Spectral moment

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In signal processing, spectral moment is a statistical measure that characterize the distribution of a signal's energy across different frequencies. It is connected to the concept of moments in statistics, but should not be confused with it.

It is used in signal processing, oceanology, stochastic processes and engineering.

Definition

File:Fluorescent lighting spectrum peaks labelled.svg

The spectral density of a fluorescent light as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.

Spectral moment is a concept that first requires the concept of the spectral density.

Spectral density describes the distribution of power or energy into frequency components $f$ composing that signal, i.e. it describes how is the energy of a wave distributed along its frequencies. When the energy of the signal (here signal can also mean water waves as in gravity waves) is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density.

For a signal $x(t)$, its energy spectral density $S(\omega )$ satisfies:

$E=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{S}_E(\omega )d\omega$

where $\omega$ is the angular frequency.

More formal definition of spectral density

The spectral density can be defined in the following way.

For a sinusoidal signal:

$\eta (x,t)=\sin (kx-\omega t)$,

The spectral density is defined as the long-time limit of the squared magnitude of its Fourier transform:

$S(\omega )=\underset{T\to \mathrm{\infty }}{\lim }\frac{1}{T}{|\hat{\eta }(\omega )|}^2,$

where $\hat{\eta }(\omega )$ denotes the Fourier transform of $\eta (t)$.

The spectral moments of the spectrum are defined as:

$M_n=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\omega }^{n}S(\omega )d\omega$.

It is defined as the weighted measure of how $S(\omega )$ is distributed along the frequency space.

Zeroth moment - variance of a stationary random process

The zeroth spectral moment of a power spectral density is defined as:

$M_0=\underset{0}{\overset{\mathrm{\infty }}{\int }}{S}_x(\omega )d\omega$

For a stationary random process, the zeroth spectral moment is equal to the mean-square value of the signal:

$M_0=\mathbb{𝔼}[x^2(t)]$

If the process has zero mean, this quantity is equal to the variance:

$\mathrm{V}\mathrm{a}\mathrm{r}(x)=\mathbb{𝔼}[x^2(t)]=\underset{0}{\overset{\mathrm{\infty }}{\int }}{S}_x(\omega )d\omega$

For a process with non-zero mean $\mu =\mathbb{𝔼}[x(t)]$, the relation becomes:

$\mathrm{V}\mathrm{a}\mathrm{r}(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{S}_x(\omega )d\omega -{\mu }^2$

One can obtain the standard deviation by taking the square root of the zeroth spectral moment:

$\sigma =\sqrt{M_0}$.

References

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