Sinc Function

A Sinc Function is a mathematical function based on sine function, [math]\displaystyle{ \mathrm{sinc}(x)=\begin{cases}\sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x = 0 \end{cases} }[/math].

• AKA: Cardinal Sine Function, Sampling Function.
• Context:
  • It ranges from being an Unnormalized Sinc Function (i.e. [math]\displaystyle{ a=1 }[/math]) to being a Normalized Sinc Function (i.e. [math]\displaystyle{ a=\pi }[/math]).
• Example(s):
  • [math]\displaystyle{ \mathrm{sinc}(x)=\dfrac{\sin(x)}{x} }[/math],
  • [math]\displaystyle{ \mathrm{sinc}(x)=\dfrac{\sin(\pi x)}{\pi x} }[/math]
• Counter-Example(s)
  • Sine Function,
  • Cosine Function.
• See: Entire Function, Mathematics, Physics, Engineering, Digital Signal Processing, Information Theory, Normalizing Constant, Integral, pi, Fourier Transform, Rectangular Function, Whittaker–Shannon Interpolation Formula.

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References

2018a

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Sinc_function Retrieved:2018-2-25.
  • In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by sinc(x), has two slightly different definitions.

    In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by

    [math]\displaystyle{ \operatorname{sinc}(x) = \frac{\sin(x)}{x}~. }[/math] In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by

    [math]\displaystyle{ \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}~. }[/math] In either case, the value at x 0 is defined to be the limiting value sinc(0) 1.

    The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of pi). As a further useful property, all of the zeros of the normalized sinc function are integer values of .

    The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.

    The only difference between the two definitions is in the scaling of the independent variable (the -axis) by a factor of . In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere and hence an entire function.

    The term sinc is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine). It was introduced by Philip M. Woodward in his 1952 paper "Information theory and inverse probability in telecommunication", in which he said the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own", and his 1953 book Probability and Information Theory, with Applications to Radar.

2018b

• (Mathworld, 2018) ⇒ Weisstein, Eric W. “Sinc Function." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/SincFunction.html Retrieved:2018-2-25.
  • QUOTE: The sinc function [math]\displaystyle{ \mathrm{sinc}(x) }[/math], also called the “sampling function” is a function that arises frequently in signal processing and the theory of Fourier transforms. The full name of the function is “sine cardinal," but it is commonly referred to by its abbreviation, “sinc”. There are two definitions in common use. The one adopted in this work defines

    [math]\displaystyle{ f(x)=\begin{cases}1 & \text{for } x = 0 \\ \sin(x)/x & \text{otherwise}\end{cases}\quad (1) }[/math]

    where [math]\displaystyle{ \sin(x) }[/math] is the sine function, plotted above.

    This has the normalization

    [math]\displaystyle{ \int_{-\infty}^{+\infty}\mathrm{sinc}(x)dx=\pi\quad (2) }[/math]. This function is implemented in the Wolfram Language as Sinc[x].

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