FT of Rectangular Pulse

The Fourier transform of a rectangular pulse produces a sinc function in the frequency domain, directly explaining why ideal low-pass filters require infinite bandwidth. This result is fundamental to understanding pulse shaping in digital communications.

Core Concept

A rectangular pulse is defined as 1 for |t| <= T/2 and 0 elsewhere, where T is the pulse duration. It is the simplest finite-duration signal. Its abrupt edges at t = +-T/2 mean it contains high-frequency components that do not decay quickly, which the sinc spectrum reflects.

The Fourier transform integral of this pulse is straightforward because the integrand is simply e^(-j2*pi*f*t) over the finite interval [-T/2, T/2]. Evaluating this integral gives T times the sinc function. The first zero of the sinc occurs at f = 1/T, so a shorter pulse (smaller T) spreads energy to higher frequencies.

This trade-off between pulse width in time and spectral width in frequency is the time-bandwidth product principle. Narrower pulses require more bandwidth. Digital communication systems use this to determine how much spectrum a data pulse occupies.

Key Formula

For x(t) = rect(t/T), the Fourier transform is X(f) = T * sinc(f*T). The sinc function is defined as sinc(x) = sin(pi*x)/(pi*x). The first zeros of X(f) occur at f = +-1/T, +-2/T, ... The peak value is X(0) = T, the area under the pulse. Note: some textbooks define sinc(x) = sin(x)/x without the pi, which shifts where zeros fall. Check the convention used.

Given: x(t) = rect(t/4), so T = 4
Find: X(f)

Step 1: Write the integral
  X(f) = integral from -2 to 2 of e^(-j2*pi*f*t) dt

Step 2: Evaluate
  X(f) = [e^(-j2*pi*f*t) / (-j2*pi*f)] from -2 to 2
       = [e^(-j4*pi*f) - e^(j4*pi*f)] / (-j2*pi*f)

Step 3: Use Euler identity  (e^(jx) - e^(-jx)) = 2j*sin(x)
  Numerator: e^(-j4*pi*f) - e^(j4*pi*f) = -2j*sin(4*pi*f)
  X(f) = -2j*sin(4*pi*f) / (-j2*pi*f)
        = sin(4*pi*f) / (pi*f)

Step 4: Rewrite using sinc
  X(f) = 4 * sin(4*pi*f)/(4*pi*f)
        = 4 * sinc(4f)

Final Answer: X(f) = 4 sinc(4f)
First zeros at f = +-0.25, +-0.5, +-0.75, ...

Exam Tip: The null-to-null bandwidth of a rectangular pulse of duration T is 2/T Hz. The main-lobe bandwidth (one-sided) is 1/T. GATE problems often ask for bandwidth given pulse duration, so memorise: BW = 1/T for main lobe, 2/T for null-to-null. Also, the energy spectral density is |X(f)|^2 = T^2 * sinc^2(fT), which is the squared sinc.

Properties Summary

• FT pair: rect(t/T) <-> T*sinc(fT), valid for any pulse duration T > 0.
• Time scaling: rect(t/T) has first zero at f = 1/T; doubling T halves the spectral width.
• Even symmetry: rect(t/T) is even in t, so X(f) is real and even in f.
• Peak value: X(0) = T, equal to the area under the time-domain pulse.
• Duality: T*sinc(tT) in time has FT equal to rect(f/T), a rectangle in frequency.
• Energy: integral |x(t)|^2 dt = T (since x = 1 over duration T), equals integral |X(f)|^2 df by Parseval.

Quick Revision

• rect(t/T) = 1 for |t| < T/2, 1/2 for |t| = T/2, and 0 for |t| > T/2.
• FT result: X(f) = T * sinc(fT) = T * sin(pi*f*T)/(pi*f*T).
• X(0) = T; zeros at f = n/T for n = +-1, +-2, ...
• Main-lobe width: 2/T (extends from -1/T to +1/T).
• Sinc side lobes decrease as 1/(pi*f*T) in amplitude.
• The FT of a pulse shifted to [0, T] is X(f)*e^(-j*pi*f*T), adding linear phase but not changing |X(f)|.
• Exam trap: confusing the sinc convention. If sinc(x) = sin(x)/x (no pi), the FT of rect(t/T) becomes T*sin(2*pi*f*T/2)/(2*pi*f*T/2) and zeros shift accordingly. Always check which sinc definition the textbook uses.