In mathematics, the sinc function, denoted by <math>\scriptstyle\mathrm{sinc}(x)\,</math> and sometimes as <math>\scriptstyle\mathrm{Sa}(x)\,</math>, has two definitions, sometimes distinguished as the normalized sinc function and unnormalized sinc function. In digital signal processing and information theory, the normalized sinc function is commonly defined by
- <math>\mathrm{Sa}(x) = \mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}.</math>
In mathematics, the historical unnormalized sinc function (or sinus cardinalis), is defined by
- <math>\mathrm{Sa}(x) = \mathrm{sinc}(x) = \frac{\sin(x)}{x}.</math>
In both cases, the value of the function at the removable singularity at zero, usually calculated by l'Hôpital's rule, is sometimes specified explicitly as the limit value 1. The sinc function is analytic everywhere. The term "sinc" is a contraction of the function's full name, the sine cardinal or sinus cardinalis.
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Properties
The zero-crossings of the unnormalized sinc are at nonzero multiples of π; zero-crossing of the normalized sinc occur at nonzero integer values. The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, <math>\scriptstyle\sin(\xi)/\xi \,=\, \cos(\xi) \,</math> for all points ξ where the derivative of sin(x)/x is zero (and thus a local extremum is reached). The normalized sinc function has a simple representation as the infinite product
- <math>\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)</math>
and is related to the gamma function <math>\Gamma(x)</math> by Euler's reflection formula:
- <math>\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)}.</math>
The continuous Fourier transform of the normalized sinc (to ordinary frequency) is <math>\mathrm{rect}(f)\,</math>.
- <math>\int_{-\infty}^\infty \mathrm{sinc}(t) \, e^{-2\pi i f t}\,dt = \mathrm{rect}(f),</math>
where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This Fourier integral, including the special case
- <math>\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \mathrm{rect}(0) = 1</math>
is an improper integral. Since
- <math>\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right|\ dx = \infty \,</math>
it is not a Lebesgue integral. The normalized sinc function has properties that make it ideal in relationship to interpolation and bandlimited functions:
- It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for k ≠ 0 and <math>\scriptstyle k\in\mathbb{Z}\,</math> (integers).
- The functions <math>\scriptstyle x_k(t)\,=\,\operatorname{sinc}(t-k) \ </math> form an orthonormal basis for bandlimited functions in the function space <math>\scriptstyle L^2(\R)</math>, with highest angular frequency <math>\scriptstyle \omega_\mathrm{H}\,=\,\pi\,</math> (that is, highest cycle frequency ƒH = 1/2).
Other properties of the two sinc functions include:
- The unnormalized sinc is the zeroth order spherical Bessel function of the first kind, <math>\scriptstyle j_0(x)</math>. The normalized sinc is <math>\scriptstyle j_0(\pi x)\,</math>.
- <math> \int_{0}^{x} \frac{\sin(\theta)}{\theta}\,d\theta = \mathrm{Si}(x) </math>
- where Si(x) is the sine integral.
- <math>\scriptstyle\lambda\, \operatorname{sinc}(\lambda x)</math> (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation
-
- <math>x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.</math>
- The other is <math>\scriptstyle\cos(\lambda t)/t</math>, which is not bounded at x = 0, unlike its sinc function counterpart.
Relationship to the Dirac delta distribution
The normalized sinc function can be used as a nascent delta function, even though it is not a distribution. The normalized sinc function is related to the delta distribution δ(x) by
- <math>\lim_{a\rightarrow 0}\frac{1}{a}\textrm{sinc}(x/a)=\delta(x).</math>
This is not an ordinary limit, since the left side does not converge. Rather, it means that
- <math>\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx
= \varphi(0),
</math> for any smooth function <math>\scriptstyle \varphi(x)</math> with compact support. In the above expression, as a approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(πx), regardless of the value of a, and approaches zero for any nonzero value of x. This complicates the informal picture of δ(x) as being zero for all x except at the point x = 0 and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the Gibbs phenomenon.
