Spectrum

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

$\overline{u^2}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{\Psi }}_u(f)\mathrm{d}f.$

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

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

The units of frequency can be cyclic such as $\mathrm{H}\mathrm{z}$ (previously called cycles per second), or they can be angular such as $\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$. The units should never be expressed as ${\mathrm{s}}^{\mathrm{-}\mathrm{1}}$ because this usage is ambiguous, even though the units of radians is technically dimensionless. The angular measures of frequency is larger than the cyclic measure of frequency 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, $f$.

If the signal is a space series, such as $u(x)$, where $x$ is the distance along a direction, then this signal also has a spectrum, but this spectrum provides the distribution of variance with respect to wavenumber, $k$. The wavenumber can be cyclic [$\mathrm{c}\mathrm{p}\mathrm{m}$] (cycles per meter) or it can be angular [$\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{m}}^{\mathrm{-}\mathrm{1}}$]. To avoid ambiguity, one should never express the units of wavenumber as ${\mathrm{m}}^{\mathrm{-}\mathrm{1}}$. The same properties apply to wavenumber spectrum, such as

$\overline{u^2}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{\Psi }}_u(k)\mathrm{d}k.$

If the signal $u$ is discretely sampled, then the upper frequency (or wavenumber) limit is reduced from $f=\mathrm{\infty }$ to the Nqyquist frequency, $f=f_N=f_s/2$, where $f_s$ is the sampling rate of the signal.