Spectrum

Short definition of Spectrum ([math]\displaystyle{ \Psi }[/math])
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 [math]\displaystyle{ u(t) }[/math], shows how the variance of this signal is distributed with respect to frequency. If the spectrum of [math]\displaystyle{ u }[/math] is [math]\displaystyle{ \Psi_u(f) }[/math], then the spectrum has the property that the variance of [math]\displaystyle{ u }[/math] is

[math]\displaystyle{ \overline{u^2} = \int_0^{\infty} \Psi_u(f)\, \mathrm{d}f \ \ . }[/math]

and the variance located between two frequencies [math]\displaystyle{ f_1 }[/math] and [math]\displaystyle{ f_2 }[/math] is

[math]\displaystyle{ \int_{f_1}^{f_2} \Psi_u(f)\, \mathrm{d}f \ \ . }[/math]

The units of frequency can be cyclic such as [math]\displaystyle{ \mathrm{Hz} }[/math] (previously called cycles per second), or they can be angular such as [math]\displaystyle{ \mathrm{rad\, s^{-1}} }[/math]. The units should never be expressed as [math]\displaystyle{ \mathrm{s^{-1}} }[/math] 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 [math]\displaystyle{ 2\pi }[/math].

Thus, the units of a spectrum, [math]\displaystyle{ \Psi }[/math] are the square of the units of [math]\displaystyle{ u }[/math] per unit of frequency, [math]\displaystyle{ f }[/math].

If the signal is a space series, such as [math]\displaystyle{ u(x) }[/math], where [math]\displaystyle{ x }[/math] 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, [math]\displaystyle{ k }[/math]. The wavenumber can be cyclic [[math]\displaystyle{ \mathrm{cpm} }[/math]] (cycles per meter) or it can be angular [[math]\displaystyle{ \mathrm{rad\, m^{-1}} }[/math]]. To avoid ambiguity, one should never express the units of wavenumber as [math]\displaystyle{ \mathrm{m^{-1}} }[/math]. The same properties apply to wavenumber spectrum, such as

[math]\displaystyle{ \overline{u^2} = \int_0^{\infty} \Psi_u(k)\, \mathrm{d}k \ \ . }[/math]

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