Sinc function

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The sinc function, denoted by [\mathrm(x)\,], has two definitions, sometimes distinguished as the normalized sinc function and unnormalized sinc function. Each is the product of a sine function and a monotonically decreasing function 1/x:

In digital signal processing and communication theory, the normalized sinc function is commonly defined by
:[\mathrm(x) = \frac]
In mathematics, the historical unnormalized sinc function (for sinus cardinalis), is defined by
:[\mathrm(x) = \frac]

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

The unnormalized sinc function is identical to the normalized sinc function above except for a missing scaling factor of π in the argument.

Contents

1 Properties
2 Relationship to the Dirac delta distribution
3 See also
4 External links

Properties

The normalized sinc function has properties that make it ideal in relationship to interpolation and bandlimited functions:

[\mathrm(0) = 1\,] and [\mathrm(k) = 0\,] for [k\ne 0\,] and [k\in\mathbb\,] (integers); that is, it is an interpolating function.
the functions [x_k(t)=\mathrm(t-k) \ ] form an orthonormal basis for bandlimited functions in the function space [L^2(\R)], with highest angular frequency [\omega_\mathrm=\pi\,] (that is, highest cycle frequency [f_\mathrm=1/2\,]).

Other properties of the two sinc functions include:

The local maxima and minima of the unnormalized sinc, [\begin\frac \end\,] correspond to its intersections with the cosine function. I.e. where the derivative of [\begin\frac \end\,] is zero (local extrema at [x = a\,]), then [\begin\frac \end = \cos(a) \,].

The unnormalized sinc is the zero^th order spherical Bessel function of the first kind, [j_0(x) = \begin\frac \end\,]. The normalized sinc is [j_0(\pi x)\,].

The zero-crossings of the unnormalized sinc are at nonzero multiples of [\pi\,]; zero-crossing of the normalized sinc [\mathrm(x) = \begin\frac \end\,] occur at nonzero integer values.

The continuous Fourier transform of the normalized sinc [\mathrm(x) = \begin\frac \end\,] (to ordinary frequency) is [\mathrm(f)\,].

:[\int_^\infty \mathrm(t)\,e^dt = \mathrm(f)],

where the rectangular function is 1 for argument between –1/2 and 1/2, and zero otherwise.

The integral

:[\int_^\infty \begin\frac \end\, dx = 1]

is an improper integral. It is not a Lebesgue integral because:

:[\int_^\infty \left|\begin\frac \end\right|\ dx = \infty \,]

[ \mathrm(x) = \frac = \prod_^\infty \left(1 - \frac\right)]

[ \mathrm(x) = \frac = \frac = \frac]

where [\Gamma(x)] is the gamma function.

Relationship to the Dirac delta distribution

The normalized sinc function can be used as a nascent delta function (see Dirac delta function), even though it is not a distribution.

The normalized sinc function is related to the delta distribution δ(x) by

[\lim_\frac\textrm(x/a)=\delta(x).]

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

[\lim_\int_^\infty \frac\textrm(x/a)\varphi(x)\,dx =\int_^\infty\delta(x)\varphi(x)\,dx = \varphi(0),]

for any smooth function [\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/(πx), regardless of the value of a. This contradicts 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.

External links

, [Sinc Function] at MathWorld.

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Sinc function

Properties

Relationship to the Dirac delta distribution

See also

External links