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9 of 14 articles

Sinc Function

sinc(t) = sin(πt)/(πt), Fourier dual of rect.

Darshan N
Updated: 7 April 2026
6 min read

The sinc function describes the exact frequency response of an ideal low pass filter. It helps determine how much bandwidth a communication channel requires.

tsinc(t)1-11
The normalized sinc function crossing zero at integer values.

Core Concept

The sinc function oscillates with a progressively decaying amplitude. It crosses zero at regular integer intervals. This behavior stems from dividing a sine wave by its linear argument.

Its maximum value occurs precisely at the origin. As time extends toward infinity, the oscillations fade entirely. This mathematical decay makes it a finite energy signal.

The most profound application lies in the Nyquist Shannon sampling theorem. Sinc interpolation perfectly reconstructs any bandlimited continuous signal strictly from its discrete samples.

Key Formula

In engineering, the normalized sinc function is sinc(x) = sin(pi*x) / (pi*x). The peak amplitude equals exactly 1 at x = 0.

Example
Given: Find the zero crossings of sinc(2t).\nFormula: sinc(x) = 0 when x is any non-zero integer n.\nSteps:\n1. Set the argument equal to an integer n: 2t = n.\n2. Solve for t to get t = n/2.\n3. Exclude the origin since sinc(0) equals 1.\nFinal Answer: Zero crossings happen at t = +-0.5, +-1.0, +-1.5...
Exam Tip: Math texts use the unnormalized form sin(x)/x, but ECE exams almost always use the normalized form sin(pi*x)/(pi*x). Verify the convention at the start of your GATE paper.

Properties Summary

  • Peak value: sinc(0) equals 1 exactly.
  • Zero crossings: Occur at all non-zero integer values of the argument.
  • Orthogonality: Integer shifted normalized sinc functions are completely orthogonal.
  • Area: The total area under the normalized sinc curve is exactly 1.
  • Transform pair: A rectangular pulse in time yields a sinc in frequency.

Quick Revision

  • Fundamental to ideal filter design.
  • Represents a brick wall filter.
  • Amplitude decays as 1/t.
  • Has infinite duration but finite energy.
  • Cannot be realized physically in real time.
  • Exam trap: Confusing the first zero crossing of the unnormalized and normalized definitions during bandwidth calculations.
#Signals & Systems

Sinc Function Quiz

Test your understanding of the sinc function and its Fourier duality with rect.

Question 1 of 3

Q1.The normalized sinc function is defined as sinc(t) = sin(pi*t)/(pi*t). What is the value of sinc(0)?

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