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Spectral density

The power spectrum S x x ( f ) {displaystyle S_{xx}(f)} of a time series x ( t ) {displaystyle x(t)} describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating x 2 ( t ) {displaystyle x^{2}(t)} over the time domain, as dictated by Parseval's theorem. The spectrum of a physical process x ( t ) {displaystyle x(t)} often contains essential information about the nature of x {displaystyle x} . For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field E ( t ) {displaystyle E(t)} as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency. However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency. Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as visible light (perceived as color), musical notes (perceived as pitch), radio/TV (specified by their frequency, or sometimes wavelength) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating a periodic signal which is not simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter. In physics, the signal might be a wave, such as an electromagnetic wave, an acoustic wave, or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in watts per hertz (W/Hz). When a signal is defined in terms only of a voltage, for instance, there is no unique power associated with the stated amplitude. In this case 'power' is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance. So one might use units of V2 Hz−1 for the PSD and V2 s Hz−1 for the ESD (energy spectral density) even though no actual 'power' or 'energy' is specified. Sometimes one encounters an amplitude spectral density (ASD), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz−1/2. This is useful when the shape of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth.

[ "Telecommunications", "Statistics", "Optics", "Cosmological perturbation theory", "Cross-spectrum", "Spectral flux", "spectral density matrix", "Whittle likelihood" ]
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