Question

According to Parseval's theorem the energy spectral density curve is equal to the area under (The question is phrased a bit oddly. "energy spectral density curve is equal to the area under..." Parseval's theorem relates energy in time domain to energy in frequency domain. The area under the Energy Spectral Density (ESD) curve *is* the total energy.)

• A. Magnitude of the signal
• B. Square of the magnitude of the signal
• C. Square root of magnitude of the signal
• D. Four times of the magnitude of the signal

Hint:

Parseval's Theorem for energy signals: \(E_x = \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega \).
Energy Spectral Density (ESD) is \(S_x(\omega) = |X(j\omega)|^2\).
Total energy is the integral of \(|x(t)|^2\) over time, or the integral of ESD over frequency (with appropriate scaling factor like \(1/(2\pi)\)).

Answer 1

The Correct Option is B

Parseval's theorem (for energy signals) states that the total energy of a signal \(x(t)\) can be calculated either by integrating the square of its magnitude in the time domain or by integrating its Energy Spectral Density (ESD), \(S_x(\omega) = |X(j\omega)|^2\), over all frequencies (divided by \(2\pi\) depending on convention). 

Parseval's theorem is given by:

\( E_x = \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega \)

Where \(X(j\omega)\) is the Fourier Transform of \(x(t)\), and \(|X(j\omega)|^2\) is the Energy Spectral Density (ESD).

The question asks: "The energy spectral density curve is equal to the area under...". This phrasing is a bit confusing, but we can clarify:

The correct interpretation should be: "The total energy of the signal is equal to the area under the Energy Spectral Density (ESD) curve in the frequency domain, AND this total energy is also equal to the integral of the square of the magnitude of the signal in the time domain."

The ESD is \(|X(j\omega)|^2\), and it represents the distribution of energy across frequency components. So, the total energy is obtained by integrating the square of the magnitude of the signal, \(|x(t)|^2\), in the time domain.

The total energy of the signal is:

\(E_x = \int_{-\infty}^{\infty} |x(t)|^2 dt\)

If the question is asking "The area under the energy spectral density curve is equal to the total energy, which is also equal to the area under the curve of [what quantity of \(x(t)\)] in the time domain?", then the answer is the square of the magnitude of the signal.

Therefore, the correct answer is:

Square of the magnitude of the signal

This aligns with the interpretation that the total energy is the integral of \(|x(t)|^2\) in the time domain.

Final Answer:

\( \boxed{\text{Square of the magnitude of the signal}} \)