Parseval's Identity

Fourier Analysis · Axiom Academy

Energy Conservation in Fourier Representations 1. What is "Energy" of a Signal? In signal processing, the energy of a signal f(x) over an interval is defined as the integral of its squared magnitude. This represents the total "power" contained in the signal. 2. Parseval's Identity Statement Parseval's Identity states that the total energy of a signal equals the sum of energies in each Fourier coefficient. The animation shows how energy is distributed across frequency components. This is a conservation law : energy in the time domain equals energy in the frequency domain. No energy is lost or gained during Fourier transformation - it's just redistributed. 4. Parseval's Formula (Real Form) For a real-valued periodic function with Fourier series coefficients a₀, aₙ, bₙ, Parseval's Identity takes this explicit form: Each term |a₀/2|², aₙ²/2, bₙ²/2 represents the energy contribution of a specific frequency component. 5. Application: Proving the Basel Problem We can use Parseval's Identity to prove the famous result that the sum of inverse squares equals π²/6. Consider f(x) = x on [-π, π]. 6. Complex Form of Parseval's Identity In the complex exponential notation, Parseval's Identity takes an elegant form using the complex Fourier coefficients cₙ:

This is the written version of the interactive lesson above. See the full Fourier Analysis course.