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If 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
Correct answer is option 'B'. Can you explain this answer?
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If a signal f(t) has energy ‘E’ the energy of the signal f...
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If a signal f(t) has energy ‘E’ the energy of the signal f...
Explanation:
Energy of a signal is given by the formula:
\[
E = \int_{-\infty}^{\infty} |f(t)|^2 dt
\]
Energy of f(2t):
When we have a signal f(t) and we consider f(2t), the energy of the signal f(2t) is given by:
\[
E' = \int_{-\infty}^{\infty} |f(2t)|^2 dt
\]
Change of variable:
Let's substitute a new variable \(\tau = 2t\), then \(dt = \frac{d\tau}{2}\).
\[
E' = \int_{-\infty}^{\infty} |f(\tau)|^2 \frac{d\tau}{2}
\]
\[
E' = \frac{1}{2} \int_{-\infty}^{\infty} |f(\tau)|^2 d\tau
\]
Comparison:
Comparing the expressions for \(E\) and \(E'\), we see that the energy of the signal f(2t) is half of the energy of the original signal f(t).
Therefore, the energy of the signal f(2t) is \(E/2\).
So, the correct answer is option B.
Energy of a signal is given by the formula:
\[
E = \int_{-\infty}^{\infty} |f(t)|^2 dt
\]
Energy of f(2t):
When we have a signal f(t) and we consider f(2t), the energy of the signal f(2t) is given by:
\[
E' = \int_{-\infty}^{\infty} |f(2t)|^2 dt
\]
Change of variable:
Let's substitute a new variable \(\tau = 2t\), then \(dt = \frac{d\tau}{2}\).
\[
E' = \int_{-\infty}^{\infty} |f(\tau)|^2 \frac{d\tau}{2}
\]
\[
E' = \frac{1}{2} \int_{-\infty}^{\infty} |f(\tau)|^2 d\tau
\]
Comparison:
Comparing the expressions for \(E\) and \(E'\), we see that the energy of the signal f(2t) is half of the energy of the original signal f(t).
Therefore, the energy of the signal f(2t) is \(E/2\).
So, the correct answer is option B.
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