GATE EE 2014 Set 1 — Question 20

MCQ+1 / -0.33MediumFourier Series (CT & DT)Fourier Analysis & SamplingSignals & Systems

Signals & Systems → Fourier Analysis & Sampling → Fourier Series (CT & DT)

Last updated 5 September 2026

Question

For a periodic square wave, which one of the following statements is TRUE?

Correct answer (C) The Fourier series coefficients exist and the reconstruction converges at most points. Solution A periodic square wave satisfies the Dirichlet conditions: it is absolutely integrable over a period, has a finite number of maxima and minima, and a finite number of finite discontinuities. Therefore, its Fourier series coefficients exist. According to Dirichlet's theorem, the Fourier series (reconstruction) converges to the function value at every point of continuity. At points of jump discontinuity, it converges to the average of the left and right limits. Since a square wave has jump discontinuities, the reconstruction converges to the original function at most points (specifically, everywhere except at the discontinuities). Turn this into a strength. Explore AI-powered practice and doubt support with Success Tracker.

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Correct answer (C) The Fourier series coefficients exist and the reconstruction converges at most points. Solution A periodic square wave satisfies the Dirichlet conditions: it is absolutely integrable over a period, has a finite number of maxima and minima, and a finite number of finite discontinuities. Therefore, its Fourier series coefficients exist. According to Dirichlet's theorem, the Fourier series (reconstruction) converges to the function value at every point of continuity. At points of jump discontinuity, it converges to the average of the left and right limits. Since a square wave has jump discontinuities, the reconstruction converges to the original function at most points (specifically, everywhere except at the discontinuities).

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