Spectral density

The power spectrum of a time series describes the distribution of 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 over the time domain, as dictated by Parseval's theorem.

Spectrum

A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word was first used scientifically within the field of optics to describe the rainbow of colors in visible light when separated using a prism. As scientific understanding of light advanced, it came to apply to the entire electromagnetic spectrum.

Spectrum has since been applied by analogy to topics outside of optics. Thus, one might talk about the spectrum of political opinion, or the spectrum of activity of a drug, or the autism spectrum. In these uses, values within a spectrum may not be associated with precisely quantifiable numbers or definitions. Such uses imply a broad range of conditions or behaviors grouped together and studied under a single title for ease of discussion.

In most modern usages of spectrum there is a unifying theme between extremes at either end. Some older usages of the word did not have a unifying theme, but they led to modern ones through a sequence of events set out below. Modern usages in mathematics did evolve from a unifying theme, but this may be difficult to recognize.