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 <math>u(t)</math>, shows how the variance of this signal is distributed with respect to frequency. If the spectrum of <math>u</math> is <math>\Psi_u(f)</math>, then the spectrum has the property that the variance of <math>u</math> is
<math>\overline{u^2} = \int_0^{\infty} \Psi_u(f)\, \mathrm{d}f \ \ .</math>
and the variance located between two frequencies <math>f_1</math> and <math>f_2</math> is
<math> \int_{f_1}^{f_2} \Psi_u(f)\, \mathrm{d}f \ \ .</math>
The units of frequency can be cyclic such as <math>\mathrm{Hz}</math> (previously called cycles per second), or they can be angular such as <math>\mathrm{rad\, s^{-1}}</math>. The units should never be expressed as <math>\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>2\pi</math>.
Thus, the units of a spectrum, <math>\Psi</math> are the square of the units of <math>u</math> per unit of frequency, <math>f</math>.
If the signal is a space series, such as <math>u(x)</math>, where <math>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>k</math>. The wavenumber can be cyclic [<math>\mathrm{cpm}</math>] (cycles per meter) or it can be angular [<math>\mathrm{rad\, m^{-1}}</math>]. To avoid ambiguity, one should never express the units of wavenumber as <math>\mathrm{m^{-1}}</math>. The same properties apply to wavenumber spectrum, such as
<math>\overline{u^2} = \int_0^{\infty} \Psi_u(k)\, \mathrm{d}k \ \ .</math>
If the signal <math>u</math> is discretely sampled, then the upper frequency (or wavenumber) limit is the Nqyquist frequency, <math>f_N=f_s/2</math>, where <math>f_s</math> is the sampling rate of the signal.
