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
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
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).
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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
and is related to the gamma function Γ(x) 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 (brickwall, meaning rectangular frequency response) lowpass filter. This Fourier integral, including the special case
is an improper integral; it is not a convergent Lebesgue integral, as
The normalized sinc function has properties that make it ideal in relationship to interpolation and bandlimited functions:
Other properties of the two sinc functions include:
where the normalized sinc is meant.
The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
for any smooth function 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.
