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GATE 2014 EC (EC1) – Question 54

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Numerical answer

For periodic $x[n]=\sin(\pi n/5)$, complex Fourier-series coefficients $a_k$ are nonzero at $k=Bm\pm1$, m any integer. Find B.

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Correct answer: 9.99 to 10.01

Explanation

Write sin(πn/5)=[exp(j2πn/10)−exp(−j2πn/10)]/(2j). Its fundamental period is10samples and its only nonzero DTFS bins are k=1and k=−1 modulo10. Thus the indices are k=10m±1, giving **B=10**.