GATE EC 2017 Set 1 — Question 8
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Signals & Systems → Continuous-Time Signals → Even & Odd Decomposition
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Question
A periodic signal has a trigonometric Fourier series expansionIf , we can conclude that
Correct answer
(A) aₙ are zero for all n and bₙ are zero for n even
Solution
The signal has two symmetry properties:
1.Odd Symmetry: . For an odd signal, the Fourier series contains only sine terms. Therefore, for all .
2.Half-Wave Symmetry (HWS): , where is the period. Here, , so the condition represents HWS. For a signal with HWS, only odd harmonics are present in the Fourier series. This means and for all even .
Combining these:- From odd symmetry: for all .
- From HWS: for even .
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