Parseval’s Theorem (Rayleigh’s Theorem)

Parseval’s theorem (also known as Rayleigh’s energy theorem in the context of Fourier transforms) states that the total energy of a signal is preserved under the Fourier transform.

One-Dimensional Form

For a function $g(x)$ with Fourier transform $G(f)$:

$$ \int_{-\infty}^{\infty} |g(x)|^2 \, dx = \int_{-\infty}^{\infty} |G(f)|^2 \, df $$

Two-Dimensional Form

For a function $g(x, y)$ with Fourier transform $G(f_x, f_y)$:

$$ \iint_{-\infty}^{\infty} |g(x, y)|^2 \, dx \, dy = \iint_{-\infty}^{\infty} |G(f_x, f_y)|^2 \, df_x \, df_y $$

Physical Interpretation

• The left side represents the total energy in the spatial (or time) domain
• The right side represents the total energy in the frequency domain
• The theorem guarantees that energy is neither created nor destroyed by the Fourier transform

Historical Note

The theorem is named after Marc-Antoine Parseval (for Fourier series) and Lord Rayleigh (for Fourier transforms). In some contexts, “Parseval’s theorem” refers specifically to Fourier series while “Rayleigh’s theorem” or “Plancherel’s theorem” refers to Fourier transforms.