## ::Spectral density

### ::concepts

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The power spectrum of a time series <math>x(t)</math> describes how the variance of the data <math>x(t)</math> is distributed over the frequency domain, into spectral components which the series <math>x(t)</math> may be decomposed. This distribution of the variance may be described either by a measure <math>\mu</math> or by a statistical cumulative distribution function <math>S(f)=</math>the power contributed by frequencies from 0 up to <math> f</math>. Given a band of frequencies <math>[a,b)</math>, the amount of variance contributed to <math>x(t)</math> by frequencies lying within the interval <math>[a,b)</math> is given by <math>S(b)-S(a)</math>.<ref name="measure">or, equivalently, by the measure of that interval, <math>\mu([a,b))</math>.</ref> Then <math>S</math> is called the spectral distribution function of <math> x</math>. Provided <math>S</math> is an absolutely continuous function,<ref name="abscontmeasure">or, equivalently, <math>\mu</math> is absolutely continuous with respect to Lebesgue measure.</ref> then there exists a spectral density function <math>S'</math>. In this case, the data or signal is said to possess an absolutely continuous spectrum. The spectral density at a frequency <math>f</math> gives the rate of variance contributed by frequencies in the immediate neighbourhood of <math>f</math> to the variance of <math>x</math> per unit frequency.

The nature of the spectrum of a function <math> x</math> gives useful information about the nature of <math>x</math>, for example, whether it is periodic or not. The study of the power spectrum is a kind of generalisation of Fourier analysis and applies to functions which do not possess Fourier transforms.

An analogous definition applies to a stochastic process <math>X(t)</math>. Furthermore, time may be either continuous or discrete.

Intuitively, the spectrum decomposes the content of a signal or of a stochastic process into the different frequencies present in that process, and helps identify periodicities. More specific terms which are used are the **power spectrum**, **spectral density**, **power spectral density**, or **energy spectral density**.

The variance of <math>x</math> has units which are the square of the units of <math>x</math>. Therefore, these are also the units of <math>\mu</math> or <math>S</math>, and so the units of the spectral density are the square of the units of <math>x</math> per unit frequency. In the case of the voltage of an electric signal, <math>x^2</math> is proportional, except that it has the wrong units, to the power of the signal (implicitly assuming a constant resistance), and so even in statistical applications which use different units, the spectral distribution function and density function are often referred to as the power spectral distribution function and the power spectral density function, although the word *power* is often omitted for brevity in contexts where no misunderstanding will arise.
The use of the power spectrum is most important in statistical signal processing and in the branch of statistics consisting of the analysis of time series. It is, however, useful in many other branches of physics and engineering, and may involve other units. Usually the data is a function of time but they may be a function of spatial variables instead.

**Spectral density sections**

Intro Explanation Preliminary conventions on notations for time series Motivating example Definition Estimation Properties Related concepts Applications See also Notes References External links

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{{#invoke:Hatnote|hatnote}}

The power spectrum of a time series <math>x(t)</math> describes how the variance of the data <math>x(t)</math> is distributed over the frequency domain, into spectral components which the series <math>x(t)</math> may be decomposed. This distribution of the variance may be described either by a measure <math>\mu</math> or by a statistical cumulative distribution function <math>S(f)=</math>the power contributed by frequencies from 0 up to <math> f</math>. Given a band of frequencies <math>[a,b)</math>, the amount of variance contributed to <math>x(t)</math> by frequencies lying within the interval <math>[a,b)</math> is given by <math>S(b)-S(a)</math>.<ref name="measure">or, equivalently, by the measure of that interval, <math>\mu([a,b))</math>.</ref> Then <math>S</math> is called the spectral distribution function of <math> x</math>. Provided <math>S</math> is an absolutely continuous function,<ref name="abscontmeasure">or, equivalently, <math>\mu</math> is absolutely continuous with respect to Lebesgue measure.</ref> then there exists a spectral density function <math>S'</math>. In this case, the data or signal is said to possess an absolutely continuous spectrum. The spectral density at a frequency <math>f</math> gives the rate of variance contributed by frequencies in the immediate neighbourhood of <math>f</math> to the variance of <math>x</math> per unit frequency.

The nature of the spectrum of a function <math> x</math> gives useful information about the nature of <math>x</math>, for example, whether it is periodic or not. The study of the power spectrum is a kind of generalisation of Fourier analysis and applies to functions which do not possess Fourier transforms.

An analogous definition applies to a stochastic process <math>X(t)</math>. Furthermore, time may be either continuous or discrete.

Intuitively, the spectrum decomposes the content of a signal or of a stochastic process into the different frequencies present in that process, and helps identify periodicities. More specific terms which are used are the **power spectrum**, **spectral density**, **power spectral density**, or **energy spectral density**.

The variance of <math>x</math> has units which are the square of the units of <math>x</math>. Therefore, these are also the units of <math>\mu</math> or <math>S</math>, and so the units of the spectral density are the square of the units of <math>x</math> per unit frequency. In the case of the voltage of an electric signal, <math>x^2</math> is proportional, except that it has the wrong units, to the power of the signal (implicitly assuming a constant resistance), and so even in statistical applications which use different units, the spectral distribution function and density function are often referred to as the power spectral distribution function and the power spectral density function, although the word *power* is often omitted for brevity in contexts where no misunderstanding will arise.
The use of the power spectrum is most important in statistical signal processing and in the branch of statistics consisting of the analysis of time series. It is, however, useful in many other branches of physics and engineering, and may involve other units. Usually the data is a function of time but they may be a function of spatial variables instead.

**Spectral density sections**

Intro Explanation Preliminary conventions on notations for time series Motivating example Definition Estimation Properties Related concepts Applications See also Notes References External links

PREVIOUS: Intro | NEXT: Explanation |

<< | >> |