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GATE 2026 EC – Question 15

Networks, Signals and Systems · Continuous-time Signals · 1 mark · Multiple choice

Two analog signals $x_1(t)$ and $x_2(t)$ ($t$ in second), are sampled at a rate $F_s = 40$ Hz, where $x_1(t)=\cos(20\pi t),\, t\ge0$, and $x_2(t)=\cos(100\pi t),\, t\ge0$. The first ten samples (starting from $t=0$) are considered for the analysis. Which of the following statements is TRUE?

  1. All of the first three samples of $x_1(t)$ are greater than the corresponding samples of $x_2(t)$.
  2. All of the last three samples of $x_1(t)$ are greater than the corresponding samples of $x_2(t)$.
  3. All of the samples of $x_2(t)$ are greater than the corresponding samples of $x_1(t)$.
  4. All of the fourth to seventh samples of $x_1(t)$ are equal to the corresponding samples of $x_2(t)$.

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Correct answer: (D) All of the fourth to seventh samples of $x_1(t)$ are equal to the corresponding samples of $x_2(t)$.

Explanation

The samples are $x_1[n]=\cos(\pi n/2)$ and $x_2[n]=\cos(5\pi n/2)=\cos(\pi n/2+2\pi n)$. The 50 Hz tone aliases onto the 10 Hz tone, so all samples are identical, and the fourth to seventh samples in particular are equal.