17.5 : 傅里叶变换的帕塞瓦尔定理

Parseval's theorem is a principle used in signal processing to calculate the energy of a signal. It allows the computation of the same energy value using either time or frequency data, demonstrating energy conservation between these two domains.

If we consider a signal's power, its energy can be calculated. Typically, a 1 Ohm resistor is used as the base for this calculation, where the power is equivalent to the square of either voltage or current.

The theorem shows energy can be determined in the frequency domain, proving that the Fourier transform conserves energy.

The theorem suggests that the total energy delivered to a 1 Ohm resistor equals the total area under the square of the signal or one over two pi times the total area under the magnitude of the Fourier transform squared.

The theorem connects time and frequency domain energies, implying the Fourier transform's squared magnitude reflects the signal's energy density.

The signal's energy can be computed directly in the time domain or indirectly from the Fourier transform in the frequency domain.