Parseval's theorem equates the total energy computed in the time domain to the total energy computed in the frequency domain. It is used to calculate signal energy directly from the spectrum when time-domain integration is difficult.

Parseval's theorem: energy computed in either domain gives the same result

Core Concept

Parseval's theorem states that the Fourier transform is an energy-preserving, or unitary, transformation. The total energy in a signal is the same whether you compute it by squaring and integrating the time waveform or by squaring and integrating the spectrum.

The quantity |X(f)|² is called the energy spectral density. It tells you how much energy is concentrated in each small frequency band. Integrating it over any frequency range gives the energy carried by those frequency components. This lets engineers calculate how much of a signal's energy falls within a given bandwidth.

For periodic signals and discrete sequences, there is an analogous result called Parseval's relation for Fourier series and for the DFT. The core idea is the same: the inner product between signals is preserved by the transform.

Key Formula

Parseval's theorem for the continuous Fourier transform: ∫ |x(t)|² dt = ∫ |X(f)|² df. For Fourier series: (1/T)∫|x(t)|²dt = Σ|cₙ|² where cₙ are Fourier coefficients and T is the period. For DFT of N-point sequence: Σ|x[n]|² = (1/N)Σ|X[k]|². The energy spectral density is S(f) = |X(f)|², measured in units of (signal unit)² per Hz.

Example

Problem: Find the energy of x(t) = e^(-at)u(t), a > 0, using the Fourier domain.

Step 1 — Find X(f):
  X(f) = 1 / (a + j2πf)

Step 2 — Compute |X(f)|²:
  |X(f)|² = 1 / (a² + (2πf)²)
           = 1 / (a² + 4π²f²)

Step 3 — Integrate energy spectral density:
  E = ∫_{-∞}^{∞} |X(f)|² df
    = ∫_{-∞}^{∞} df / (a² + 4π²f²)

Step 4 — Use standard integral ∫ dx/(b²+x²) = (π/b) ... scaling:
  Let u = 2πf, du = 2π df, so df = du/(2π)
  E = (1/2π) ∫_{-∞}^{∞} du / (a² + u²)
    = (1/2π) · (π/a)
    = 1/(2a)

Verification via time domain:
  E = ∫₀^∞ e^(-2at) dt = 1/(2a)  — matches.

Final Answer: E = 1/(2a)

Exam Tip: Parseval's theorem is often the fastest way to evaluate an integral like ∫ sinc²(f)df or ∫ df/(a²+f²) — recognise it as an energy computation in disguise and use the known transform pair instead of direct integration. For DFT, remember the factor of 1/N on the frequency side — it differs from the continuous case and is a common source of error in numerical problems.

Properties Summary

Energy equality: ∫|x(t)|²dt = ∫|X(f)|²df — FT preserves signal energy.
Energy spectral density: S(f) = |X(f)|² — energy per unit bandwidth.
Band energy: energy in band [f₁,f₂] = ∫_{f₁}^{f₂} |X(f)|² df + symmetric negative band.
Fourier series version: (1/T)∫|x(t)|²dt = Σ|cₙ|².
DFT version: Σ|x[n]|² = (1/N)Σ|X[k]|² — note factor of 1/N.
Cross-energy version: ∫x(t)y*(t)dt = ∫X(f)Y*(f)df — inner products preserved.

Quick Revision

FT is a unitary transform — energy is invariant.
Energy spectral density S(f) = |X(f)|² has units (signal²/Hz).
Integrate S(f) over any band to find energy in that band.
Parseval converts hard time-domain integrals into frequency-domain ones using known pairs.
DFT Parseval has a 1/N factor; continuous FT Parseval does not.
Periodic signals use power (energy/period), not total energy — use Fourier series version.
Cross-energy generalisation uses conjugate: ∫x(t)y*(t)dt = ∫X(f)Y*(f)df.
Exam trap: applying the continuous Parseval formula to DFT without including the 1/N scaling factor.

#Signals & Systems