Spectrum: Difference between revisions
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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>\ | 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} \ | <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 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{ | 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. | ||
These two measures of frequency differ by a factor of <math>2\pi</math}. | These two measures of frequency differ 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>. | ||
Revision as of 20:46, 13 July 2021
| 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_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. These two measures of frequency differ by a factor of [math]\displaystyle{ 2\pi\lt /math}. Thus, the units of a spectrum, \lt math\gt \Psi }[/math] are the square of the units of [math]\displaystyle{ u }[/math] per unit of frequency, [math]\displaystyle{ f }[/math].
