Sinc Function - Properties

Properties

The zero crossings of the unnormalized sinc are at nonzero multiples of π, while 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.

A good approximation of the x-coordinate of the n-th extremum with positive x-coordinate is

x_n \approx (n+\tfrac12)\pi - \frac1{(n+\frac12)\pi}

where odd n lead to a local minimum and even n to a local maximum. Besides the extrema at xn, the curve has an absolute maximum at ξ0 = (0,1) and because of its symmetry to the y-axis extrema with x-coordinates −xn.

The normalized sinc function has a simple representation as the infinite product

and is related to the gamma function by Euler's reflection formula:

Euler discovered that

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

is an improper integral and not a convergent Lebesgue integral, as

The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled 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) (k integer) 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, . The normalized sinc is
where Si(x) is the sine integral.
  • λ sinc(λ x) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation
The other is cos(λ x)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
where the normalized sinc is meant.

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