier integral (instead of using angular frequency) and the sinc() won't need a 2 π in the denominator. also we should completely clean the links to go strait to the Sinc function page. and then ask the admins for some file deletes. r b-j 06:11, 8 June 2006 (UTC)

You missed at least one. I will start a list here:

Diffraction#Quantitative_analysis_of_single-slit_diffraction --Bob K 11:45, 8 June 2006 (UTC)
Diffraction#Quantitative_analysis_of_N-slit_diffraction

changed in both cases. that article did not (until now) have a linked reference to Sinc function or derivatives so i didn't know it was there. even in that article, changing to normalized sinc simplified the equations. r b-j 16:38, 8 June 2006 (UTC)

I reverted this change. The physics literature (such as Hecht's Optics) uses the mathematical definition of sinc without pi in it. It's nice to have consistency in wikipedia, but the articles should match the relevant literature even if that makes things inconsistent. Pfalstad 22:41, 8 June 2006 (UTC)

I concur with the reversion. Consistency is nice, but consistency with the optics literature for an optics article is more important than consistency with the rest of the wikipedia. We don't get to decide that theirs only one important definition of sinc, if standard practice has two. One can always add a note to say it's the unnormalized definition... Dicklyon 03:28, 9 June 2006 (UTC)

[edit] move request to admins

Rbj, you request that an admin do a move. I'm an admin, and I'm happy to do a move with deletion if you like. But is it ready to go? Is there anything left at sinc function or sinc function (normalized) which needs to be merged back? For example, the disambig page has a graph which is in neither particular page. Is there any desire to keep that graph in the article? What about the stuff about orthonormal bases that's in the normalized article but not the unnormalized article? -lethe ^{talk +} 06:23, 3 June 2006 (UTC)

I've done the move. -lethe ^{talk +} 06:47, 3 June 2006 (UTC)

i made a stab at merging the stuff. more wordsmithing and editing will have to be done. thanks for doing the move. i would like to eventually kill the sinc function (normalized) and sinc function (unnormalized) pages, but i have to find the places where they are referenced and fix that before requesting AfD. r b-j 06:54, 3 June 2006 (UTC)

Redirects are cheap, why bother? -lethe ^{talk +} 06:56, 3 June 2006 (UTC)

'cause i'm anal-retentive and i hate dangling loose ends. r b-j 06:59, 3 June 2006 (UTC)

Well, you can find all the articles which go through the redirects by Special:Whatlinkshere/Sinc function. After you bypass all the redirects, I'll be happy to delete them for you. By the way, I guess you've merged whatever content there was from sinc function (normalized), but not the picture from the disambig page. If you want it, it's in the article history, which I've undeleted after the merge, in case you think it has a place in the article (I personally do not though). -lethe ^{talk +} 07:05, 3 June 2006 (UTC)

Another thing, I notice that you plan to standardize all of wikipedia's uses of the sinc function to be the same. I'm not sure whether that's a good idea, but it might be. I would suggest that you put your choice of convention at Wikipedia:WikiProject Mathematics/Conventions, so that future editors might know what the preferred meaning of sinc(x) is at Wikipedia. I will opine that I don't think it should be disallowed to use the other convention, so long as the author of any particular article is clear on what convention he is using. On the other hand, consistency is always good. -lethe ^{talk +} 07:22, 3 June 2006 (UTC)

[edit] Two different functions?

Omegatron's attempt at showing both on the same graph

I hope everyone will come to realize at some point during this debate that changing your units doesn't give you two different functions, does not give you two different graphs, does not require two different Wikipedia articles, and does not require disambiguation. There is only one function here. -lethe ^{talk +} 14:58, 3 June 2006 (UTC)

I agree that they belong in the same article, but sin(x) and sin(2x) are, indeed, different functions. Same here. If we're going to have two different definitions of the function, we need to show both graphs. — Omegatron 15:03, 3 June 2006 (UTC)

If you insist, let's at least show them both in one graph. -- Jitse Niesen (talk) 15:43, 3 June 2006 (UTC)

I don't think you can see the shape as clearly with them overlapping. Maybe show a graph of just the normalized alone, and then a graph of both on the same plot (which shows the unnormalized zero crossings more clearly)?

Here's my attempt at showing both on the same graph. — Omegatron 17:15, 3 June 2006 (UTC)

Excellent ! --Bob K 11:33, 4 June 2006 (UTC)

There is already a graph showing them both, which I've removed. But why do you need to see them both? They are, after all, the same graph! Just because you change the symbol that looks like a 1 on the x-axis into a symbol that looks like a π does not mean you've got a new function. Rather, it just means you changed your scale. If you need to see a new graph for every scale, then you should be reading graph of a function, rather than this article. -lethe ^{talk +} 20:36, 3 June 2006 (UTC)

they are the same function qualitatively, but different quantitatively and that difference created some contention regarding what is the real sinc function. there is not enough difference (a scaling factor of π) to justify two articles for essentially the same thing, but they're not precisely the same thing and those that prefer the historical definition over the one that is practical for communications and signal processing will not like it (and have already express as much) if this historical definition isn't presented in some way co-equally although i don't want to see occurances of $\mathrm{sinc}(x) \$ floating around in equations of other articles in WP that do not clearly mean the same thing and, i think we got some concession and consensus that, at least in other articles where $\mathrm{sinc}(x) \$ is used, it means $\frac{\sin(\pi x)}{\pi x} \$ . but if we only graph the normalized sinc function, the proponents of the historical definition will have a legitimate reason to say that the two definitions are not presented as co-equal.

i want to say that my first stab at a merge is just a first stab, i expect some significant reorganizing and/or wordsmithing (from others, i'm sorta outa ideas) to this. —Preceding unsigned comment added by rbj (talk • contribs)

Yeah, I'm not sure if it's better to show two graphs or show them both on the same graph. It's kind of cluttered, since each wants to be on its own number line (integers vs multiples of pi), and I tried to show that with the grid lines. I guess it's ok. — Omegatron 02:22, 4 June 2006 (UTC)

you have to tell us how to create consistant looking graphs like that, O. i got MATLAB and an old version of Powerpoint on the Mac. i have a student version of MATLAB and dunno what on a PC (i hate PCs).r b-j 02:43, 4 June 2006 (UTC)

It's just gnuplot and Inkscape. Instructions and source code are on the image description page. I'm always working towards the Ideal Graph; more info is on Wikipedia:How to create graphs for Wikipedia articles. You can use GNU Octave, a MATLAB clone, and output directly to gnuplot, too, as described on that page. Any suggestions on improving the graphs are appreciated, and on improving the coverage of that page. — Omegatron 04:21, 4 June 2006 (UTC)

i guess i'm so Mac-centric and, despite them saying it's good on virtually every OS, there is no gnuplot for the Mac that i know of. BTW, there might be a mathematical mistake on your example Image:Exponentialchirp.png. when you want the instantaneous frequency of a sinusoid, you take its argument and differentiate it (w.r.t. t) rather than divide by t. so the argument of your exponential chirp should be the integral of 3^t rather than 3^t t. r b-j 05:19, 4 June 2006 (UTC)

Yeah, I know. Something about instantaneous frequency not being the same as the f in sin(2πft), right? But in a chirp are you varying the instantaneous frequency or the f variable? The examples in the article are my fault, too, but were more easily changed than the graph. I've been meaning to fix it but haven't invested the thought/time required. — Omegatron 14:32, 4 June 2006 (UTC)

if you want ordinary (non-angular) frequency, you always calculate the derivative (w.r.t. time) of the argument of sin(θ(t)), which is θ'(t) and divide by 2π. that's the consistent and accurate rule. it is not the same as dividing θ(t) by 2πt except in the case of θ(t) = 2πft. r b-j 18:55, 4 June 2006 (UTC)

MacOSX is a unix, so of course it can run gnuplot. You can install it using fink:

$ fink install gnuplot

and run it under Apple's X11 implementation. -lethe ^{talk +} 05:36, 4 June 2006 (UTC)

where do i get X11? i got Inkscape a while back but it wouldn't run (didn't tell me why) and after investigation i think i figgered out that i needed X11, but i couldn't find where to get it (too many hits on Google). is it free? where can i get it? personally, although i have one OSX mac, i feel it's really more bother than my older OS9 (that has lotsa old, but working software on it that would cost $4digits to replace). i need someone to tell me where i can get this stuff because it just doesn't seem clear. i am OS10.2.8, is that too old? r b-j 18:55, 4 June 2006 (UTC)

Apple's X11 comes with the OS. It's on the install disk, though it does not install by default. However, during the time of Jaguar (10.2), it was still under development, and only available as a development download. I didn't find that beta release still available for download on the Apple X11 website, but I didn't look too hard, maybe it's still there somewhere. If you can't find a version of Apple's implementation to run on your 4 year old OS, you should download XDarwin, the predecessor, which I think has continued development. And get the OroborosX window manager to give it that mac "look and feel". But can I just add that MacOSX is a huge shift in direction for Apple, and it went through a very rapid development cycle because it took a lot of time to bring it up to speed. By sticking with Jaguar, you're stuck to the beginning of that shift, and if you're not willing or able to do the shift with Apple, then yes, you probably should have stuck with OS9. MacOSX is gonna take time to get right. -lethe ^{talk +} 20:16, 4 June 2006 (UTC)

I found a download page. This is Apple's X11 v 1.0, while Tiger comes with v 1.1, so it's a little old, but newer than the beta download. The page lists Panther as a requirement, but maybe it'll run on your Jaguar box anyway. Keep your fingers crossed. -lethe ^{talk +} 20:24, 4 June 2006 (UTC)

doesn't work (but i forgot to cross my fingers - i hope that didn't blow it). .pkg installer senses immediately my OS ain't new enough and declares i need 10.3 or later. r b-j 21:10, 4 June 2006 (UTC)

And does it seem like XDarwin wants $60 for their software? Well, you could try fink. I think fink will compile X11 for your system. But it may be more trouble than it's worth. The best idea would be to upgrade to Tiger. -lethe ^{talk +} 21:44, 5 June 2006 (UTC)

There are many places where there is some tension between different conventions. We do not have to give each convention equal coverage. It suffices to simply mention both conventions. Sticking to them for too long is more confusing than it is helpful. Look to error function. They managed to get along with 3 different graphs for different normalization choices. It seems to me that the double graph is far too cluttered to be useful. For example, can you read the roots off of that graph? I offer you a completely neutral graph. With no axes, one cannot tell what normalization it has. -lethe ^{talk +} 02:32, 4 June 2006 (UTC)

no one disagrees as to the vertical scaling of the sinc function. and not having some meaning to where it crosses zero in the graph is, missing something. i think have both on a single graph is the best solution.

well, i just took a look at error function and that mess illustrates precisely what i want to avoid here. there are two mathematically inconsistent uses of erf(). i think the definition where erf(0) = 1/2 is best, but i am not willing yet to join that fight. but they should have the definition that makes most compact sense for use in probability theory and then mention the historical or alternative definition where erf(0) = 0 for completeness. there is some similarity to the issue here in sinc land.

the article (sinc) needs unification of properties. other than the two properties resulting from "normalization" (which are nice and that's why it's a better definition, IMHO) that the normalized sinc lands on zero at non-zero integers and has an integral of 1, all other qualitative properties apply to both sinc() functions. we need to make the article say it as such and the properties of the sinc() need to be all combined with the exception of mentioning those two advantageous properties for the normalized sinc and that the unnormalized sinc is the historical definition, has the present property 1. and is the zero^th order spherical Bessel function of the first kind (did i say that right?). i'll think about it, but am happy if others stab at this. maybe one of us will connect, kill it, and get it over with. r b-j 02:43, 4 June 2006 (UTC)

[edit] i have AfD the other sinc function pages ...

... and it looks like they acted pretty quickly, but there is a left over turd of some of the old talk pages. i don't get how this AfD works completely, but i tried to follow the directions. r b-j 17:21, 8 June 2006 (UTC)

Why? They should be kept as redirects. — Omegatron 17:48, 8 June 2006 (UTC)

Yes. And furthermore I see no point in changing a link such as "sinc function (normalized) | sinc function" to just "sinc function". The casual reader sees the same thing in either case. So you are needlessly destroying a bit of information that might actually be useful to somebody. --Bob K 23:02, 8 June 2006 (UTC)

But it's not really information, is it? It's misinformation. At best, it's simply evidence of the fact that at some point, a Wikipedian made a bad decision, which is not something our readers need to know. -lethe ^{talk +} 01:01, 9 June 2006 (UTC)

Wrong on both counts:

Knowing that disambiguation is considered a "bad idea" might prevent a recurrance, not that I would mind.
And a link such as "sinc function (normalized) | sinc function" assures me that it's the normalized definition that applies. That is information. The article itself does nothing of the kind; i.e. it is ambiguous, just the way you like it.

--Bob K 02:05, 9 June 2006 (UTC)

the sinc function article itself is a little ambiguous due to "political" reasons (the non-engineers who would prefer the unnormalized sinc) with no more than two qualitatively identical but quantitatively different definitions. my aim was so that, in equations throughout WP, every occurace of $\operatorname{sinc}(x) \$ meant the same thing quantitatively. and i think we successfully argued that it should be the normalized sinc definition. what we do is say "normalized sinc function " if need be (i don't think the word "normalized" would be necessary in all cases, but if you want to add the word where it is missing, fine with me) and in these cases we can use $\operatorname{sinc}(x) \$ in the equations.

in the other cases (not very many) we say "unnormalized sinc function " in the article text and explicitly use $\frac{\sin(x)}{x} \$ without any use of $\operatorname{sinc}() \$ in the equations. in the latter case, we always include the word "unnormalized" just before the link: sinc function. this way, the math notation is always consistent and accurate ( $\operatorname{sinc}(x) \$ always means $\frac{\sin(\pi x)}{\pi x} \$ ), the concept of the sin(x)/x species of function (whichever scaling) is always associated with the sinc function article. this is the best we can do to avoid confusion and convey information and do this right. r b-j 03:03, 9 June 2006 (UTC)

Since there is an ambiguity in normalization, every usage anywhere in the encyclopedia should be explicit about which normalization it is using. If they do that, then hiding information inside a redirect or piped link is not necessary (I also prefer this solution to rbj's policy of enforcing that everyone use the same form). Furthermore, doing so is against the MoS. It's a sensible rule; if we start allowing information to be hidden in our links, then we leave our readers high and dry when we print this encyclopedia out on paper. You can't see that hidden information in the print version. Anyway, I've deleted the redirects. -lethe ^{talk +} 03:27, 9 June 2006 (UTC)

Paper? Now who's kidding? And I just read Manual_of_Style#Wikilinking, and I think you might need a refresher. But don't get me wrong. I am happier with the current outcome than with disambiguation. I just didn't think it was possible. Disambig was my incremental solution to Wikipedia's narrow-minded view and usage of sinc. Rbj's persistence made it happen, but I like to think of my own role as catalytic, at least. Apparently I provided the common enemy that made unity seem more tolerable than before. And for the record, I did not say that the extra information in the links is "necessary". I only said that its destruction is unnecessary. Rbj described it as "anal", and actually that makes more sense than any thing else. And I respect his honesty, if that's what it was. We're all just trying to have a little fun here, so Rbj, if that's what floats your boat, please be my guest. --Bob K 05:15, 9 June 2006 (UTC)

A paper wikipedia may seem like a distant fantasy, but it is a goal, and we should not encourage practices which are detrimental to that goal. So no, I'm not kidding. If you can't find the rule about including hidden information in your links, check WP:PIPE (easter eggs). Given your rather misguided behavior in this situation, I find it rather silly that now you are advising me that I need a refresher in WP policy. -lethe ^{talk +} 05:35, 9 June 2006 (UTC)

Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_paper_encyclopedia --Bob K 16:32, 19 August 2006 (UTC)

The example of a no-no I found at WP:PIPE (not MoS) is a link whose appearance is "exceptions" and then takes the reader to "Thomas Bowdler". But Rbj is worried about this link: "sinc function (normalized) | sinc function", whose appearance is "sinc function" and took the reader to "sinc function", via re-direction. No harm, no foul. --Bob K 12:57, 9 June 2006 (UTC)

So did you want to hide information inside the redirect? Or did you merely think it was harmless to do so? -lethe ^{talk +} 13:10, 9 June 2006 (UTC)

Actually I hid nothing. After I created the sinc function (normalized) article, I changed some [[sinc function]] links to normalized [[sinc function (normalized) | sinc function]], because I decided that normalized sinc function looks a little better than sinc function (normalized). I considered renaming the article, but this solution seemed less disruptive. As you can see, both forms contain the same information. It was just a cosmetic rearrangement. --Bob K 13:52, 9 June 2006 (UTC)

A little while ago, you were arguing against deleting the redirects because they contain information. -lethe ^{talk +} 13:57, 9 June 2006 (UTC)

It's redundant information. Is there a "rule" against that too? And speaking of rules, here is a refresher straight from the MoS: "Clear, informative, and unbiased writing is always more important than presentation and formatting." Those were my motives for everything that I did, whether you happen to agree with it or not. I took a very biased sinc function article and tried to clarify the situation and de-bias it. Unlike Rbj who thinks that the unnormalized sinc should be outlawed, I happen to agree with you that "every usage anywhere in the encyclopedia should be explicit about which normalization it is using". When I changed ambiguous [[sinc function]] links to normalized [[sinc function (normalized) | sinc function]], that is exactly what I was doing. --Bob K 16:24, 9 June 2006 (UTC)

[edit] yea! PAR! that looks really nice!

i'm pretty happy with the article they way it is and will just accept that the Optics guys will use sinc(x) differently than i would prefer. i am now trying to make two new articles for Zero-order hold and First-order hold and to get rid of a poor rendition of the two in another article i'd like to see deleted. i'm in a sorta "weed the garden" mode now. r b-j 02:18, 12 June 2006 (UTC)

[edit] Question

Anybody have any idea how to find the Fourier Transform of Sinc²(πax) using the Convolution Theorem? I cant find any refs that actually show examples of how to use the Conv Theorem in FTs.

Any thoughts?--Zereshk 21:53, 15 September 2006 (UTC)

The transform of sinc times sinc is rect convolved with rect, which is the triangle function. In the frequency domain, the triangle is scaled to go to zero at twice the frequency at which the sinc's spectrum goes to zero, which you can think of as a frequency doubling effect. If you want equations, let us know. Dicklyon 23:57, 15 September 2006 (UTC)

Assuming you mean the unnormalized sinc function it's row 12 with $t \rightarrow x\,$ and $a \rightarrow a\,$ (not $a \rightarrow \pi a\,$ ) --Bob K 05:02, 16 September 2006 (UTC)

Very helpful indeed. Thank You both. I now understand it a bit more clearly.--Zereshk 01:31, 21 September 2006 (UTC)

[edit] 2D Sinc?

I reverted the addition of the nice 3D picture of the 2D sinc as being off topic and not all that useful. I cannot think of any use of the 2D sinc; can anyone? Dicklyon 04:56, 18 November 2006 (UTC)

I assume that it applies to image filtering and filter design analogous to signal processing filters. --Bob K 14:46, 18 November 2006 (UTC)

Sounds like a plausible guess, but it is not so. The impulse response of a circularly symmetric sharp lowpass filter is an Airy function, not a 2D since. And the impulse response of a square sharp lowpass filter is a separable product of sincs, not a radial sinc. Dicklyon 16:46, 18 November 2006 (UTC)

[edit] Integral proof

May someone proof $\int_{-\infty}^\infty \begin{matrix}\frac{\sin(\pi x)}{\pi x} \end{matrix}\, dx = 1$ ? Thanks in advance, --Abdull 19:31, 4 December 2006 (UTC)

It's a trivial consequence of the Fourier transform relationship to the rect function, which is easier to show in the other direction, as done here: Nyquist-Shannon#Mathematical_basis_for_the_theorem. Dicklyon 20:23, 4 December 2006 (UTC)

Ah, thank you for the hint! Once I understood what you meant it really was trivial. Bye, --Abdull 20:12, 7 December 2006 (UTC)

why put this proof that everyone agrees is trivial into the article? r b-j 04:19, 8 December 2006 (UTC)

I agree. A few words to say it follows from the transform relationship at f=0 would be more than adequate. Dicklyon 05:07, 8 December 2006 (UTC)

I did it because it was not trivial for me in the first moment (else i wouldn't have asked). Here is my edit to the article that was removed, in case of someone has similar problems seeing it right ahead:

$\int_{-\infty}^\infty \mathrm{sinc}(t)\,e^{-2\pi i 0 t}dt = \int_{-\infty}^\infty \mathrm{sinc}(t)\,e^{0}dt = \int_{-\infty}^\infty \mathrm{sinc}(t)\cdot 1 dt = \int_{-\infty}^\infty \mathrm{sinc}(t) = \mathrm{rect}(0) = 1$

--Abdull 21:19, 20 December 2006 (UTC)

[edit] Table

Is the table really necessary? It doesn't really provide anything useful to the reader that they couldn't work out with a calculator, or by looking at the graph. It also breaks up the flow of the article. Oli Filth 08:50, 5 March 2007 (UTC)

I agree. And by the way, the column labelled $20\log_{10} \mbox{sinc}(x)\,$ should be labelled $20\log_{10} |\mbox{sinc}(x)|\,$ or $10\log_{10} \mbox{sinc}^2(x).\,$

--Bob K 11:26, 5 March 2007 (UTC)

I've bitten the bullet and removed the table. Oli Filth 22:27, 5 March 2007 (UTC)