Plancherel's Theorem

Fourier Analysis · Axiom Academy

Energy Conservation in Fourier Analysis For any square-integrable function f , the total energy in the time domain equals the total energy in the frequency domain, up to a constant factor: This remarkable identity shows that the Fourier transform is an isometry: it preserves the L² norm. The factor of 1/(2π) depends on the convention used for the Fourier transform definition. In signal processing, |f(t)|² represents the instantaneous power at time t . Integrating over all time gives the total energy. Plancherel's theorem tells us this energy can be computed equivalently in either domain: This dual perspective is invaluable: we can analyze energy content by studying either the signal itself or its frequency components. The quantity |F(ω)|² is called the power spectral density . It describes how the signal's energy is distributed across different frequencies: A large value of |F(ω₀)|² indicates that the frequency ω₀ contributes significantly to the total energy. This decomposition allows us to identify dominant frequencies in signals ranging from audio waveforms to electromagnetic radiation. 4. Applications in Signal Analysis Plancherel's theorem has far-reaching applications across science and engineering:

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