Spectrum

Short definition of Spectrum
Shows how the variance of a signal is distributed with respect to frequency or wavenumber

This is the common definition for Spectrum, but other definitions maybe discussed within the wiki.

The spectrum of a signal, say $u(t)$, shows how the variance of this signal is distributed with respect to frequency. If the spectrum of $u$ is ${\mathrm{\Psi }}_u(f)$, then the spectrum has the property that the variance of $u$ is

$\overline{u^2}={\int }_0^{\mathrm{\infty }}{\mathrm{\Psi }}_u(f)\mathrm{d}f.$

and the variance located between two frequencies $f_1$ and $f_2$ is

${\int }_{f_1}^{f_2}{\mathrm{\Psi }}_u(f)\mathrm{d}f.$

The units of frequency can be cyclic such as $\mathrm{Hz}$ (previously called cycles per second), or they can be angular such as $\mathrm{rad}{\mathrm{s}}^{-1}$. The units should never be expressed as ${\mathrm{s}}^{-1}$ because this usage is ambiguous, even though the units of radians is technically dimensionless. These two measures of frequency differ by a factor of $2\pi$.

Thus, the units of a spectrum, $\mathrm{\Psi }$ are the square of the units of $u$ per unit of frequency, $f$.