Parseval's Theorem

• categories: Functional analysis, Signal processing, Theorem

Definition

Parseval’s theorem is a fundamental result in Fourier analysis that states that the total energy of a signal in the time domain is equal to the total energy in the frequency domain. It establishes a relationship between a function and its Fourier transform.

---

Continuous-Time Version

Let $f(t)$ be a square-integrable function with Fourier transform $\hat{f}(\omega )$. Parseval’s theorem states:

$${\int }_{-\infty }^{\infty }|f(t){|}^{2}dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|\hat{f}(\omega ){|}^{2}d\omega .$$

• The left-hand side represents the energy in the time domain.
• The right-hand side represents the energy in the frequency domain.

Discrete-Time Version

For a discrete signal $x[n]$ with discrete-time Fourier transform (DTFT) $X(\omega )$, Parseval’s theorem states:

$$\sum _{n=-\infty }^{\infty }|x[n]{|}^2=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }|X(\omega ){|}^{2}d\omega .$$

Discrete Fourier Transform (DFT) Version

For a sequence $x[n]$ of length $N$ with DFT $X[k]$, Parseval’s theorem takes the form:

$$\sum _{n=0}^{N-1}|x[n]{|}^2=\frac{1}{N}\sum _{k=0}^{N-1}|X[k]{|}^2.$$

Here:

• ${\sum }_{n=0}^{N-1}|x[n]{|}^2$ is the total energy of the signal in the time domain.
• $\frac{1}{N}{\sum }_{k=0}^{N-1}|X[k]{|}^2$ is the total energy in the frequency domain.

Key Properties

1. Energy Conservation:
   Parseval’s theorem shows that energy is conserved between the time and frequency domains.

2. Orthonormal Basis:
   The Fourier transform decomposes a signal into an orthonormal basis of sinusoidal functions, preserving the signal’s energy.

3. Implication for Power Spectral Density:
   Parseval’s theorem is often used in signal processing to analyze power spectral densities.