sinc2

Sinc2, commonly written as sinc^2(x), refers to the square of the sinc function. In many engineering contexts, the normalized sinc is defined as sinc(x) = sin(πx)/(πx), so sinc^2(x) = [sin(πx)/(πx)]^2. Depending on convention, some texts use the unnormalized form sinc(x) = sin x / x, in which case sinc^2 is sin^2 x / x^2. The squared form is nonnegative, even, and attains its maximum value of 1 at x = 0, with zeros at all nonzero integer values when using the normalized definition.

Key properties include that the integral of sinc^2 over the real line depends on the sinc convention. With the normalized sinc, ∫_{-∞}^{∞} sinc^2(x) dx = 1. The function is square-integrable and decays approximately as 1/x^2 for large |x|. The Fourier transform of sinc^2 is a triangular function: it yields a piecewise linear spectrum that rises to a maximum at zero frequency and falls to zero beyond a unit-width band, reflecting the standard relationship that squaring in the time domain corresponds to convolution of the corresponding frequency content.

Applications and usage include its role as an interpolation or reconstruction kernel in signal processing, where sinc-shaped kernels arise in ideal bandlimited interpolation. The squared form often serves as a smoother or window-like kernel in smoothing, kernel density estimation, or filter design. It is also used in theoretical analyses of spectral leakage and in the study of the auto-correlation of rectangular pulse shapes.

Notes on variants: because the area and transform depend on the chosen sinc normalization, it is important to specify whether sinc(x) = sin(πx)/(πx) or sin x / x is being used when comparing results.