16.5 : パーセバルの定理

Parseval's theorem states that if a function is periodic, then the average power of the signal over one period equals the sum of the squared magnitudes of all the complex Fourier coefficients.

To validate Parseval's theorem, assume that the function has a complex Fourier series of a standard form. Substituting this and solving further gives the proof of the theorem.

Interestingly, Parseval's theorem can also be expressed in terms of the Fourier coefficients of the trigonometric Fourier series.

In audio processing, Parseval's theorem is used to compare the energy of an original sound wave with its compressed version.

The engineering interpretation of this theorem provides practical insights. If the function represents an electrical signal, such as current or voltage, then the square of the function represents the instantaneous power in a 1-ohm resistor.

This theorem also relates the energy dissipated in the resistor during one period to the Fourier series, providing two different expressions - one in terms of the trigonometric Fourier series and another in terms of the amplitude phase Fourier series.