Sincsquared

Sincsquared refers to the square of the sinc function. The term sinc is used with two common conventions. In mathematics and many engineering texts, sinc(x) = sin(x)/x for x ≠ 0 with sinc(0) defined as 1. In normalized form used in signal processing, sinc(x) is often defined as sin(πx)/(πx). Sincsquared then means [sinc(x)]^2 in the respective convention, or [sinc(πx)]^2 when using the normalized form.

Basic properties of sinc squared include: it is an even, nonnegative function with a maximum at x = 0 where the value is 1 in either convention. Its zeros occur at nonzero arguments corresponding to the zeros of the sine function: for the unnormalized version, at x = nπ (n ≠ 0); for the normalized version, at x = n for all nonzero integers. It decays as roughly 1/x^2 for large |x|, and near zero it is smooth since the sinc function approaches 1.

Integrals and transforms provide further context. For the unnormalized sinc, ∫_{-∞}^{∞} sinc^2(x) dx = π. With the normalized sinc, ∫_{-∞}^{∞} sinc^2(x) dx = 1. In the frequency domain, the Fourier transform of sinc is a rectangle function; the Fourier transform of sinc^2 is a triangular function. This reflects the time–frequency trade-off: a narrow time-domain feature corresponds to a broader frequency-domain spread, and vice versa.

Applications of sinc squared appear in signal processing and optics. It describes the amplitude distribution from a rectangular aperture in one dimension (the diffraction pattern), and its squared magnitude often models the intensity pattern. Sincsquared is also used as a smoothing kernel and in idealized low-pass filter representations in various analytical contexts.