Rectangular Pulse

The rectangular pulse is the simplest finite-energy signal and the standard model for a digital bit pulse in baseband communication systems. Its Fourier transform, the sinc function, directly determines how much bandwidth a data channel requires.

Core Concept

The rectangular pulse is defined as A for |t| <= T/2 and zero otherwise. It is an even function because it is symmetric about t = 0. The width T and amplitude A fully characterise it. Engineers write it as A rect(t/T), where rect(.) is the unit rectangular function that equals 1 when its argument has magnitude less than 1/2.

The Fourier transform of rect(t/T) is T sinc(f T) where sinc(x) = sin(pi x) / (pi x). The sinc function has nulls at all nonzero integer multiples of 1/T in frequency. This means a narrower pulse (small T) has a wider spectrum, and a wider pulse (large T) has a narrower spectrum. This time-bandwidth trade-off appears in every signal processing textbook.

The rectangular pulse is a finite-energy signal. Its energy is A^2 T because integrating A^2 over the interval [-T/2, T/2] gives exactly A^2 T. Using Parseval's theorem, the same result must follow from integrating |A T sinc(f T)|^2 over all frequency — a useful double-check in exam problems.

Key Formula

rect(t/T) = 1 for |t| <= T/2, else 0. Fourier transform: F{A rect(t/T)} = A T sinc(f T) where sinc(x) = sin(pi x)/(pi x). First null of spectrum at f = 1/T Hz. Signal energy: E = A^2 T. Parseval check: integral |X(f)|^2 df = A^2 T^2 integral sinc^2(fT) df = A^2 T.

Given: x(t) = 2 rect(t / 4). Find Fourier transform, first null frequency, and signal energy.

Parameters: A = 2, T = 4

Step 1: Apply Fourier transform pair
  F{A rect(t/T)} = A T sinc(f T)
  X(f) = 2 * 4 * sinc(4f)
       = 8 sinc(4f)

Step 2: First null frequency
  sinc(4f) = 0 when 4f = +/- 1, +/- 2, ...
  First null: 4f = 1  =>  f = 1/4 = 0.25 Hz

Step 3: Signal energy
  E = A^2 * T = (2)^2 * 4 = 4 * 4 = 16 J

Step 4: Verify with Parseval (optional check)
  integral |8 sinc(4f)|^2 df = 64 * (1/4) = 16  [since integral sinc^2(aT)df = 1/T]

Final Answer:
  X(f) = 8 sinc(4f)
  First null at f = 0.25 Hz
  Energy E = 16 J

Exam Tip: Two sinc definitions coexist in textbooks and GATE papers — sinc(x) = sin(pi x)/(pi x) (normalised) and sinc(x) = sin(x)/x (unnormalised). The Fourier transform pair rect(t/T) <-> T sinc(f T) uses the normalised definition. If a problem states sinc(x) = sin(x)/x, the transform becomes (T/pi) Sa(pi f T) where Sa is the sampling function. Always check which convention the question uses before substituting numbers.

Properties Summary

• Definition: rect(t/T) = 1 for |t| <= T/2, zero otherwise; even function.
• Fourier transform: A T sinc(f T); mainlobe width = 2/T Hz.
• First null: at f = 1/T; null-to-null bandwidth = 2/T.
• Energy: E = A^2 T (direct integration of |x(t)|^2).
• Time-bandwidth product: T * (1/T) = 1 (constant, illustrating time-bandwidth trade-off).
• Duality: T sinc(t T) in time transforms to rect(f/T) in frequency.
• Parseval: integral |x(t)|^2 dt = integral |X(f)|^2 df = A^2 T.

Quick Revision

• rect(t/T): width T centered at origin, amplitude 1.
• Fourier transform: T sinc(f T); sinc(x) = sin(pi x)/(pi x).
• First spectral null at f = 1/T.
• Narrower pulse => wider spectrum; wider pulse => narrower spectrum.
• Energy = A^2 T.
• Duality theorem swaps rect and sinc between time and frequency.
• Exam trap: confusing the normalised sinc(x) = sin(pi x)/(pi x) with the unnormalised sin(x)/x. Always confirm the definition used in the problem before writing the Fourier transform pair.