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.
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The spectrum of a signal, say , shows how the variance of this signal is distributed with respect to frequency. If the spectrum of is , then the spectrum has the property that the variance of is
and the variance located between two frequencies and is
The units of frequency can be cyclic such as (previously called cycles per second), or they can be angular such as . The units should never be expressed as because this usage is ambiguous, even though the units of radians is technically dimensionless. These two measures of frequency differ by a factor of Failed to parse (syntax error): {\displaystyle 2\pi</math}. Thus, the units of a spectrum, <math>\Psi} are the square of the units of per unit of frequency, .
