Sinc Function
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The sinc function is a fundamental function in signal processing and Fourier analysis. It arises naturally as the Fourier transform of a rectangular pulse.
Definition
The normalized sinc function (used in signal processing):
$$ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} $$with $\text{sinc}(0) = 1$ by L’Hôpital’s rule.
The unnormalized sinc (used in mathematics):
$$ \text{sinc}(x) = \frac{\sin(x)}{x} $$Properties
| Property | Value |
|---|---|
| $\text{sinc}(0)$ | 1 |
| $\text{sinc}(n)$ for $n \in \mathbb{Z} \setminus \{0\}$ | 0 |
| $\int_{-\infty}^{\infty} \text{sinc}(x) \, dx$ | 1 |
| $\int_{-\infty}^{\infty} \text{sinc}^2(x) \, dx$ | 1 |
Fourier Transform Pair
The sinc and rect functions form a fundamental transform pair:
$$ \mathcal{F}\{\text{rect}(t)\} = \text{sinc}(f) $$$$ \mathcal{F}\{\text{sinc}(t)\} = \text{rect}(f) $$Role in Sampling Theory
The sinc function is the ideal interpolation kernel for bandlimited signals. The Whittaker-Shannon sampling theorem states that a bandlimited signal can be perfectly reconstructed from its samples using sinc interpolation:
$$ x(t) = \sum_{n=-\infty}^{\infty} x[n] \, \text{sinc}\left(\frac{t - nT}{T}\right) $$In Optics
The sinc² intensity pattern appears in Fraunhofer diffraction from a rectangular slit. The zeros of sinc determine the positions of dark fringes.