GATE EC 2026 Set 1 — Question 58
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Signals & Systems → Continuous-Time Signals → Fourier Series
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Question
Let and represent two sinusoids for a positive integer and .Which of the following statements about and is/are valid?
Correct answer
(A) x₁(t) and x₂(t) are orthogonal to each other over 0 ≤ t < 1/n.; (C) x₂(t) is a harmonic of x₁(t).; (D) x₁(t) and x₂(t) are non-orthogonal to each other over 0 ≤ t < 1/(2n).
Solution
Given and .
Let , then .
At . At .
The integral becomes . Thus, statement (A) is correct.
Thus, statement (B) is incorrect.
1.Harmonic Relationship:
The fundamental frequency of is . The fundamental frequency of is . Since , is the second harmonic of . Thus, statement (C) is correct.2.Orthogonality over :
Two signals are orthogonal over an interval if .Let , then .
At . At .
The integral becomes . Thus, statement (A) is correct.
3.Orthonormality:
Orthonormality requires the signals to be orthogonal and have unit energy. The energy of over is:Thus, statement (B) is incorrect.
4.Orthogonality over :
The signals are non-orthogonal over this interval. Thus, statement (D) is correct.Continue learning with Success Tracker
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