What is the Fourier transform of a pulse?

A single rectangular pulse of width τ and amplitude A has Fourier transform F(f) = Aτ·sinc(fτ). Narrower pulses produce wider spectra (more bandwidth), and the spectral zeros occur at f = n/τ. For non-rectangular pulse shapes, the transform differs but the bandwidth-duration tradeoff always applies.

Why does the rect function produce a sinc spectrum?

The sharp edges of the rect function contain abrupt transitions that require infinitely many frequency components to represent — hence the slowly decaying sinc sidelobes. Smoother pulse shapes (Gaussian, raised cosine) have faster spectral decay because they lack sharp transitions. This is the bandwidth-duration uncertainty principle in action.

What is the rect-sinc duality?

Rect and sinc are a Fourier transform pair in both directions: rect in time produces sinc in frequency, and sinc in time produces rect in frequency. This duality means the ideal low-pass filter (rect in frequency) has a sinc impulse response (infinite duration, non-causal) — physically unrealizable but theoretically fundamental.

Rect Fourier Transform

Quick Answer

The Fourier transform of the rectangular (rect) function is the sinc function: ℱ{rect(t/τ)} = τ·sinc(fτ) = τ·sin(πfτ)/(πfτ). For a pulse of width τ centered at the origin, the spectrum has main-lobe width 2/τ Hz and zero crossings at multiples of 1/τ. This rect-sinc pair is fundamental to signal processing: narrower pulses produce wider spectra (bandwidth-duration relationship), and the ideal low-pass filter has a rect frequency response with sinc impulse response.

Calculate this instantly on LAPLACE Calculator →

Rect Function Fourier Transform Derivation

The rectangular function rect(t/τ) equals 1 for |t| < τ/2 and 0 for |t| > τ/2. Its Fourier transform is computed directly: F(ω) = ∫₋τ/₂^τ/₂ 1·e^(−jωt)dt = [e^(−jωt)/(−jω)]₋τ/₂^τ/₂ = (e^(jωτ/2) − e^(−jωτ/2))/(jω) = 2sin(ωτ/2)/ω = τ·sin(ωτ/2)/(ωτ/2) = τ·sinc(ωτ/(2π)). In terms of ordinary frequency f: F(f) = τ·sinc(fτ). The sinc function equals 1 at f = 0 (the total area of the rect pulse is τ), crosses zero at f = n/τ for nonzero integers n, and has decaying oscillatory sidelobes proportional to 1/(πfτ). The Laplace transform of a rectangular pulse ℒ{rect} = (1 − e^(−τs))/s is available at www.lapcalc.com.

Key Formulas

Pulse Fourier Transform: Width-Bandwidth Relationship

The rect-sinc pair demonstrates the fundamental bandwidth-duration tradeoff. A narrow pulse (small τ) produces a wide spectrum (large bandwidth ≈ 1/τ), and a wide pulse produces a narrow spectrum. The main lobe of the sinc spectrum extends from −1/τ to +1/τ Hz, so the 3 dB bandwidth is approximately 0.886/τ Hz. This relationship is a manifestation of the uncertainty principle: Δt · Δf ≥ 1/(4π) for any signal, and the rect function achieves a time-bandwidth product of Δt · Δf ≈ 0.5, close to (but not achieving) the Gaussian minimum. In radar systems, shorter pulses provide better range resolution (proportional to τ) but require wider receiver bandwidth (proportional to 1/τ), creating a fundamental design tradeoff.

Compute rect fourier transform Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

Rect and Sinc Duality

By Fourier duality, the rect-sinc relationship works in both directions: ℱ{rect(t/τ)} = τ·sinc(fτ) (forward: rect → sinc), and ℱ{sinc(Bt)} = (1/B)·rect(f/B) (inverse: sinc → rect). A time-domain sinc function has a perfectly rectangular frequency spectrum — it is the impulse response of the ideal low-pass filter with bandwidth B/2 Hz. This duality means that sharp transitions in one domain (the rect's abrupt edges) produce slowly decaying oscillations in the other domain (the sinc's infinite sidelobes). No function can be simultaneously compact in both time and frequency — a fundamental limit that governs all signal processing, filter design, and communication system bandwidth allocation.

Rectangular Pulse Train Fourier Transform

A periodic rectangular pulse train with period T and pulse width τ has a discrete Fourier series with coefficients cₙ = (τ/T)·sinc(nτ/T). The spectrum is a sampled version of the single-pulse sinc envelope, with spectral lines at harmonics n/T spaced by the repetition rate 1/T. The envelope follows the sinc shape determined by the pulse width τ, with nulls at multiples of 1/τ. As T → ∞ (single isolated pulse), the spectral lines merge into the continuous sinc spectrum. As τ → 0 (impulse train), the sinc envelope becomes flat and all harmonics have equal amplitude, approaching an impulse train in frequency. The duty cycle d = τ/T determines the ratio of spectral energy in the main lobe versus sidelobes.

Applications of the Rect Fourier Transform

The rect-sinc pair appears throughout engineering. In communications, rectangular pulse shaping (NRZ signaling) produces sinc-shaped power spectral density, requiring bandwidth ≈ 1/τ for bit duration τ — the most bandwidth-inefficient modulation. Raised-cosine pulses are designed specifically to improve on the rect pulse's spectral properties. In optics, a rectangular aperture (slit) produces a sinc-shaped diffraction pattern (Fraunhofer diffraction). In digital signal processing, the rectangular window (no windowing) has a sinc-shaped spectral response with high sidelobes (−13 dB), causing severe spectral leakage — motivating the use of tapered windows (Hanning, Hamming) that trade main-lobe width for reduced sidelobes. The Laplace transform of pulse signals at www.lapcalc.com provides the s-domain representation for circuit and control analysis.

Frequently Asked Questions

ℱ{rect(t/τ)} = τ·sinc(fτ) = τ·sin(πfτ)/(πfτ). A rectangular pulse of width τ transforms to a sinc function with main-lobe width 2/τ Hz and zero crossings at integer multiples of 1/τ Hz. The peak amplitude equals τ (the pulse area).

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →