GATE EE 2015 Set 1 — Question 38
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Signals & Systems → Fourier Analysis & Sampling → Fourier Series (CT & DT)
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Question
The signum function is given byThe Fourier series expansion of has
Correct answer
(D) only cosine terms with odd numbered harmonics.
Solution
Let .
Functions with half-wave symmetry contain only odd harmonics ().
Combining these, the Fourier series expansion has only cosine terms with odd numbered harmonics.
1.Even/Odd Symmetry: Since is an even function, is also an even function (). Even functions contain only cosine terms in their Fourier series expansion ().
2.Half-Wave Symmetry: Check if . The period of is , so .
.Functions with half-wave symmetry contain only odd harmonics ().
Combining these, the Fourier series expansion has only cosine terms with odd numbered harmonics.
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