A theorem stating that a unitary Fourier transform preserves the L2 energy (norm) of a signal: the integral (or sum) of the squared magnitude in the time domain equals the integral (or sum) of the squared magnitude in the transform domain, up to the transform's chosen scaling convention.

Principle

Fourier transforms (when defined as unitary) are energy-preserving linear isometries between appropriate function spaces, allowing energy computations in whichever domain is more convenient.

Demonstration

Illustrative scenario → For a square‑integrable pulse x(t) with Fourier transform X(f) under unitary convention: compute ∫|x(t)|^2 dt and ∫|X(f)|^2 df; both integrals match. Recognition → Numerical energy computed in time and frequency domains coincide. Action → Use the frequency-domain computation to evaluate SNR under spectral shaping. Consequence → Designers may analyze power, energy or SNR equivalently in time or frequency, simplifying spectral-design and filter calculations.

Misapplication

Assuming equality without attention to transform conventions or signal class: using a non‑unitary Fourier convention, a DFT with different scaling, or a non‑square‑integrable signal without adjusting constants leads to incorrect energy comparisons.

Consequence

Permits moving energy and orthogonality calculations across domains; underpins Parseval‑based proofs of orthogonality, filter energy, and spectral estimation methods.

Reversal

The theorem requires the transform to be unitary and the signal to belong to the applicable space (e.g., L2 or finite-length sequences). For transforms that are not norm-preserving (certain Laplace conventions, non‑normalized DFT), explicit scaling factors are required.

Boundary

Holds for unitary Fourier transforms on appropriate spaces (continuous-time L2, discrete-time l2, or normalized DFT). It does not apply without modification to non-unitary integral transforms, nonlinear transforms, or generalized functions unless the transform and domain are extended with appropriate definitions.

Semantic Tension

Energy preservation ↔ Transform convention: the practical equality of energies depends on choosing and tracking transform normalization; failing to do so creates apparent contradictions between domains.

Synthesis

Parseval's Theorem is a practical bridge: with correct transform conventions and domain assumptions, energy and orthogonality are invariant under Fourier transform, enabling domain choice for analysis while demanding careful bookkeeping of normalization.