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GATE 2018 EE – Question 48

Signals and Systems · Representation of continuous and discrete time signals, shifting and scaling properties · 2 marks · Multiple choice

Consider the two continuous-time signals defined below:

$x_1(t)=|t|$ for $-1\le t\le1$ (0 otherwise), and $x_2(t)=1-|t|$ for $-1\le t\le1$ (0 otherwise).

These signals are sampled with a sampling period of $T=0.25$ seconds to obtain discrete-time signals $x_1[n]$ and $x_2[n]$, respectively. Which one of the following statements is true?

  1. The energy of $x_1[n]$ is greater than the energy of $x_2[n]$.
  2. The energy of $x_2[n]$ is greater than the energy of $x_1[n]$.
  3. $x_1[n]$ and $x_2[n]$ have equal energies.
  4. Neither $x_1[n]$ nor $x_2[n]$ is a finite-energy signal.

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Correct answer: (A) The energy of $x_1[n]$ is greater than the energy of $x_2[n]$.

Explanation

The samples of $x_1$ are $1,0.75,0.5,0.25,0,0.25,0.5,0.75,1$, whose squares add up to $3.75$. The samples of $x_2$ are $0,0.25,0.5,0.75,1,0.75,0.5,0.25,0$, whose squares add up to $2.75$. So $x_1[n]$ has greater energy.