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 [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(f) }[/math], then the spectrum has the property that the variance of [math]\displaystyle{ u }[/math] is
[math]\displaystyle{ \overline{u} = \int_0^{\infty} Psi(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(f)\, \mathrm{d}f \ \ . }[/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].
