In signal processing, the power spectrum ${\displaystyle S_{xx}(f)}$ of a continuous time signal ${\displaystyle x(t)}$ describes the distribution of power into frequency components ${\displaystyle f}$ composing that signal. Fourier analysis shows that any physical signal can be decomposed into a distribution of frequencies over a continuous range, where some of the power may be concentrated at discrete frequencies. The statistical average of the energy or power of any type of signal (including noise) as analyzed in terms of its frequency content, is called its spectral density.

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 (PSD, 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 PSD then refers to the spectral power distribution that would be found, 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 ${\displaystyle x^{2}(t)}$ over the time domain, as dictated by Parseval's theorem.

The spectrum of a physical process ${\displaystyle x(t)}$ often contains essential information about the nature of ${\displaystyle x}$. For instance, the pitch and timbre of a musical instrument can be determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field ${\displaystyle E(t)}$ as it oscillates at an extremely high frequency. Obtaining a spectrum from time series data such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not directly captured 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.

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 density of the signal as a function of frequency. Power spectral density is commonly expressed in the SI unit watt per hertz (W/Hz).

When a signal is defined in terms of only a voltage varying in time, for instance, there is no specific power associated with a given voltage. 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 the unit V²⋅Hz⁻¹ for the PSD. Energy spectral density (ESD) would have the unit V²⋅s⋅Hz⁻¹, since energy is power multiplied by time (e.g., watt-hour).

In the general case, the unit of PSD will be the ratio of unit of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD with the unit m²/Hz. In the analysis of random vibrations, the unit g₀²⋅Hz⁻¹ may be used for the PSD of acceleration, where g₀ denotes standard gravity.

Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x(t) will remain unspecified, but the independent variable will be assumed to be that of time.