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 <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. 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>.
