Parseval's theorem

Parseval’s theorem: the total energy (or average power) of a signal computed in the time domain equals the total energy computed in the frequency domain. Energy is conserved across the Fourier representation. It’s the “rotation-preserves-length” statement applied to function-space orthogonal decompositions.

Fourier series form

For a periodic signal $x(t)$ with Fourier coefficients $c_x[k]$:

$\frac{1}{T}{\int }_T|x(t){|}^{2}dt={\sum }_{k=-\infty }^{\infty }|c_x[k]{|}^2.$

Left side is the average power $P_x$ of $x$ over one period; right side is the sum of squared magnitudes of all harmonic coefficients. So power equals the sum of harmonic powers.

Fourier transform form

For a general (aperiodic) signal $x(t)$ with Fourier transform $X(f)$:

${\int }_{-\infty }^{\infty }|x(t){|}^{2}dt={\int }_{-\infty }^{\infty }|X(f){|}^{2}df.$

Left side is the total signal energy $E_x$; right side is the energy spectral density integrated over all frequencies. Energy in time = energy in frequency.

Why both forms work

Both are instances of the same orthogonality principle: an orthonormal basis preserves the inner product. For Fourier series, the basis is $\{(1/\sqrt{T})e^{j2\pi kt/T}{\}}_k$; the coefficients in this basis are $\sqrt{T}{c}_x[k]$, and the squared length of $x$ in time equals the sum of squared coefficient magnitudes. For the Fourier transform, the “basis” is a continuum of complex exponentials, and the sum becomes an integral.

Practical uses

Sanity checking. If you’ve computed Fourier coefficients (or a Fourier transform) and Parseval’s identity is way off, there’s an error. The two integrals must be the same number.

Estimating power. For a signal with most of its energy in a few dominant coefficients, you can estimate average power without integrating the full time-domain signal: sum $|c_x[k]{|}^2$ over the few significant $k$ values.

Spectral occupancy. $|X(f){|}^2$ is the energy spectral density of an energy signal, the energy per unit frequency. Integrate it over a band to get the energy in that band, a basic quantity in communications and audio engineering.

A quick example

For $x(t)=A\cos (2\pi f_{0}t)$, the Fourier coefficients (period $T=1/f_0$) are $c_x[\pm 1]=A/2$, all others zero. Parseval’s identity gives

${\sum }_k|c_x[k]{|}^2=2\cdot (A/2{)}^2=A^2/2.$

Compare to $P_x=A^2/2$ from the direct integral (worked out in Signal power). ✓

Generalization

The general statement, sometimes called the Plancherel theorem or Rayleigh’s energy theorem, says the inner product of two signals equals the inner product of their transforms:

${\int }_{-\infty }^{\infty }x(t)y^{\ast }(t)dt={\int }_{-\infty }^{\infty }X(f)Y^{\ast }(f)df.$

Parseval’s theorem is the special case $x=y$. The general form shows up when correlating two signals, common in communication receiver design and statistical signal processing.