Question

A signal $f(t)$ has energy $E$, the energy of the signal $f(2t)$ is equal to

• A. $E$
• B. $E/2$
• C. $2E$
• D. $4E$

Hint:

Time compression increases frequency but reduces energy by the same scaling factor.

Answer 1

The Correct Option is B

Step 1: Recall the definition of signal energy.
The energy of a continuous-time signal $f(t)$ is defined as:
\[ E = \int_{-\infty}^{\infty} |f(t)|^2 \, dt \]
Step 2: Apply time scaling property.
For a time-scaled signal $f(at)$, the energy becomes:
\[ E_a = \int_{-\infty}^{\infty} |f(at)|^2 \, dt \]
Step 3: Perform change of variable.
Let $u = at \Rightarrow dt = \frac{du}{a}$. Hence,
\[ E_a = \frac{1}{|a|} \int_{-\infty}^{\infty} |f(u)|^2 \, du \]
Step 4: Substitute the given value.
Here, $a = 2$. Therefore,
\[ E_{f(2t)} = \frac{1}{2} E \]
Step 5: Final conclusion.
Thus, the energy of the signal $f(2t)$ is $E/2$.