Sinc function

In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by sinc(x), has two slightly different definitions.

In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by

In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by

In either case, the value at x = 0 is defined to be the limiting value sinc(0) = 1.

The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of π). As a further useful property, all of the zeros of the normalized sinc function are integer values of x.

The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.

The only difference between the two definitions is in the scaling of the independent variable (the x-axis) by a factor of π. In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere and hence an entire function.

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