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Is it possible, by any means, to restore the signum of sinc(x) after being transformed to its absolute value, abs(sinc(x))? How it got to abs() is irrelevant, I only want to know if the reverse is possible.

user164048
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  • Well, the smallest value ${\rm sinc}(x)$ ever attains is about $-0.217$, so if $|{\rm sinc}(x)|\ge0.218$ then you know ${\rm sinc}(x)$ must be positive. But otherwise, no. –  Jul 16 '14 at 08:33

2 Answers2

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Observe that $\operatorname{sinc }x=\frac{\sin x}{x}$ has its zeros at the nonzero integer multiples of $\pi$, and also observe that all of those zeros are simple. Thus knowing that $\operatorname{sinc x}$ is positive around $x=0$, we can determine the signum in the following way:

  1. Given $x \in \mathbb{R}$ determine in which of the two families of intervals it lies: $\{[-\pi,\pi],[2\pi,3\pi],[-3 \pi,-2 \pi],\dots\}$ and $\{[\pi,2 \pi],[-2 \pi, -\pi],\dots \}$.

  2. If it lies in the first family the signum is $+1$, otherwise it is $-1$.

If you feel like working a bit harder you can express the sign using the nearest integer function.

user1337
  • 24,381
  • This is so simple it's embarrassing! I know I tried once thinking of going from the center (the sinc is actually shifted with -N/2) but not like this. And it can be done with mod() fairly easy, without encumbering too much the already fluffy formulas. This needs chewing, I think I can have a true IDFT now. If this leads to the result, this one will be the answer. Thank you. – user164048 Jul 16 '14 at 10:14
  • It took me a while to find the old file but, unfortunately, it's not that easy. The resulting sinc can even be a sum of sinc functions having different frequencies, but those are more exotic cases. Even so, there is no clear interval during which the function goes to zero (which is why I used differentiation) because the IDFT is done via a custom, arbitrary frequency domain function, which means, sadly, I can't make it like this. Still, this is the answer for the mathematical problem so I'll mark it. Unfortunately I can't upvote, I need 15 rep... Have a beer instead :-) – user164048 Jul 16 '14 at 12:31
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There is no way to restore the exact singum if you only know the absolute value, but in general the inverse of $y = |x|$ is $x = \pm y$

ThreeFx
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  • I remember that one of the closer results I got was to differentiate and then use a function similar to the DFLOP (I'm with electronics, mostly): for a clock input one period is logic 1 then logic low, etc. It worked, within tolerances since the sinc was truncated to a variable N periods. But that required realtime processing and I was interested in something more mathematical with, well, really "by any means". – user164048 Jul 16 '14 at 08:55
  • @user164048 Could you then please explain the exact reason for first taking the absolute value and then trying to restore the signum? – ThreeFx Jul 16 '14 at 09:00
  • Well, the result of a whole complex thing (lots of $e^i$) was supposed to give a clean sinc, but I couldn't operate with $i$ in the program so I used sin^2+cos^2. It's really not a critical matter, but it got stuck on my neuron since two years ago. Today it resurfaced (an nth time) and I simply couldn't bear not to ask :-) In a shorter answer, it is related to the DFT/IDFT – user164048 Jul 16 '14 at 09:06