The GATE Grind

GATE 2026 EC – Question 58

Networks, Signals and Systems · Continuous-time Signals · 2 marks · Multiple select

Let $x_1(t)=\cos(2\pi nt)$ and $x_2(t)=2\sin(4\pi nt)$ represent two sinusoids for a positive integer $n$ and $-\infty<t<\infty$. Which of the following statements about $x_1(t)$ and $x_2(t)$ is/are valid?

  1. $x_1(t)$ and $x_2(t)$ are orthogonal to each other over $0\le t<1/n$.
  2. $x_1(t)$ and $x_2(t)$ are orthonormal to each other over $0\le t<1/n$.
  3. $x_2(t)$ is a harmonic of $x_1(t)$.
  4. $x_1(t)$ and $x_2(t)$ are non-orthogonal to each other over $0\le t<1/(2n)$.

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Show answer and explanation

Correct answer: (A) $x_1(t)$ and $x_2(t)$ are orthogonal to each other over $0\le t<1/n$.; (C) $x_2(t)$ is a harmonic of $x_1(t)$.; (D) $x_1(t)$ and $x_2(t)$ are non-orthogonal to each other over $0\le t<1/(2n)$.

Explanation

Over $[0,1/n]$, $\int\cos(2\pi nt)\sin(4\pi nt)dt=\tfrac12\int[\sin(6\pi nt)+\sin(2\pi nt)]dt=0$, so they are orthogonal but not normalized (energies $1/2n$ and $2/n$). $x_2$ has frequency $2n$, the 2nd harmonic of $n$. Over $[0,1/2n]$ the integral equals $\tfrac12[1/(3\pi n)+1/(\pi n)]\ne0$, so they are non-orthogonal there.