McClure School of Emerging Communication Technologies Ohio University

Fourier Transform of a Rectangular Pulse

Interactive explorer — adjust T, A, and the carrier frequency fc

Duration T (s) 1.0
Amplitude A 1.0
Rectangular pulse   x(t) = A · rect(t / T)
Spectrum shown
|A·T·sinc(πfT)|
Peak |X(0)|
1.00
First null (Hz)
1.00

Fourier Transform of a Modulated Pulse

Carrier frequency fc (Hz) 2.0
Modulated pulse   y(t) = A · rect(t / T) · cos(2πfct)
Spectrum shown
|½A·T·[sinc((f−fc)T)+sinc((f+fc)T)]|
Lobe peaks ±fc
±2.0 Hz
Lobe nulls
fc ± 1/T
ℹ️ Both x(t) and y(t) are real and even functions of t (symmetric about t = 0), so their Fourier transforms are purely real — the imaginary part is identically zero everywhere.
Both pulses span t ∈ [−T/2, T/2]. The rectangular pulse spectrum is independent of fc and uses a fixed frequency axis scaled only by T. The modulated spectrum axis adapts to show the lobes at ±fc.
Multiplying by cos(2πfct) shifts the spectrum: Y(f) = ½[X(f−fc) + X(f+fc)], producing two sinc lobes at ±fc, each half the height of the original. Because both signals are real and even, Im{X(f)} = Im{Y(f)} = 0.