Question

Consider the two continuous-time signals defined below:
These signals are sampled with a sampling period of T = 0.25 second to obtain discrete-time signals $x_1\left[n\right]$ and $x_2\left[n\right]$, respectively. Which one of the following statement is true?

• A. The energy of $x_1\left[n\right]$ is greater than the energy of $x_2\left[n\right]$
• B. Neither $x_1\left[n\right]$ nor $x_2\left[n\right]$ is a finite energy signal.
• C. The energy of $x_2\left[n\right]$ is greater than the energy of $x_1\left[n\right]$
• D. $x_1\left[n\right]$ and $x_2\left[n\right]$ have equal energies.

Answer 1

The correct option is A The energy of $x_1\left[n\right]$ is greater than the energy of $x_2\left[n\right]$

$x_1\left(t\right)=\{\begin{array}{c}|t|,-1\le t\le 1\\ 0,otherwise\end{array}$


$T_s\text{= sampling time-period}$
$=0.25sec$

$x_1\left(n\right)=1,0.75,0.5,0.25,0,0.25,0.5,0.75,1$

$Now,x_2\left(t\right)=\{\begin{array}{c}1-|t|,-1\le t\le 1\\ 0,otherwise\end{array}$

$x_2\left(n\right)=0,0.25,0.5,0.75,1,0.75,0.5,0.25,0$

Since $x_1\left(n\right)$ is having one more non-zero sample of amplitude '1' as compared to $x_2\left(n\right)$. Therefore, energy of $x_1\left(n\right)$ is greater than energy of $x_2\left(n\right)$.