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I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: $$\sin(x)/x=\text{sinc}(x) \hspace{0.2in}\textbf{unnormalized sinc function}$$

And for the normalized sinc we have: $$\sin(\pi x)/\pi x = \text{sinc}(x) \hspace{0.2in}\textbf{normalized sinc function}$$

My question is: If we have something like: $\dfrac{\sin\left(\frac{200\pi x}{500}\right)}{200\pi x }$, if we divide and multiply the equation by $500$, will this convert to be: $500\,\text{sinc}(x)$? It just a tad bit confusing about what constants stay if inside the argument of the sine (if any).

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It is definitely a bad idea to be using one name for two different functions... what normalization are you most comfy with? –  J. M. Apr 19 '12 at 9:38
    
@J.M.: What do you mean by using one name for two different functions? I'm more so used to the normalized form. –  night owl Apr 19 '12 at 9:55
    
Okay, but in you post both the normalized and unnormalized functions are called $\mathrm{sinc}$... that, good sir, is always liable to cause confusion. –  J. M. Apr 19 '12 at 9:58
    
@J.M.: Please see here: en.wikipedia.org/wiki/Sinc_function –  night owl Apr 19 '12 at 10:00
    
I'm perfectly aware of that. My point was that if you're going to be using both normalized and unnormalized forms in your question, you'd do well to distinguish them with some mark, no? –  J. M. Apr 19 '12 at 10:01

2 Answers 2

up vote 3 down vote accepted

In the engineering literature, those who define the Fourier transform as $$X(\omega) = \int_{-\infty}^{\infty} x(t)\exp(-i\omega t)\ \mathrm dt$$ tend to use the unnormalized version, while those who define the Fourier transform as $$X(f) = \int_{-\infty}^{\infty} x(t)\exp(-i2\pi f t)\ \mathrm dt$$ tend to use the normalized version. My personal preference is for the normalized version because the zeroes of the function are the nonzero integers, but, like most people, I have learned to live with both definitions and figure out which one an author is using even if it is not explicitly stated. There are, of course, zealots who say that people who use the convention they do not happen to prefer are in a state of sin.

Incidentally, I would like to say that my preference for the definition of the sinc (or sine cardinal) function is $$\mathrm{sinc}(x) = \begin{cases}\frac{\sin(\pi x)}{\pi x}, & x \neq 0,\\ 1, & x = 0,\end{cases}$$ and not simply $\mathrm{sinc}(x) = \sin(\pi x)/(\pi x)$ the way the OP and Wikipedia states it.

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I like the connection of "[. . .]zealots who say that people who use the convention they do not happen to prefer are in a state of sin." and "sinc (or sine cardinal)". Great answer. Although, if I may ask, (feel free to ignore my ignorance) what is the purpose of those two different conventions for the Fourier transform? You left me quite curious. –  000 Apr 19 '12 at 11:08
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@Limitless There are more than two conventions for the Fourier transform and different areas of study tend to use different ones because the "normalizations" suit the conventions in that area. The second version in my answer has the nice property that it is a unitary transformation and thus Parseval's theorem is an affirmation of this property: $$\int_{-\infty}^{\infty}|X(f)|^2\ \mathrm df = \int_{-\infty}^{\infty}|x(t)|^2\ \mathrm dt$$ –  Dilip Sarwate Apr 19 '12 at 11:22
    
Thanks for that explanation. I was always a bit intrigued by Fourier transforms in general. –  000 Apr 19 '12 at 11:25
    
Very nice answer. :) –  night owl Apr 29 '12 at 11:37

$$\frac{\sin\left(\frac{200}{500}\pi x\right)}{200\pi x} = \frac{1}{500} \frac{\sin\left(\frac{200}{500}\pi x\right)}{\frac{200}{500}\pi x}=\begin{cases} \frac{1}{500}\operatorname{sinc}\left(\frac{200}{500}\pi x\right) & \rm nonnormalized~sinc~convention \\ \color{White}X \\ \frac{1}{500}\operatorname{sinc}\left(\frac{200}{500}x\right) & \rm ~~~normalized~sinc~convention. \end{cases}$$

If you're doing mathematical writing and utilize a standard function that has multiple different versions (this happens a lot with normalizations, see also the Fourier transform), what you need to do is pick one of the conventions, and begin by making explicit to the reader which convention you have chose to use. Which version you pick is of course up to you, but a lot of the time there might be pros or cons to the different versions, depending on context and motivations.

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