Spectrum: Difference between revisions
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{{DefineConcept | {{DefineConcept | ||
|parameter_name=<math>\Psi</math> | |||
|description=Shows how the variance of a signal is distributed with respect to frequency or wavenumber | |description=Shows how the variance of a signal is distributed with respect to frequency or wavenumber | ||
|article_type=Fundamentals | |article_type=Fundamentals | ||
}} | }} | ||
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>\ | 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} = \int_0^{\infty} | <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 | and the variance located between two frequencies <math>f_1</math> and <math>f_2</math> is | ||
<math> \int_{f_1}^{f_2} | <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>. | 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 reduced from <math>f=\infty</math> to the Nqyquist frequency, <math>f=f_N=f_s/2</math>, where <math>f_s</math> is the sampling rate of the signal. | |||
Latest revision as of 21:25, 13 July 2021
| 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.
