Sincs

Sincs, or sinc functions, are a family of real-valued functions that play a central role in analysis and signal processing. They come in two common definitions: the unnormalized sinc, sinc(x) = sin x / x with sinc(0) defined as 1 by the limit; and the normalized sinc, sinc(x) = sin(π x)/(π x) with the same value at 0. The term sinc is a contraction of “sine cardinal.”

Key properties and conventions

The two definitions differ by a scaling of the argument, but both share essential features: they are even functions, oscillatory, and decay approximately as 1/|x|. They have zeros at x = nπ for the unnormalized version and at x = n for the normalized version, for nonzero integers n. Sinc(0) = 1 in both conventions. Because of their slow decay and infinite extent, sinc functions are non-compact and cannot serve as an exact, finite-extent filter.

Fourier transform and relation to rectangles

A fundamental aspect of sinc functions is their relationship with rectangular (rect) functions in Fourier analysis. In the standard signal-processing convention, the Fourier transform of a rectangular spectrum yields a sinc time-domain function, while the transform of a sinc in time produces a rectangular spectrum. This duality underpins the Nyquist–Shannon sampling theorem and idealized reconstruction kernels.

Applications and limitations

Sinc functions are central to ideal interpolation, reconstruction of bandlimited signals, and certain digital-to-analog conversion schemes. In practice, ideal sinc filters are non-causal and infinite in extent, so real implementations use truncated or windowed approximations. The concept remains foundational in understanding sampling, bandwidth, and the trade-offs involved in practical filter design.

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