Sinc function: Wikis

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In mathematics, the sinc function, denoted by sinc(x) and sometimes as Sa(x), has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by

$\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}.\,\!$

It is called normalized because its Fourier transform is the rectangular function and its square integral is unity. In mathematics, the historical unnormalized sinc function is defined by

$\mathrm{sinc}(x) = \frac{\sin(x)}{x}.\,\!$

In both cases, the value of the function at the removable singularity at zero is sometimes specified explicitly as the limit value 1.[1] The sinc function is analytic everywhere.

The term "sinc" is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine).

Properties

The normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale from x = −6π to 6π.

The zero crossings of the unnormalized sinc are at nonzero multiples of π; zero crossings 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, sin(ξ)/ξ = cos(ξ) 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

$\frac{\sin(\pi x)}{\pi x} = \lim_{m\to\infty}\prod_{n=1}^m \left(1 - \frac{x^2}{n^2}\right)\,\!$

and is related to the gamma function Γ(x) by Euler's reflection formula:

$\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)}.\,\!$

Euler discovered that

$\frac{\sin(x)}x=\cos\left(\frac{x}2\right)\cdot\cos\left(\frac{x}4\right)\cdot\cos\left(\frac{x}8\right)\cdots.$

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect(f),

$\int_{-\infty}^\infty \mathrm{sinc}(t) \, e^{-2\pi i f t}\,dt = \mathrm{rect}(f),\,\!$

where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the sinc filter is the ideal (brick-wall, meaning rectangular frequency response) low-pass filter. This Fourier integral, including the special case

$\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \mathrm{rect}(0) = 1\,\!$

is an improper integral; it is not a convergent Lebesgue integral, as

$\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right|\, dx = +\infty.$

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 nonzero integer k.
• The functions xk(t) = sinc(t−k) form an orthonormal basis for bandlimited functions in the function space L2(R), with highest angular frequency ωH = π (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, $\scriptstyle j_0(x)$. The normalized sinc is j0(π x).
• $\int_{0}^{x} \frac{\sin(\theta)}{\theta}\,d\theta = \mathrm{Si}(x) \,\!$
where Si(x) is the sine integral.
$x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.\,\!$
The other is cos(λ x)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
• $\int_{-\infty}^{\infty} \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \,\! \rightarrow \int_{-\infty}^{\infty} sinc^2(x)\,dx = 1 \,\!$.

where the normalized sinc is meant.

• $\int_{-\infty}^{\infty} \frac{\sin^3(\theta)}{\theta^3}\,d\theta = \frac{3\pi}{4} \,\!$
• $\int_{-\infty}^{\infty} \frac{\sin^4(\theta)}{\theta^4}\,d\theta = \frac{2\pi}{3} \,\!$

Relationship to the Dirac delta distribution

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:

$\lim_{a\rightarrow 0}\frac{1}{a}\textrm{sinc}(x/a)=\delta(x).$

This is not an ordinary limit, since the left side does not converge. Rather, it means that

$\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx = \varphi(0),$

for any smooth function $\scriptstyle \varphi(x)$ 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/(π a x), 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.