Sinc Function — Definition, Formula & Examples

The sinc function is defined as

\operatorname{sinc}(x) = \frac{\sin(x)}{x}

for

x \neq 0

and

\operatorname{sinc}(0) = 1

. It produces a damped oscillation that appears naturally in Fourier transforms, diffraction patterns, and signal reconstruction.

The unnormalized sinc function is the continuous function $\operatorname{sinc}: \mathbb{R} \to \mathbb{R}$ given by $\operatorname{sinc}(x) = \frac{\sin x}{x}$ for $x \neq 0$ , extended to $x = 0$ by the limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ , making it continuous everywhere. The normalized variant, common in engineering, replaces $x$ with $\pi x$ : $\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$ .

Key Formula

\operatorname{sinc}(x) = \begin{cases} \dfrac{\sin x}{x}, & x \neq 0 \\[6pt] 1, & x = 0 \end{cases}

Where:

$x$ = Any real number (the input to the function)
$\sin x$ = The ordinary sine function evaluated at x (radians)

How It Works

The sinc function oscillates like

\sin(x)

but its amplitude decays as

1/|x|

for large

|x|

. Its zeros occur at every nonzero integer multiple of

\pi

(unnormalized) or at every nonzero integer (normalized). A key result in calculus is

\int_{-\infty}^{\infty} \frac{\sin x}{x}\,dx = \pi

, known as the Dirichlet integral. The function is even, meaning

\operatorname{sinc}(-x) = \operatorname{sinc}(x)

, and it has a global maximum of

1

at the origin.

Worked Example

Problem: Evaluate sinc(x) at x = π/2, x = π, and x = 0.

At x = π/2: Substitute into the formula. Since sin(π/2) = 1:

\operatorname{sinc}\!\left(\frac{\pi}{2}\right) = \frac{\sin(\pi/2)}{\pi/2} = \frac{1}{\pi/2} = \frac{2}{\pi} \approx 0.6366

At x = π: Since sin(π) = 0 and x ≠ 0, the function equals zero. This is the first positive zero of sinc.

\operatorname{sinc}(\pi) = \frac{\sin(\pi)}{\pi} = \frac{0}{\pi} = 0

At x = 0: The formula sin(x)/x is indeterminate at 0, so use the defined value (equivalently, L'Hôpital's rule confirms the limit is 1).

\operatorname{sinc}(0) = 1

Answer: sinc(π/2) = 2/π ≈ 0.637, sinc(π) = 0, and sinc(0) = 1.

Visualization

Why It Matters

In signal processing and electrical engineering, the normalized sinc function is the ideal low-pass filter impulse response, central to the Nyquist–Shannon sampling theorem. In physics, the sinc function describes single-slit diffraction intensity patterns in optics.

Common Mistakes

Mistake: Confusing the unnormalized sinc(x) = sin(x)/x with the normalized sinc(x) = sin(πx)/(πx).

Correction: Check which convention your course or textbook uses. Mathematics texts typically use sin(x)/x; engineering and DSP texts typically use sin(πx)/(πx), whose zeros fall at nonzero integers.