Talk:Rectangular function

A perfect example of a rectangular function that I think should be added is:

$R(x)=\lim _{n\to \infty }(\tan \circ \cos )^{n}(x)$

We already have a much simpler perfect example. Less is more. --Bob K 00:39, 6 June 2006 (UTC)

Nope. — Omegatron 00:42, 8 June 2006 (UTC)

Am I going mad, or is the definition of this simple function just completely wrong? Shouldn't it be

R(x) = 0, if x < -1/2
R(x) = 1, if -1/2 <= x <= 1/2
R(x) = 0, if x > 1/2

????—Preceding unsigned comment added by 213.162.107.11 (talk • contribs)

Please sign your entries with "~~~~"

Is it the values at x = ± ½ that you are concerned about? --Bob K 04:20, 11 July 2006 (UTC)

I am used to seeing this definition in the literature:

$\mathrm {rect} (t)={\begin{cases}1&{\mbox{if }}|t|<{\frac {1}{2}}\\0&{\mbox{if }}|t|\geq {\frac {1}{2}}\end{cases}}$

Though I have also seen the current definition ($\mathrm {rect} (\pm 1/2)=1/2$). I just added in a short blurb about the various definitions. --Rabbanis 18:54, 5 August 2006 (UTC)

---

I added the "As long as the function is motivated by the time-domain experience of it..." clause. I thought it contains insight but I could be mistaken. 77.30.95.216 (talk) 12:35, 7 November 2008 (UTC)

I would like also to comment that it might be better to follow the same convention used in the Fourier Transform article, which uses real frequency for the independent variable in the frequency domain, rather than the other two conventions using angular frequency. But still, I am not a specialist, I just pursue what may be more coherent with the rest of articles, and what may be more intuitive. 77.30.95.216 (talk) 12:51, 7 November 2008 (UTC)

This is not angular frequency:

$\int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\mathrm {sinc} (f),\,$

--Bob K (talk) 13:14, 7 November 2008 (UTC)

(blush) Right. Please revise what I have written. It may be inappropriate for the context or simply not needed. I am no specialist. Actually, this is the first time I edit an article, ever :^S. 77.30.95.216 (talk) 15:53, 7 November 2008 (UTC)

My lecturers all call this thing a top-hat function. Is this a normal word for it? I spent about fifteen minutes trying to find this page. Possibly useful redirect? —Preceding unsigned comment added by 82.6.96.22 (talk) 18:53, 7 November 2010 (UTC)

The midpoint need not be 1/2 for the Fourier Transform of rect to be sinc.

I haven't read the rest of user:Jens VF history, but I deleted a clearly false claim at the beginning of the article. 69.5.112.154 (talk) 22:58, 2 November 2022 (UTC)

The fact that the Fourier integral (of sinc in this case) converges to the midpoint wherever the resulting function jumps is a well-known theorem in Fourier theory, see "The value of the integral is f(t), whenever f(t) is continuous, and is equal to the average of left and right limits whenever f(t) has jumps." in Kaplan, Operational Methods for Linear Systems (1962) THEOREM 2, p.241. The particular case of the Fourier transform of sinc is given as an example. Jens VF (talk) 10:38, 24 May 2024 (UTC)

Last equation

Shouldn't the last equation be ${\mathcal {F}}\delta (t)=1,$ ? or conversely ${\mathcal {F^{-1}}}\delta (f)=1/{2\pi },$ Groovamos (talk) 22:22, 7 August 2023 (UTC)

Remove vague text?

I can't attach any definite meaning to this discussion in the current version ..

Note that as long as the definition of the pulse function is only motivated by its behavior in the time-domain experience, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans.

Would anyone object to cutting the whole sentence (as 'not directly understandable by humans') ? Tdent (talk) 21:39, 27 September 2023 (UTC)

Non-Analitic Smooth Approximation

for a real valued $1\leq m\to \infty$:

$f(x)={\begin{cases}1,\quad x=0\\0,\quad |x|\geq 1\\{\dfrac {1}{1+\exp \left({\frac {m(1-2|x|)}{x^{2}-|x|}}\right)}},\quad -1<x<0\vee 0<x<1\end{cases}}$

can approximate smoothly the rectangular function as $m$ increase with perfectly flat intervals.

I believe it should be added but I got suspended for doing it once without consulting it here before (and I pretty sure nobody reads this already so I am not sure how Wikipedia grows - lets hope for some supper user to read it) 45.181.122.234 (talk) 20:03, 23 September 2024 (UTC)