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). \[ E_x = \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega \] 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 confusing. It should likely 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 itself is \(|X(j\omega)|^2\). The options seem to refer to what \(x(t)\) term is integrated in the time domain to get energy. The total energy is \(E_x = \int_{-\infty}^{\infty} |x(t)|^2 dt \). So, the total energy (which is the area under the ESD curve, perhaps scaled by \(1/(2\pi)\)) is obtained by integrating the "square of the magnitude of the signal" \(|x(t)|^2\) in the time domain. Option (b) "Square of the magnitude of the signal" aligns with this interpretation. If the question is asking what quantity's area (when plotted against frequency) gives the ESD at a particular frequency, that doesn't make sense. The ESD *is* the curve. 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". Given the options, this seems to be the intended interpretation. \[ \boxed{\text{Square of the magnitude of the signal}} \]