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.

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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 $\mathrm{\Psi }(f)$, then the spectrum has the property that the variance of $u$ is

$\bar{u}=\underset{0}{\overset{\mathrm{\infty }}{\int }}Psi(f)\mathrm{d}f.$

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

$\underset{f_1}{\overset{f_2}{\int }}Psi(f)\mathrm{d}f.$

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