Sinc function

The sinc function (normalized) is

$\mathrm{sinc}(x)=\frac{\sin (\pi x)}{\pi x},$

Image: Normalized sinc function, CC BY-SA 3.0

with $\mathrm{sinc}(0)=1$ defined by the L’Hôpital limit, and zeros at every nonzero integer $x$.

Two conventions

The normalized sinc (above) has zeros at integer $x$ and is the one used in signal processing and in this course. The unnormalized sinc $\sin x/x$ has zeros at integer multiples of $\pi$ and is more common in physics and pure math. Whenever you see “sinc” in this vault and in Continuous-Time Signals and Systems, it’s the normalized one. Other contexts may differ.

Picture

• Main lobe from $-1$ to $+1$ (where the first zeros sit), with a peak of $1$ at $x=0$.
• Side lobes that decay in envelope as $1/x$.
• Alternating sign: positive on $(0,1)$, negative on $(1,2)$, positive on $(2,3)$, etc.

The slow $1/x$ decay is what makes sinc a difficult thing to truncate cleanly — its tails are long, and chopping them off produces ringing artifacts (closely related to the Gibbs phenomenon).

Where sinc shows up

The Fourier transform of a rectangle is a sinc:

$\mathrm{rect}(t/w)\stackrel{\mathcal{F}}{\leftrightarrow }w\mathrm{sinc}(wf).$

And dually, the Fourier transform of a sinc is a rectangle:

$\mathrm{sinc}(t/w)\stackrel{\mathcal{F}}{\leftrightarrow }w\mathrm{rect}(wf).$

These are the most-used Fourier pairs in the course. The sinc and rect together encode the time-bandwidth tradeoff: a narrow rect (good time localization) has a wide sinc spectrum (poor frequency localization), and vice versa.

The sinc also appears:

• In the impulse response of an ideal lowpass filter — the inverse transform of a brick-wall rectangle in frequency is a sinc in time.
• In sinc interpolation for reconstructing a sampled signal — each sample contributes a sinc pulse centered at the sample time.
• In the Fourier series coefficients of a periodic train of rectangles (the rect-train pair).

sinc²

The square ${\mathrm{sinc}}^2(x)$ is positive everywhere, with main lobe peaking at $1$ and side lobes much smaller than for sinc alone (since squaring suppresses values less than 1). It is the Fourier transform of the unit triangle:

$\mathrm{tri}(t)\stackrel{\mathcal{F}}{\leftrightarrow }{\mathrm{sinc}}^2(f).$

This follows from $\mathrm{tri}(t)=\mathrm{rect}(t)\ast \mathrm{rect}(t)$ and the convolution theorem (convolution in time = multiplication in frequency).

Key values

• $\mathrm{sinc}(0)=1$ (by L’Hôpital, since $\sin (\pi x)/(\pi x)\to 1$ as $x\to 0$).
• $\mathrm{sinc}(n)=0$ for every nonzero integer $n$.
• $\mathrm{sinc}$ is even: $\mathrm{sinc}(-x)=\mathrm{sinc}(x)$.