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Rectangle Function — Definition, Formula & Examples

The rectangle function (often written rect) is a piecewise function that equals 1 inside a specified interval and 0 outside it. It models ideal on/off signals and serves as a building block in Fourier analysis and signal processing.

The rectangle function rect⁡(t)\operatorname{rect}(t) is defined as rect⁡(t)=1\operatorname{rect}(t) = 1 for ∣t∣<12|t| < \tfrac{1}{2}, rect⁡(t)=12\operatorname{rect}(t) = \tfrac{1}{2} for ∣t∣=12|t| = \tfrac{1}{2}, and rect⁡(t)=0\operatorname{rect}(t) = 0 for ∣t∣>12|t| > \tfrac{1}{2}. It can equivalently be expressed using Heaviside step functions as rect⁡(t)=H ⁣(t+12)−H ⁣(t−12)\operatorname{rect}(t) = H\!\left(t + \tfrac{1}{2}\right) - H\!\left(t - \tfrac{1}{2}\right).

Key Formula

rect⁡(t)={0,∣t∣>1212,∣t∣=121,∣t∣<12\operatorname{rect}(t) = \begin{cases} 0, & |t| > \tfrac{1}{2} \\[4pt] \tfrac{1}{2}, & |t| = \tfrac{1}{2} \\[4pt] 1, & |t| < \tfrac{1}{2} \end{cases}
Where:
  • tt = The independent variable (often time or a normalized coordinate)

How It Works

To create a rectangular pulse of width τ\tau centered at t0t_0, substitute rect⁡ ⁣(t−t0τ)\operatorname{rect}\!\left(\frac{t - t_0}{\tau}\right). The argument scaling stretches the unit-width pulse to width τ\tau, and the shift moves its center to t0t_0. Because the function has sharp transitions, its Fourier transform is a sinc function: F{rect⁡(t)}=sinc⁡(f)\mathcal{F}\{\operatorname{rect}(t)\} = \operatorname{sinc}(f). This rect–sinc duality is one of the most important pairs in Fourier analysis.

Worked Example

Problem: A radar system transmits a pulse that is on from t=2 μst = 2\,\mu s to t=6 μst = 6\,\mu s and off elsewhere. Express this pulse using the rectangle function.
Find center and width: The pulse runs from 2 to 6, so its center is the midpoint and its width is the difference.
t0=2+62=4 μs,τ=6−2=4 μst_0 = \frac{2 + 6}{2} = 4\,\mu s, \quad \tau = 6 - 2 = 4\,\mu s
Write the rect expression: Substitute into the shifted and scaled rectangle function.
p(t)=rect⁡ ⁣(t−44)p(t) = \operatorname{rect}\!\left(\frac{t - 4}{4}\right)
Verify: At t=3 μst = 3\,\mu s: the argument is (3−4)/4=−0.25(3-4)/4 = -0.25. Since ∣−0.25∣<0.5|-0.25| < 0.5, the function equals 1. At t=0t = 0: the argument is −1-1, and ∣−1∣>0.5|-1| > 0.5, so the function equals 0.
Answer: The radar pulse is p(t)=rect⁡ ⁣(t−4 μs4 μs)p(t) = \operatorname{rect}\!\left(\dfrac{t - 4\,\mu s}{4\,\mu s}\right).

Why It Matters

In electrical engineering and physics courses, the rectangle function is the standard model for ideal pulses and finite-duration signals. Its Fourier transform (the sinc function) explains bandwidth limitations in communication systems and the ringing artifacts seen in signal reconstruction.

Common Mistakes

Mistake: Forgetting that the width parameter goes in the denominator, not the numerator, of the argument.
Correction: To get a pulse of width τ\tau, write rect⁡(t/τ)\operatorname{rect}(t/\tau). Placing τ\tau in the numerator would compress the pulse instead of stretching it.