Parseval's theorem relates a signal's energy in the time domain to what in the frequency domain?

• A. The integral of the square of the time-domain signal.
• B. The derivative of the Fourier transform.
• C. The maximum magnitude of the spectrum.
• D. The sum of the squared magnitudes of the Fourier coefficients.

Answer: (D)

Parseval's theorem shows that the energy of a signal, when measured in time, is the same as the energy measured in the frequency domain. When the signal is represented by its Fourier coefficients, this energy shows up as the sum of the squared magnitudes of those coefficients. Each coefficient indicates how much energy is carried by a particular frequency, and adding the contributions from all frequencies gives the total energy. The exact expression depends on the normalization convention, but the key idea is that the time-domain energy equals the sum of |X|^2 (the squared magnitudes of the Fourier coefficients) in the frequency domain. The other options don’t describe this energy relation in the frequency domain.