sinc

Sinc is a mathematical function used especially in signal processing and analysis. There are two common conventions for defining it. The normalized sinc is sinc(x) = sin(π x)/(π x) with sinc(0) defined as 1. The unnormalized version is sinc(x) = sin x / x, also with the value at 0 taken as 1 by limit. Both conventions share similar shapes but differ by a scaling of the argument.

The function is even and decays roughly as 1/x for large |x|. It has zeros at every nonzero integer for the normalized form (x = ±1, ±2, …) and at x = ±π, ±2π, … for the unnormalized form. The integral of the normalized sinc over the real line equals 1: ∫_{-∞}^{∞} sinc(x) dx = 1. However, the absolute integral ∫_{-∞}^{∞} |sinc(x)| dx diverges, since the tails behave like 1/x.

In the frequency domain, the sinc function arises as the Fourier transform of a rectangular function. This makes it the impulse response of an ideal low-pass filter: the time-domain sinc corresponds to a perfect rectangular spectrum, and vice versa. Because of its infinite extent and noncausality, the ideal sinc is of theoretical use but impractical for real systems. It also serves as the reconstruction kernel in the Whittaker–Shannon sampling theorem, yielding the exact reconstruction of a bandlimited signal from its samples via a sum of shifted sinc functions.

The term sinc, short for sine cardinal, is widely used in engineering and mathematics to denote this family of functions.